I I1 iu( u r~ u ir i^ c rr ~e ~r rr r, u uc r rr u -~~ Lr ruw *~ C r ir rr*r * c 2. h Ih: y c aa* ri cc rr Zr S" " " ~ rsr rr * c; ir 3 "i ~ *.. r-r, r n* *~*-d k rrV rr,. uf -CYY~*) I: i e, 3 i, Xk ~rca ~rr h rr.* inr;cr ii rr.. L* CI i~~; ~,. c. '.r~r e t: 1:~r* I*r 'I u u c u " -U1 ~. 11" Jr,~r I~ '*: A 3 I~ rU W - ~, ~" i c ~, Yii i ~"r cc~c ~~ "1 ~ c -~ ~~~* ~c *r *i r* Y "?~" r Ir:~x~ cr cc IL WI ~ rr *)"* PLANE TRIGONOMETRY BY FLETCHER DURELL, PH.D. HEAD OF THE MATHEMATICAL DEPARTMENT THE LAWRENCEVILLE SCHOOL NEW YORK CHARLES E. MERRILL CO. 44-60 EAST TWENTY-THIRD STREET 1911 Durell's Mathematical Series Plane Geometry 341 pages, 12mo, half leather..75 cents Solid Geometry 213 pages, 12mo, half leather.. 75 cents Plane and Solid Geometry 514 pages, 12mo, half leather... $ 1.25 Plane Trigonometry 184 pages, 8vo, cloth...... $1.00 Plane Trigonometry and Tables 298 pages, 8vo, cloth... $1.25 Plane Trigonometry, with Surveying and Tables In preparation... Plane and Spherical Trigonometry, with Tables 351 pages, 8vo, cloth...... $1.40 Plane and Spherical Trigonometry, with Surveying and Tables In preparation... Logarithmic and Trigonometric Tables 114 pages, 8vo, cloth... 75 cents Copyright, 1910, by Charles E. Merrill Co. [3] PREFACE THE principal object in writing this book has been the same as that which has governed the author in writing other mathematical text-books; viz., to bring out the fundamental utilities which underlie and grow out of the principles presented. Not only is the fundamental source of new power in Trigonometry frequently emphasized, but each new process is taken up, not arbitrarily, but for the sake of the economy or new power which it gives. Among other special features of the book, the following may be mentioned: Under each case in the solution of triangles two groups of examples are given; one with the degree divided sexagesimally, and the other with the degree divided decimally. The inclusion of the examples in terms of the decimally divided degree meets the new requirements of Harvard, Yale, and Princeton. A chapter is given on logarithms and their properties. Practical examples are included in this chapter which are not only interesting in themselves, but which afford a review of and a correlation with other branches of mathematics. When use is made of the line equivalents of the trigonometric ratios, it is specially shown that such treatment is merely a convenient substitute for the ratio treatment, and the method of this substitution is shown and its processes carefully safeguarded. A chapter is given in which the applications of trigonometry are reduced to a system. 4 PREFACE The subject-matter of the text-book is enlivened and made more vital and human by a chapter on the history of trigonometry. Attention is also called to the method in which logarithmic work is arranged. This form of tabulation is used, for instance, in the designing room in the United States Navy Department and by engineers in general. Among the advantages of this method of arranging logarithmic work are the following: (1) It abbreviates the work by omitting the equality marks. (2) It includes within itself the actual numbers whose logarithms are being used. (3) It facilitates the correction of mistakes by including and presenting in order all the steps of a logarithmic reduction. (4) The arrangement of the work is such that after the pupil has acquired facility in logarithmic computation, some of the steps in the tabulation may be omitted without changing the general form of tabulation. The author wishes to express his especial indebtedness to Mr. Howard Smith of the Hill School, Pottstown, Pa., to whom most of the examples are due, and who has made important suggestions concerning other parts of the work. The writer is also under obligation to his colleague, Mr. J. H. Keener, to whom the examples in the General Review Exercise are mainly due. Professor William Betz of the East Rochester High School, Rochester, N.Y., Dr. Henry A. Converse of the Polytechnic Institute, Baltimore, Md., and Professor William H. Metzler of Syracuse University have also aided the writer by important corrections and suggestions. FLETCHER DURELL. LAWRENCEVILLE, N.J., January 10, 1910. TABLE OF CONTENTS CHAPTER I PAGE LOGARITHMS..... 7 CHAPTER II DEFINITIONS. TRIGONOMETRIC FUNCTIONS.. 24 CHAPTER III RIGHT TRIANGLES........... 52 CHAPTER IV GONIOMETRY........ 73 CHAPTER V GONIOMETRY (Continued).......... 93 CHAPTER VI OBLIQUE TRIANGLES...... 107 CHAPTER VII PRACTICAL APPLICATIONS..... 131 CHAPTER VIII CIRQULAR MEASURE. GRAPHS OF TRIGONOMETRIC FUNCTIONS. 142 CHAPTER IX INVERSE TRIGONOMETRIC FUNCTIONS....... 152 CHAPTER X COMPUTATION OF TABLES. TRIGONOMETRIC SERIES.... 157 CHAPTER XI HISTORY OF TRIGONOMETRY.... 162 5 PLANE TRIGONOMETRY CHAPTER I LOGARITHMS 1. The logarithm of a number is the exponent of that power of another number, taken as the base, which equals the given number. Thus, 1000 = 103, hence log 1000 =3, 10 being taken as the base; again, if 8 be taken as the base, 4 =83, hence log 4-=-; also, if 5 be taken as the base, log 125 = 3, log -- = -2, etc. The base used is sometimes stated in the context as above; but, when desirable, it is indicated by writing it as a small subscript to the word log. Thus the above expressions might be written, logo 1000 = 3; log8 4 = -; log 125 = 3; log, -= -2; etc. In general, by the definition of a logarithm, number = (base)l~g'rithm, or N= B'; hence logBN= 1. 2. Uses or Utility of Logarithms. One of the principal uses of logarithms is to simplify numerical work. For instance, by logarithms the numerical work of multiplying two numbers is converted into the simpler work of adding the logarithms of these numbers. To illustrate this principle we may take the simple case of multiplying two numbers which are exact powers of 10, as 1000 and 100. Thus 7 8 TRIGONOMETRY 1000= 103 100 = 102 hence 1000 X 100 = 10 = 10000, the multiplication being performed by the addition of exponents. Similarly, if 384 = 102.58433+ and 25= 10139794+, 384 may be multiplied by 25 by adding the exponents of 102.58433 and' 101.39794+, thus obtaining 103-9822+, and then getting from a table of logarithms the value of 10398227+, viz. 9600. In like manner, by the use of logarithms, the process of dividing one number by another is converted into the simpler process of subtracting one exponent, or log, from another; the process of involution is converted into the simpler process of multiplication; and the extraction of a root into the simpler process of division. The saving of labor effected by the use of logarithms can be increased by committing to memory the logs of certain much used numbers as of 2, 3, - 9, Vr,, 1/ 2V, 1/3, etc. 7r Also by use of the slide rule, the practical use of logarithms is reduced to sliding one rod along another and reading off the number corresponding to the terminal position of one end of a rod. If the teacher can find time, it will be a useful exercise to teach the class the use of the slide rule in connection with the study of this chapter. 3. Systems of Logarithms. Any positive number except unity may be made the base of a system of logarithms. Two principal systems are in use: 1. The Common (or Decimal) or Briggsian System, in which the base is 10. This system is used almost exclusively when logarithms are employed to facilitate numerical computations. LOGARITHMS 9 2. The system termed Natural or Napierian, in which the base is 2.7182818+. This system is generally used in algebraic processes, as in demonstrating the properties of algebraic expressions, etc. EXERCISE I 1. Give the value of each of the following: log3 9, log3 27, log4 64, log4 -, log3 1, l og3 -8l, log o10 1 0, og1.0, log10 001. 2. Also of log2 32, log2 -32 log12 128 log4 8, logs 16. 3. Simplify log2 4 + logs 9 + log10.1 - log -9. 4. Write out the value of each power of 2 up to 220 (thus 21 = 2, 22 = 4, 23 = 8, etc.) in the form of a table. 5. By means of this table multiply 32 by 8, converting the multiplication into an addition of exponents. 6. In like manner convert each of the following multiplications into an addition: 32 x 16; 64 x 32; 1024 x 16; 512 x 64. 7. Also convert each of the following divisions into a subtraction: 1024 16; 512 - 64; 32768 1024. 8. Also convert each of the following involutions into a multiplication: (32)3; (64)2; (32)4. 9. Also convert each of the following root extractions into a division: -/64; -/1024; /4096. 10. Let the pupil make up two examples like those in Ex. 6; in Ex. 8; in Ex. 9. 11. Let the pupil construct a table of powers of 3 and make up similar examples concerning it. COMMON SYSTEM 4. Characteristic and Mantissa. If a given number, as 384, be not an exact power of the base, its logarithm, as 2.58433+, consists of two parts, the whole number called the characteristic, and the decimal part called the mantissa. To obtain a rule for determining the characteristic of a given number (the base being 10), we have, 10,000 = 104, hence log 10,000 = 4; 1000 = 103 hence log 1000 = 3; 100 = 102, hence log 100 = 2; 10 = 101, hence log 10 = 1. 10 TRIGONOMETRY Hence any number between 1000 and 10,000 has a logarithm between 3 and 4; that is, the log consists of 3 and a fraction. But every integral number between 1000 and 10,000 contains four digits. Hence every integral number containing four figures has 3 for a characteristic. Similarly every number between 100 and 1000, and therefore containing three figures to the left of the decimal point, has 2 for a characteristic; every number between 10 and 100 (that is, every number containing two integral figures) has 1 for a characteristic; and every number between 1 and 10 (that is, every number containing one integral figure) has 0 for a characteristic. Hence, the characteristic of an integral or mixed number is one less than the number of figures to the left of the decimal point. 5. Characteristic of a Decimal Fraction. 1 =10~.. log 1 = 0;.1 = 1-1.. log. 1;.0 1 = = 10-... log.01= —2; 100 1-02 1 1.001 = 1 = l10-... log.001 - 3, etc. lO00 10a Hence the logarithm of any number between.1 and 1 (as of.4 for instance) will lie between -1 and 0 and hence will consist of - 1 plus a positive fraction; also the logarithm of every number between.01 and.1 (as of.0372 for instance) will be between - 2 and - 1, and hence will consist of - 2 plus a positive fraction; and so on. Hence, the characteristic of a decimal fraction is negative, and is numerically one more than the number of zeros between the decimal point and the first significant figure. There are two ways in common use for writing the characteristic of a decimal fraction. Thus, (1) log.0384 =2.58433, the minus sign being placed over the characteristic 2, to show that it alone is negative, the mantissa being positive. LOGARITHMS 11 Or (2) 10 is added to and subtracted from the log, giving log.0384 = 8.58433 - 10. In practice the following rule is used for determining the characteristic of the logarithm of a decimal fraction: Take one more than the number of zeros between the decimal point and the first significant figure, subtract it from 10, and annex -10 after the mantissa. EXERCISE 2 Give the characteristic of: 1. 452. 6..08267. 11. 7. 2. 16730. 7. 1.0042. 12. 6267.3. 3. 767.5. 8. 7.92631. 13..000227. 4. 64.56. 9..007. 14. 100.58. 5. 9.22678. 10..0000625. 15. 23.7621. 16. How many figures to the left of the decimal point (or how many zeros immediately to the right) are there in a number the characteristic of whose logarithm is 3? 2? 5?? 0? 4? 8-10? 7 -10? 9-10? 17. Can you make up a rule for fixing the decimal point in the number which corresponds to a given logarithm? 6. Mantissas of numbers are computed by methods, usually algebraic, which lie outside the scope of this book. After being computed the mantissas are arranged in tables, from which they are taken when needed. In this connection it is important to note that The position of the decimal point in a number affects only the characteristic, not the mantissa, of the logarithm of the number. Thus, if log 6754 = 3.82956 6754 103s82956 log 67.54 = log 10- = log 10-2 = log 101.82956 = 1.82956. In general log 6754 = 3.82956 log 675.4 = 2.82956 log 67.54 = 1.82956 log 6.754 = 0.82956 log 0.6754 = 9.82956- 10 log 0.06754 = 8.82956 - 10, etc. 12 TRIGONOMETRY 7. Direct Use of a Table of Logarithms; that is given a number, to find its logarithm. For methods in detail see Introduction to Logarithmic Tables (Arts. 2-5 and 17). EXERCISE 3 Using five-place tables find the logarithm of numbers: each of the following 1. 7627. 10..00672. 19. 17 2. 6720. 1..000007. 20. 42 3. 82. 12. 400000. 21..0( 4. 7862. 13. 14.6235. 22..01 5. 75. 14..00226725. 23. 32 6. 157. 15. 87. 24. 32 7. 36278. 16..76. 25. 32 8. 67.222. 17..000125. 26. 32 9. 3.3427. 18. 100.25. 27. 3.2 28. Commit to memory the mantissa for each of tI 2, 3, 5, 7r. Then write at sight the log of each of the fc 3000, 50, 100 7, 20,.002,30,.0005, 17r-.3,.2, 10 A, 20,000. 1~00~.6287. W. )0001. 186789. }679. 67.9. 6.79..679. 2679. ie following: )llowing, 200, Using four-place tables, find the logarithm of each of the following: 29. 12.67. 36. 24.68. 43..000036775. 30. 762.8. 37..11116. 44..0026382. 31. 42.68. 38. 11.685. 45. 28966. 32. 1.2267. 39..0012678. 46. 19.572. 33..0263. 40. 965.3. 47..8625. 34..0012678. 41. 1.4676. 48..0100267. 35. 1.0026. 42. 1.7628. 49. 2.225. 50. Work Ex. 28 for four-place tables. 8. Inverse Use of a Table of Logarithms; that is, given a logarithm, to find the number corresponding to it (called its antilogarithm). See Introduction to the Logarithmic Tables (Arts. 6 and 17). LOGARITHMS 13 EXERCISE 4 Using five-place tables, find the antilogarithm of each of the follow. 1. 2. 3. 10. 11. 12. 13. 14. 1.41863. 4. 7.68416. 7. 6.59068. 2.19756. 5. 9.22321-10. 8. 5.74706-10. 0.98349. 6. 6.42857-10. 9. 8.00400. Find log of 2.34578. 15. Find antilog of 3.21678. Find antilog of 2.34578. 16. Find antilog of 6.00371. Find log of 1.03678. 17. Find log of 6.00371. Find antilog of 1.03678. 18. Find antilog of 4.98672. Find log of 3.21678. 19. Find log of 4.98672. Find the number corresponding to each of the following logarithms, using four-place tables. 20. 1.4082. 23. 9.1546-10. 21. 2.7332. 24. 2.0326. 22. 3.2335. 25. 1.0135. 32. Find antilog of 2.3041. 33. Find log of 2.3041. 34. Find log of 0.4975. 26. 8.0283-10. 29. 2.6575. 27. 7.1170-10. 30. 4.3490-10. 28. 5.0019-10. 31. 2.8177. 35. Find antilog of 0.4975. 36. Find antilog of 1.6924. 37. Find log of 1.6924. COMPUTATIONS BY USE OF LOGARITHMS -9. Properties of Logarithms used in Numerical Computations. It is shown in algebra that a. ax = a+~Y; and also that (a)P = ap. Using these properties of exponents, it can be shown that 1. log (mn) = log m + logn. 3. logm =p logm. 2. log () = log - log n. 4. log/m= log P For m = 10..~. log m = x. n =10Y... log n =y..'. mn = 1Ox+ or log mn= x + y = log m + log n. Also = 10 = 10-y, or log m = x - y = log m - log n. n 10Y p (1) (2) 1-4 TRIGONOMETRY Also mp =(10)P = 10P... log mr =px =p. log m, (3) and {/m=10... log Vm== logg (4) P p Hence: I. To multiply numbers: Add their logarithms and find the antilogarithm of the sum. This will be the product of the numbers. II. To divide one number by another: Subtract the logarithm, of the divisor from the logarithm of the dividend and obtain the antilogarithm of the difference. This will be the quotient. III. To raise a number to a required power: Multiply the logarithm of the number by the index of the required power and find the antilogarithm of the product. IV. To extract the required root of a number: Divide the logarithm of the number by the index of the required root and find the antilogarithm of the quotient. Ex. 1. Multiply 561.75 by.03286 by the use of logarithms. log (561.75 x.03286)= log 561.75 + log.03286 log 561.75 = 2.74954 log.03286 = 8.51667 -10 antilog 1.26621 = 18.4591, Product. The following, however, is the arrangement of work used by many practical computers. It has the advantage of showing all the steps in a complex logarithmic computation. (See p. 12, etc.) 561.75 log 2.74954.03286 log 8.51667- 10 Answer = 18.4591 log 1.26621 Observe that "561.75 log 2.74954" reads "561.75, its log is 2.74954," etc. Ex. 2. Compute the amount of $1 at 5 per cent compound interest for 20 years. LOGARITHMS 15 The amount of $ 1 at 5% for 20 years = (1.05)20. 1.05 log 0.02119; 20 log 0.42380 Amount = 2.65338 log 0.42380. If the student will compute the value of (1.05)20 by continued multiplication, and compare the labor in such a process with that involved in the above process, he will have a good illustration of the usefulness of logarithms. Ex. 3. Extract approximately the cube root of 532.768. 532.768 log 2.72653 ~ log 0.90884. Root = 8.1066 log 0.90884. 10. Cologarithm. In operations involving division, instead of subtracting the logarithm of the/divisor, it is usual to add its cologarithm. The cologarithni of a number is obtained by subtracting the logarithm of the number from 10-10. Hence adding the cologarithm of the divisor gives the same result as subtracting its logarithm. The use of the cologarithm saves figures, and gives a more orderly and compact statement of the work. The cologarithm of a number is obtained directly from a table of logarithms by the following rule: Subtract each figure of the logarithm of the given number from 9 except the last significant figure, which subtract from 10. Ex. 1. Find the colog of 37.16. log 37.16 = 1.57008. Hence, colog 37.16 = 8.42992 - 10. Ex. 2. Divide 52678 -by 37.16 by the use of the cologarithm of the divisor. 52678 log 4.72163 37.16 log 1.57008 colog 8.42992 - 10. Quotient = 1417.58 log 3.15155. 11. In the extraction of the root of a decimal number it is best to add to and subtract from the logarithm of the decimal 16 TRIGONOMETRY number such a multiple of 10 that the last term of the quotient shall be 10. Ex. Extract the seventh root of.0854329..0854329 log 8.93162 - 10 60 -60 7)68.93162 - 70 Root =.703667 log 9.84737 - 10 12. Computations involving Negative Numbers. In computing, by the use of logarithms, the value of expressions containing one or more negative factors, first, determine the sign of the result; second, determine the magnitude of the result by treating all the factors as if they were positive and using logarithms. -876 Ex. Compute - 87 795' The result must be negative, since a negative number divided by a positive number gives a negative quotient. The magnitude of the result is determined by computing 876 the value of 876 795' EXERCISE 5 Compute by means of five-place logarithms the value of each of the following: 1. 85 x 627. 5. 45 x 27.68 x.0967 x 4.2678. 2. 26.27 x 52.67. 6. (2.67)3. 3. 8.25 x 25675. 27.8675 1768 18.678 4. 211.6' 8. (.5278)7. 9. {/156.78. Also, if you can, extract the cube root of 156.78 without the use of logarithms. About how much more work in this process than in the logarithmic process? Which process is more likely to be accurate, the long or the short one? 10. -~/.86785. Also extract the square root of the square root of LOGARITHMS 17.86785. About how much longer is this process than the logarithmic work? 11. / —76.526. 12. ~/-.00021. 13. /-.00062367 x 7.867. Find the compound interest on: 14. $ 15375 for 20 years at 6 %. Make the computation without the use of logs. What fraction of the work is avoided by the use of logs? 15. $ 323.50 for 12 years at 8%. 16. In 1623 the Dutch bought Manhattan Island from the Indians for $ 24. What would this sum amount to at the present time, if it had been placed on interest at 6%, the interest to be compounded annually? 17. By aid of the logs committed to memory in Ex. 28, page 12, 200 100 7 300 x 500 compute each of the following: 376; 58; 18. Also obtain the colog of 43560 (the number of square feet in an acre) and use it to find the area in acres of a field 200 ft. x 300 ft.; one 300 ft. x 500 ft.; one 1000 ft. x 2000 ft. Using four-place logarithms, compute the value of the following: 19. 1.2634 x 26.42. /2293 24. 22.9 20..001467 x 96.8 x 47.37. 2' 16.91 21. 556.85 x.00016277 x 4.6..001666 22. (12.67)3. 25.00042635' 23. (3.176)7. 26. ~/42.67 x.10126 x 9.2. 27. -7/0000073. 28. Work Exs. 17 and 18 by the four-place tables. 29. Why are four-place logarithmic tables sufficiently accurate for the work of a carpenter or land surveyor? Find the compound interest on: 30. $ 359.67 for 8 years at 6%. 31. $ 100 for 37 years at 4%. 32. $4962.75 for 16 years at 5%. Try to compute this without the use of logs. About how much longer is the process without logs? Which process is more likely to be accurate? 13. Complex Computations. By the use of the properties of logarithms demonstrated in Art. 9, the value of a complex numerical expression may be computed. 18 TRIGONOMETRY Ex. 1. Compute /57 x 52 by the use of logarithms. 67' x 52 215 1 215 lo 67g2 I 6O 5 = -(log 215 + colog 67 + colog 52). log d67.x 52 log 67 x 52 Before looking up the logarithm of any number in the table, it is important to make a scheme or outline of the work, leaving blank the places which are to be filled in by logs taken from the table. Thus the preliminary outline for Ex. 1 would be as follows: 215 log........ 67 log...... colog........ 52 log...... colog.... 2)..... Answer =...... log.... After looking up and inserting the logarithms and completing the computation, the work will appear as follows: 215 log 2.33244 67 log 1.82607 colog 8.17393 - 10 52 log 1.71600 colog 8.28400 - 10 2)18.79037 - 20 Anszoer =.248422 log 9.39519 - 10 One advantage of the above method of tabulating logarithmic work is that without essential change in the form of the tabulating, the work may be presented in the above complete form, or in a more condensed form (at the option of the teacher), as by omitting the logs of 67 and 52 and giving only their respective cologs in the tabulation. V/21.8. /.03678 Ex. 2. Compute.2875367 by the use of logarithms..28756 21.8 log 1.33846 i log 0.66923.03678 log 8.56561 -10 1 log 9.52187 - 10.28756 log 9.45873 - 10 colog 0.54127 Answer = 5.39975 log 0.73237 14. Exponential Equations. An exponential equation is one in which the unknown quantity occurs in the exponent of some term or factor, as a" = b. An equation of this kind can often be solved by the use of logarithms. Ex. Find the value of x in the equation.33= 2. LOGARITHMS 19 Taking the logarithm of each member of the equation, x log.3 = log 2. Henee* x = log 2 0.30103 0.30103 log.3 9.47712 -10 - 0.52288 =-575 A EXERCISE 6 Using five-place tables, compute the value of the following: (Do not fail to make an outline of the work in each example before looking up any logarithms.) 1. 2.82 x V/.0071725 3..59 x 2209.92678 47 x.3481 2. (4/.26728)3 4. V(.19678)2 - (.072567)2. (.06756)2. (V278.2 x 2.578)2 _-/.00231 x x/76.19 6. 267.85 x 7 x.000925 x 468.765 (21.67)2 x.00096725 X /567.256 7. Using the logarithms committed to memory in Ex. 28, Exercise 3, compute each of the following: /300 x 500 _/ 300 r 200 x 30; - ~ 3.1416 37 r ' 8. If there are 39.37 inches in a meter, convert the following into feet: 500 meters; 7294 meters; 300 meters (height of Eiffel Tower). What logs used in the first of these computations could be retained and used in the other computations? Solve for x: 9. 6x = 67. 1l. 2.8- =.1967. o1. 142x+3 = 2167. 12..85" =.01978. * If the teacher prefers, the remainder of the work for this example may be arranged as follows: log x + log (log.3) = log (log 2)... log x = * log 2 —1 log.3. 2 log 0.301031. log 9.47861 -- 10..3 log 9.47712 -10 (or-.52288) 1 log (-) 9.71840- 10 colog 0.28160. x = -.5757+ log 9.76021 - 10. 20 TRIGONOMETRY 13. Find the side of a square whose area is equal to that of a parallelogram whose base is 22.678 and whose altitude is 17.375. 14. Find the side of a square whose area is equal to that of a circle whose radius is 13.56. 15. Calculate the value of K in the equation, 1K= Vs(s - a)(s - b)(s- c), when s =, and a = 17.6, b 21.675, c = 26.427. 2 16. Calculate the value of b in the equation, b = V/a2 - c2, when a =.17623 and c=.12673. (Use b= V/(a + c)(a- c), etc.) 17. Find the volume of a sphere whose radius is 14.7, if V= 4 7rR3 and 7r = 3.1416. 18. Givent=8, a = 32.17, finds, if s=-tt2. 19. Given s= a +c and a=.1732, b =.14326, c.2242, find 2 h, if h = - - s(sa) (s - b)(s- c). c 20. Given R = 14.16 and 7r = -2, find X, if S = 4 7rR2. 21. Given r = -2- and D= 23.8, find V, when V-= - rD3. 22. In how many years will $1 at compound interest at 5 % amount to $ 25? Using four-place tables, compute the value of the following: 23. 25. 126.781. 97 V7 x 51.8 12.97 \ 16.78 24..3756 x.265 26. -(125)2 -(67)2..227 x.1678 47.326 1 55400 x 8 *.10021V 123456 x.007 28. /.2167 x /21.67 x.16765 32.77 1.76364 29. { </12.673 (26.72)2 13. (36.27)1 x.01267 J Solve for x: 30. 2 = 19. 32. 19.383x = 81672.,31 42-3 = 11+1. 33..17" =.4782. LOGARITHMS 21 34. Find the side of a square whose area is equal to that of a rectangle whose base is 17.628 and whose altitude is 8.263. 35. Find the volume of a sphere whose radius is 1.1124, using V = 4 rrR3 and r = 2 2 36. Given t = 12 and g = 32.17, find s, if s = ~gt2. 37. Work Exs. 16-19 above by the use of four-place tables. 38. Work Exs. 7 and 8 above by the use of four-place tables. GENERAL PROPERTIES OF SYSTEMS OF LOGARITHMS 15. The logarithm of unity in any system of logarithms is zero. For, if a be the base, 1= a~... log 1= 0. 16. The logarithm of the base in any system of logarithms is unity. For a = a1...log, = 1. 17. The logarithm of zero in any system whose base is greater than unity is negative infinity; that is, as the number approaches 0, the logarithm approaches negative infinity. I 1 For, since a> 1, = = = c-a... log = - o. But in any system whose base is less than unity, the logarithm of zero is positive infinity. For, since a < 1, 0 = a..-. log, 0 = oo. 18. Logarithm of a Product, Quotient, Power, and Root in any system. If a be taken as the base, and mn and n be any two numbers, it can be shown in a manner similar to that used in Art. 9 that 1. log n mn = loga, m + log, n. 2. loga, n- = log m - loga n. [Let the pupil supply the 3. log Ip = P proof. See Art. 9; use 3. loga mp = p log, m. a for 10.] - logam 4. log,/m - logm —. P 22 TRIGONOMETRY 19. Changing the Base of a System of Logarithms. Given the logarithm of a given number, r, to a base a, to find the logarithm of r to another base k, we use the following formula: l log'r- r logk r log k' log, k For, let logk r = x. Then =...... (1) by definition of a logarithm. Take the logarithm of each member of (1) to base a, then x log, k = loga r. Hence, = log r loga log, r or logr r= -l log,, k It follows as a special case that if r = a, log, a = log or log, a. log, k = 1. Ex. Find the logarithm of.7 to the base 5. By the formula just proved, log,7 = log,.7 = 9.84510 -10 logo 5 0.69897 - -0 1549 - 0.2216 +, Ans. 0.69897 EXERCISE 7 In working the first twelve examples in the following exercise use four-place tables in solving the even-numbered examples, and five-place tables in solving the odd-numbered examples. Find the value of: 1. log5 60. 5. log x/V5. 9. log2.7261. 2. log6 9.3. 6. log8018. 10. log.21.08275. 3. log3.726.2. 7. log1.8.17362. 11. log1.2.9267. 4. log4.93. 8. log.8.2631. 12. log, V3.1416. LOGARITHMS 23 Find without the use of tables: 13. log327. 15. log9 - 17. log.125. 14. log232. 16. log6 8. 18. log2.0625. 19. Find the base of the system of logarithms in which the log of 16=4. 20. If the log of 27= 4, find the base. 21. If= 2the log of 5, find the base. 22. Given the log of 5-1 =-, find the base. 23. If the log of 64 = 1.2, find the base. 24. In how many years will a sum of money double itself at 4 % compound interest? at 6 %? 25. If $1520 amounts to $10,701.46 in 40 years at compound interest, what is the rate per cent? 26. Who invented logarithms, and when (see p. 169)? Find out all you can about this man and the way in which he inve.nted logarithms. 27. What nation first divided the circle into 360 degrees, and one degree into 60 minutes? CHAPTER II DEFINITIONS. TRIGONOMETRIC FUNCTIONS 20. Source of New Power. Illustrations. A spring of water is situated at the point A and a house at B. It is desired to find the length of a pipe needed to connect B with A, A and B being separated by a swamp. How can the length of the pipe be determined without going through the swamp? A \R~ An|~A aU~ ~ ~ O~~~~ to~_ = 5.2 90 3~20'/ D 20' C 390 yd. B C 540 yd. B D FIG. 1. FIG. 2. FIG. 3. If the swamp is situated as in Fig. 1, so that a point C can be taken where CA and CB form a right angle, then CA and CB can be measured and the length of AR computed by the methods of plane geometry. Let the pupil compute AB of Fig. 1. But if the swamp is situated as in Fig. 2, the above method of computing AB cannot be followed. However, if we take a convenient point C in Fig. 2 and measure the lines AC, CB, and the Z C, the distance AB can be computed provided woe have a table giving the ratios of the sides of all possible right triangles. Thus from this table we form the triangle given (on enlarged scale) in Fig. 3. Then by the properties of similar triangles we have the proportion 10: 5.2 = 420 yd.: AD. 24 TRIGONOMETRIC FUNCTIONS 25 From this proportion AD is obtained; afterward AB may be computed from the right triangle ADB by geometry. Hence the source of new power in trigonometry is a set of tables giving the ratio of each pair of sides in all possible right triangles. By the aid of such tables it will be found that we are able to find the unknown parts of many triangles which cannot be solved by ordinary A geometry. Thus it will be found that if one side AB (Fig. 4) and any two angles (as A and B) of a triangle be known, the B other sides (AC and CB) may be corn- A puted. By this method, for instance, the FIG. 4. distance from the earth to the moon is computed. (For other illustrations of the new power given by trigonometry see Chapter VII.) 21. Trigonometry, as first considered, is that branch of mathematics which determines the remaining parts of a triangle from certain given parts. Thus it will be found that if any three parts of a triangle are given, provided one of them is a side, the remaining parts may be determined. Later the word trigonometry comes to have a more extended meaning so as to cover the theory of the functions of angles in general wherever these angles may be found. Hence it comes to include much of the theory of wave motion and therefore of particular cases of wave motion, as of sound, light, and electricity. It also becomes largely algebraic in nature. Plane Trigonometry treats of plane triangles. See if you can find the derivation of the word trigonometry. 22. Trigonometric Functions of an Acute Angle. The fundamental tools or instruments used in trigonometry are the functions of an angle now to be described and defined. 26 TRIGONOMETRY From any point B in one side of an acute angle BAC let fall a perpendicular BC to the other side, forming the right triangle ABC. EB/ B A Z - A b 0 FIG. 5. FIG. 6. BC Then the ratio is termed the sine of the angle A. AB Similarly, AC AC AB cosine A = A, cotangent A = cosecant A - B BC ABAC tangent A = — C secant A A, versed sine A = 1 - -C AC' A C AB coversed sine A 1 - B AB' or, in general, in a right triangle: The sine of an acute angle is the ratio of the opposite leg to the hypotenuse. The cosine is the ratio of the adjacent leg to the hypotenuse. The tangent is the ratio of the opposite leg to the adjacent leg. The cotangent is the ratio of the adjacent leg to the opposite leg. The secant is the ratio of the hypotenuse to the adjacent leg. The cosecant is the ratio of the hypotenuse to the opposite leg. The versed sine is 1 minus the cosine. The coversed sine is 1 minus the sine. These eight ratios are called the trigonometric ratios, or the trigonometric functions. The versed sine and the coversed sine are used so little in TRIGONOMETRIC FUNCTIONS 27 elementary work that we confine our attention mainly to the other six functions. Hence when we speak of the " six functions" we mean the first six trigonometric functions as given above. The abbreviations sin, cos, tan, cot, sec, csc, vers, covers, are ordinarily used for the eight functions. The cosine, cotangent, cosecant, and coversed sine are termed the co-functions of the sine, tangent, secant, and versed sine respectively. In the above triangle (Fig. 6), denoting the side AB by c, AC by b, and BC by a, we have sin A =a sec A = c b cos A = csc A =c c a tan A vers A =,1 -- b c cot A = b covers A = 1 -a a c Similarly b c sin B = b sec B _ c a cos B = C tan B -= a cot B - b Or using abbreviations, sin of either acute Z = IoPP,. hyp. cos of either acute Z = ~ adj. hyp. tan of either acute Z = 1 opp. I adj.' csc B=b a vers B 1 -- covers B = 1 - b c cot of either acute Z = I adj. J opp. sec of either acute Z= hyp. I adj. csc of either acute Z = hyp. 1- opp. 28 TRIGONOMETRY The method of indicating a power of a trigonometric function is shown by the following example: for " the square of the sine of the angle A," that is, for " (sin A)2," we write "sin2A." How then would "the cube of cos A" be written? " The nth power of tan A?" In this book unless the contrary is stated, in the right triangle ABC, the letter C is supposed to be placed at the vertex of the right angle. 23. Utility of the Trigonometrical Ratios. It will be found that the numerical value of the above trigonometrical ratios for every angle from 0~ to 90~ may be computed and arranged in tables whence they may be taken and used when needed. These numerical values are used by what is virtually the geometrical principle of similar triangles in solving triangles. Later, however, they become units and elements which can be variously grouped and used in many kinds of algebraic processes. 24. The value of a trigonometric function of an angle EB depends only on the size of the angle, not on the length of the lines chosen to B'v form the ratios. Thus, by similar triangles (in Fig.7), B'C BC B"C" FA I CC C" sin A = AR = -AR etc. FIG. 7. AB" AB AB' 25. Given two sides of a right triangle, to compute the trigonometric functions for both acute angles of the triangle. Ex. If in a right triangle a = 3, and b = 4, find c and the trigonometric ratios of each acute angle. The hypotenuse c = V32 + 4 = 25 = 5 Hence sin A = sin B = 3 cos A = - cos B = tan A = - tan B = -- --- etc. etc. FIG. 8. TRIGONOMETRIC FUNCTIONS 29 In studying trigonometry (and indeed in all mathematical work) the pupil should make the capital letter a in the printed form A and not in the script form 6. In other words, he should make the small and capital letters as unlike as possible, and hence make them unlike in shape as well as in size. The reason for this is that the small and capital letters have entirely different meanings; and if as written by the pupil they have the same shape, the pupil is continually mistaking the small letter for the large, and vice versa. Similarly the capital letter c should always be written in the form,g and not C. EXERCISE 8 1. Write the functions of the acute angle B (Fig. 6) in terms of a, b, c. (Let the teacher invert the triangle in various ways.) 2. Construct a right triangle in which a 8, b =6, c =10, and write out the functions of A in this triangle; also of B. Determine the value of the functions of A in the rt. A ABC, whose sides are a, b, c, if: 3. a=6, b=8. 6. a =39, b = 80. 4. a=8, b = 5. 7. a=.09, c-.41. 5. a=12, c=13. 8. b=12, c=16.9. 9. Find the value of the functions of B in Exs. 3-8. 10. In Ex. 2 find the value of (1) sin A tan A. (4) 1 + tan2 A. (7) tan - sin A (2) sin2 A + cos2 A. (5) sec2 A -tan A. cos A (3) sin A cscA. (6) tan A cot A. (8) cos A sec A. By the use of squared paper construct the angle whose 11. Tangent =3. 16. sine = - 12. Tangent= 1. 17. cosine =2 0 13. Tangent =1. 18. secant = V3. 14. Tangent = 4. 19. cosecant = 5. 15. Tangent _= V3. 20. Construct with a protractor an angle of 23~. Then construct a right triangle with sides of convenient length having 23~ for one of its angles. Measure the sides of this right triangle and hence find sin 23~. Compare this value with the value of sin 23~ given in Table V. Determine and test cos 23~ and tan 23~ in the same way. 30 TRIGONOMETRY 21. Treat 37~ in the same way; also 52~. 22. On Fig. 2 (p. 24) compute the numerical value of AD; then of CD and DB; then of AB. 23. On Fig. 3, what is the value of sin A'? 24. On Fig. 6, if AB = 125, Z B = 27~, and sin 27~ =.454, compute AC. 25. Can you suggest some practical problem similar to that given in Art. 20, which could be solved by trigonometry and not by geometry? What is the source of new power in trigonometry which enables us to do this? 26. If by the methods of trigonometry we are able to solve any triangle in which one side and any two angles are given, suggest some practical problem which could be solved by this means (and not by geometry). In a rt. A, given: 27. a = Vpj + q', b = V/2pq, find sin A and cos A. 28. a = 2 ran, c = m2 + n2, determine sin A, sec A, and tan A. 29. b = 2pq, c =p2 + q2, find tan A, sin A, csc A. 30. a = -m2 + Mnn, b = V/rn + n2, find all the functions of B. 31. If a = 2/nzn and c = m + n, find all the functions of B. 32. If a = 60 and c = 61, find sec A, tan B, cot B, sin A. 33. If b = 2.64 and c = 2.65, find the functions of B. 34. If a = 2 b, find the functions of A. 35. If b = c, find the functions of A. 36. If a + b = 4 c, find the functions of B. 37. If a- b = -7 c, find the functions of A. 38. Find the functions of B, if a = 4 d and b = 3 d. By use of squared paper construct a rt. A, given: 39. c = 4 and tan A =. 40. b = 3 and sin A= 3. 41. Find b if cos A =.36 and c = 4.5. 42. On Fig. 8, sin A = what? cos B = what? Does sin A= cos B? In like manner, show that cos A = sin B, tan A = cot B, cot A = tan B, sec A = csc B, csc A = sec B. 43. Show the same on Fig. 6. TRIGONOMETRIC FUNCTIONS 31 44. In Fig. 6, since c is the hypotenuse, it is evident that it is greater than either leg. Hence sin A, or -, is always less than 1. C What other function of A is always less than 1? Which functions of A are always greater than 1? Which may be either greater or less than 1? 45. Which of the six functions are always proper fractions? improper fractions? may be either proper or improper fractions? Verify this on Fig. 8. 46. If A is any acute angle, is it correct to say that secA is always greater than sin A? Why? 47. The values of which of the six functions of A (on Fig. 6) have c for a denominator? a? b? 48. How many of the above examples can you work at sight (i.e. for how many can you give results without the use of pencil and paper)? 26. Functions of the Complement of an Angle. From Fig. 6 (page 26). sin A -; also cos B = a c c Hence, sin A = cos B, or sin A = cos (90~ - A), since B = 90 - A. Let the pupil show in like manner that cos A = sin B = sin (90~ - A), tan A = cot B = cot (90~ - A), and sec A = csc B = csc (90~ - A). Hence, in general, Any trigonometric function of an angle is equal to the cofunction of the complement of the angle. By the use of this property, Any trigonometric function of an angle between 45~ and 90~ can be reduced to the function of an angle between 0~ and 45~. Thus, sin 88~ 10' = cos 1~ 50'. 32 TRIGONOMETRY EXERCISE 9 Express each of the following trigonometric functions as a function of the complementary angle: 1. sin 60~. 5. csc 21~ 24' 30'K 2. cos 15~. 6. sec 84~ 16'. 3. tan 65~ 24'. 7. sin 89~ 59'. 4. cot 55036'. 8. cos 1 18'. 9. Given tan 60~ = V3, find cot 30~. 10. Given sin 30~ = find cos 60~. x 11. Given cos A= -, find sin (90~ — A). y 12. Given sin A==p, find cos (90~ - A). 13. How many of the examples in this exercise can you work at sight? RELATIONS OF TRIGONOMETRIC FUNCTIONS OF AN ANGLE 27. Three pairs of reciprocals exist among the trigonometric functions of an acute angle, viz.: sin and csc cos and sec tan and cot For B / ax..-. sin A x cscA 1. c a // a 0cC a b. -a b 1... cos A x secA =1. c b ax b. A b C xbi... tanA x cotA = 1. FIG. 9. b a 28. Four equations connect the trigonometric functions of an acute angle in important ways. For, from Fig. 9, & +b. a o (1) a2 + b2 = C2.... *. (1) TRIGONOMETRIC FUNCTIONS 33 Dividing (1) by c2, -=2 + b = 1o 2 2 that is, sin2 A + cos2A = 1. Dividing (1) by b2, a2 C2 a-+1=-, or (-)2+1=; that is, tan2 A +1 = sec2 A. Let the student prove in like manner that cot2 A +1 = csc2 A. Also from Fig. 9. a a b b c c that is, sin A tan A = cos A 29. Hence nine (or more) formulas give important values for the trigonometric functions. For from the results of Arts. 27 and 28 we readily obtain, for instance, sin A = V1 -cos' A. cos A = i/ - sin2 A. sin A tan A= sin A cosA tan A = cot A cos A cot A= -os A sin A sec A= - cos A 1 csc A = sin A vers A = 1 - cos A. covers A = 1 - sin A. 30. One trigonometric function of an angle being given, the other functions may be found in either of two ways. ALGEBRAIC METHOD. By use of the formulas of Art. 29 and equations of Art. 28. 34 TRIGONOMETRY Ex. 1. If sin A= 2 find the other trigonometric functions of A. cos A= 1-sin2A =1 -- v = 5/ = V/. sinA 2 /5 2 2 tanA = 3 =cos A 3 3 / cotA= 1 -v= tan A 2 seA=1 1 5 3 3 sec A = -- = 1 - = - = - -/5 cos A 3 V5 5 1 2 3 cscA= 1 — l!sin A 3 2 vers A = 1 -cos A = 1 - V5. covers A = 1 - sin A = 1 - = -1 Ex. 2. If tan x= 2, find the other functions of x. sec2 x = 1 + tan2 x. (Art. 28.) *. sec2 = 1 +4 =5. sec x = V5. 1 1 _ 1 Cosx= - = -5. sec ax/5 5 sin - = Vi - C2 = - _ = = - 2 V5, etc. GEOMETRIC METHOD. This consists of constructing a right triangle by use of the given function and deriving the required functions from the right triangle. Ex. 3. Given sin A = obtain the other trigonometric functions of A by use of the right triangle. Construct a right triangle whose hypotenuse is 3 and altitude is 2, as ABC. Then AC = -32 - 22 = V9 - 4 = 5. Then from'the figure by the definitions of the trigonometric ratios S 2 cos A=; tan A = = /5; cotA=5 |secA= - 3A5; cscA -; vers 1- -5 se e A = 3 3 sc A; vers = 3 A =- C /- 0 5 3 FIG. 10. covers A = 1 - - = -. TRIGONOMETRIC FUNCTIONS 35 As the sides of a right triangle are all positive in sign, in studying the trigonometry of the right triangle we neglect the ~ sign usually placed before a square root radical sign, and take any square root radical as normally plus..When we come to study angles in general, as in Chapters IV and V, it will be necessary carefully to consider whether the sign before a given radical sign is to be taken as + or - (see Art. 61). EXERCISE 10 Find by means of the formulas the values of the other functions of A, given:. sill A = 1 5 17' 2. tan A = 12 5' 3. sec A= -491 4. cos A = - 3-. 5. cot A = m. 6. csc A = 5. 7. sin A = 0. 8. cos A = 0. 9. tan A=0. 10. sinA= 1. 11. sec A = oo. 12. sinx=5 5p. Find by geometric methods (squared paper may be used to advantage in constructing diagrams) the other functions of A (or x), given: 13. tan A =. 14. cos A = 15. 15. csc A = -. 15' 16. cot A =. 17. sin A =. 18. sec A = 4. 19. tan A= m. 20. sin A = 1 ^/2. 21. cos x = 1. Find by both methods the other functions of the angle named when: 22. cs A — 41 23. ta A= 2 mn mn2 - 22 24. cotA=V2 +1. 25. sin A = 1. 26. tan 221i~ = /2 - 1. 27. cos A = —. 28. sec A = V6 -V/2 29. cos A =. 30. cot 15~ = 2 + V3. Express each of the other trigonometric functions of A in terms of: 31. sin A. 38. Given sin A = -, find cot A. 32. cosA. 39. Given cos A = 39 find csc A. 33. tan A. 40. Given tan A = V3, find sin A 34. cot A. 41. Given csc A =, find sec A. 35. sec A. 42. Given sec A= -2 —, find cot A. 36. csc A. 43. Given cot A = -2 - 1, find cos A. 37. vers A. 44. Given tan A = V6, find csc A. 36 TRIGONOMETRY 45. Transform the expression sin2 A + cos A so that the only trigonometric function contained in it shall be cos A. 46. Transform (1 + tan2 A) sec A so that it shall contain only cos A. 47. Transform (tan A + cot A) sec A cos A so that it shall contain only sin A and cos A. 48. Transform the equation cos2 x - sin2 x = sin x so that it shall contain only sin x. 49. Transform tan x = 2 + cot x so that it shall contain only tan x. 50. Which of the six functions are always less than 1? Which are always greater than 1? Which may be either greater or less than 1? How can you use this principle in testing the accuracy of examples like Exs. 1-30 of this Exercise? 51. How many of the above examples can you work at sight? 31. Trigonometric Identities. As stated in algebra, an identity is an equality which is true for all values of the unknown quantity (or quantities) contained in it. Thus (x + 2)(x - 2) = x - 4 is an identity, since it is true for all values of x, as for x =0, 1, 2, 3,.., or -1, -2, etc. An equation proper (or a conditional equation) is an equality which is true only for a certain special value (or values) of the unknown quantity (or quantities). Thus x2 - x = 2 x- 2 is true only when x = 1 or 2, and hence is an equation proper, or conditional equation. The equality mark used in equations is =, and that used in identities is =. However, in elementary mathematics it is customary to use the mark = for both equations and identities and let the context decide whether we are dealing with an identity or an equation. Similarly in geometry the word " circle " is sometimes used to denote an area and sometimes a line (the circumference), the context deciding in each case what is meant. So 8" may mean either 8 inches or 8 seconds of angle, etc. Relations of identity among trigonometrical functions may be proved in either of two ways. FIRST METHOD. By use of the formulas for the functions given in Arts. 28 and 29 (and particularly those which reduce the function to sine and cosine) an expression may TRIGONOMETRIC FUNCTIONS, 37 be proved identical with another, by reducing one of the given expressions directly to the form of the other. Ex. 1. Prove cot2 A cos2 A = cot2 A- cos2 A. COS2 A 2 cot2 A cos2 A = A cos2 A sin2 A (1 - sin2 A) cos2 A sin2 A cos2 A sin2 A cos2 A sin2 A sin2 A = cot2 A - cos2 A. Instead of proving an identity by reducing one member of the identity to the form of the other, it is sometimes more advantageous to reduce both expressions to a common third form, and hence infer their identity by Ax. 1. Thus we may start with cot2 A cos2 A = cot2 A - cos2 A and transform it as follows: cos2 A 2 os A= A, sin2 A sin2 A cos4 A cos2 A - cos2 A sin2 A or sin2 A sin2 A cos4 A _ cos2 A (1-sin2 A) sin2 A sin2 A cos4A cos4 A sin2 A sin2 A Since the last is plainly an identity, we infer that cot2 A cos2 A = cot2 A - cos2 A is also an identity. SECOND METHOD. By use of the values of the functions obtained by applying the definitions of the functions to the right triangle (Art. 22, Fig. 6). Ex. 2. Prove sin A cot A. cos A tan2 A 38 TRIGONOMETRY Substitute a for sin A; for cos A; a for tan A; b for cot A. Then c c b a a sin A c b sin A _ c = b _ cot A. cos A tan2 A b a2 a c b2 EXERCISE 11 Prove each of the following identities: (In the solution of identities, the first of the two methods given above is to be preferred, since its use helps fix in mind the fundamental equations and formulas given in Arts. 28 and 29.) 1. cos A tan A = sin A. 5. sin A = cos A tan A. 2. sinA sec A = tan A. 6. I+cosA sinA sin A 1 - cos A 3. cos A csc A = cot A. 1 sin A cos A 7. -sin _ cos A 4. cos A = sin A cot A. cos A 1 - sin A 8. sin2 A-cos2 A =2 sin2 A-1. 9. (1 - sin2 A) tan2 A = sin2 A. 10. (tan A + cot A) sin A cos A = 1 11. (1 - sin2 A) csc2 A = cot2 A. 12. (sin A + cos A)2 = 1 + 2 sin A cos A. 13. (sin A + cos A)2 + (sin A - cos A)2 = 2. 14. (csc2 A -1) sin2 A = cos2 A. 1 sin A cosA =secAcscA cos A sin A 16. cot2 os2A. 1 + cot2 A 17. tan A + cot A = sec A cse A. sec2 A + csc2 A 18. tan A+ cot A =se A cs2 A sec A csc A 19. siti A - cos4 A = sin2 A - cos2 A. sin A cos A 20. - = sin A + cos A. 1- cotA 1 - tanA 21. 1-cosA 2 21. - cos A= csc A- cot A. 1 + cos A TRIGONOMETRIC FUNCTIONS 39 22 - tan A 1 -tan A 1 + cot A cot A- 1 23. cotA+tan A = sin A cos A 24. tan2 A - sin2 A = tan2 A sin2 A. 25. csc4 A-2 csc2 A = cot4 A-1. 26. sec4 A (1 - sin4 A) = 2 tan2 A + 1. cse A 27. s = cos A. tan A + cot A 1 -- cot2 A 28. =1o2 sil2 A - cos2A. 1 + cot2 A 2. cot A- cos A cot A cos A 29. -- cot A cos A cot A + cos A 30. 1 — cot4 A= 2 csc2 A - csc4 A. 31. /1 - sin2 A tan A = sin A. 32. sin6 A + cos6 A = 1 - 3 sin2 A cos2 A. 33. cos3 A - sin3 A = (cos A - sin A) ( + sin A cos A). 34. Reduce tan6 x sec4 x to the form (tan8 x + tan6 x) sec2 x. Transform: 35. tan8 x into (tan6 x - tan4 x + tan2 x- )sec2 x + 1. 36. secl0 y into sec2y (1 + 4 tan2y + 6 tan4 y + 4 tan6 y + tan8 y). ^ —;-. cos x 37. 1 + sini n to 38. into sec2 x - sec x tan x. 1 + sin x 39. sin x into sec2 x + sec x tan x. cos2 X 40. See if you can make up or discover any other trigonometrical identities for yourself. 41. How many of the above examples can you work at sight? TRIGONOMETRIC FUNCTIONS OF PARTICULAR ANGLES 32. Functions of 45~. The trigonometric functions of 30, 45~, and 60~ are used so frequently that it is of service to determine their values and commit these values to 40 TRIGONOMETRY memory. It is helpful to notice that we determine these values in each case by the use of a right angle, the hypotenuse of which is taken as 1. Let ABC (Fig. 11) be an isosceles right triangle, the hypotenuse of / =12- which, AB, is 1. Then, by geome/45~ try, each leg is V2_/ (for Z B= 45~, A/ -i C..A C = BC; but AC2 + BCl = 12, FIG. 11..~ 2 2 = 12, etc.). By the definitions of the trigonometric functions, sin 45~ = (I Vg2) I = t V/2. cos 45~ = (IV/2) -1 = V2. 2 2 -1 1. csc 45~ 1 /22 /1. 33. Functions of 30~ and 60~. Let ABD (Fig. 12) be an iec,45 =a 1 = 2 V2. 2 V2 csc 45 =1 22. 2 V2 33. Functions of 300 and 6NO. Let ABD (Fig. 12) be an equilateral triangle in which the length of one side is 1. Let AC be -BD. Then, by geometry 60- Z BAD = 60~, A -30 and ZBA C = 30. Also AC bisects BD, hence BC =. A -VA-'' -, / - iV = 3. AC=VAB BC - ~1 = - = '3'. FIG. 12. Then in the right triangle ABC, sin 30~ = 1 cos 30~ = 2/V. TRIGONOMETRIC FUNCTIONS 41 1 1 tan 30~ 2 _ _ 2-A3 V3 3 cot 30~ 21 =V3. 1 2 sec30~ =_= 2= 2/3. \/3 V3 3 csc30~ =2. 2 Let the pupil write out in like manner the functions of 60~ (that is, of Z ABC in the A ABC). Of the results obtained in Arts. 32 and 33 those which are most used may be conveniently arranged in a table thus: 30~ 45~ 60~ sin A V W cos IV3 V2A ~ tan V3/ 1 V3 34. Functions of 0~. Let ABC (Fig. 13) be a right triangle in which the hypotenuse AB = 1 and the angle BAC is small and is diminished and made to approach 0~ as a limit. Then - if AB remains fixed in length, BC A - i c approaches zero and A C approaches 1. FIG. 13. At the limit, 0 1 sin 0~ = 0. sec 0 = 1 1 1 1 1 cos 0~ = 1. csc 00 = -. 1 0O 0 tan O~ = =0. vers0~= 1-1=-0. cot 00 = = o. covers = 1 - = 1. 42 TRIGONOMETRY 35. Functions of 90~. Let triangle in which BAC is approaches 90~ as a B length; hence BC AC approaches 0. A At the limit, 1 -1 A S C FIG. 14. I sin 90~ = - = 1. 1 0 cos 90~ = -= 0. tan 90= =. tan 900 = - = oo. 0 ABC (Fig. 14) be a right nearly a right angle and limit. AB remains fixed in approaches 1 as a limit and sec 90~ = = 0o 1 csc 90~ = - = 1. vers 90~ = 1 - 0 = 1. covers 900 = 1 - 1 = 0. 0 cot 90 = = 0. I 1 The results obtained in Arts. 34 and 35 may be conveniently arranged in a table thus: 0~ 900 sin 1 cos 1 0 tan 0. o cot co 0 sec 1 oo csc co 1 36. Representation of the Trigonometric Functions of an Acute Angle by Lines. If a quadrant of a circle OAB be drawn with center 0 and radius OB B - P" equal to 1, the sine of any angle A OP' is / M M'l _ M'P'. 1 /I /\ OP' 1 Similarly the sine of z A OP = -P, and sine of z A OP" = M"P". o M" M '- MA In other words the sine of any angle IG. 15. A OP in a quadrant whose radius is 1 is represented by the perpendicular let fall from P upon the radius OA. TRIGONOMETRIC FUNCTIONS Hence it is easy to see that, since MP is the sine of z AOP, if AOP becomes very small and 0, MFP 0, and at the limit sin 0~ = 0. Also if zAOP" increases and ' 90~, sin Z AOP" or M"P" OB or 1. Hence at the limit sin 90~ = 1. Similarly cos Z A OP' = M = OM - OM.' Hence also OP' 1 cos LAOP = OM, cos ZAOP" = OM." In other words the cosine of any angle AOP in a quadrant whose radius is 1 is represented by the part of OA intercepted between 0 and the foot of the line representing the sine. Hence cos 0~ = OA or 1, and as / A OP changes from 0~ to 90~, the cosine changes from 1 to 0. Similarly, (Fig. 16), AT AT tan AOT= OA AT. OA 1 OT OT - - se A OT= T - - OT. OA 1 T cot AOT== tanzB OR 1 BR _BlR = = = BR. o - A OB 1 o.A csc / A OT = sec B FOR16 OR OR... ~-= OR. OB 1 B - The various lines which represent the trigonometric functions of an acute N- angle AOP may be combined in a single figure (Fig. 17). Let the pupil find the lines on the figure which 0 M A represent vers z AOP and covers FIG. 17. z AOP. 37. Tables of Trigonometric Functions of Angles from 0~ to 90~ called Natural Functions. By methods which will be explained later (see Art. 116) the values of the trigonometric 44 TRIGONOMETRY functions for angles of every degree and minute from 0~ to 90~ may be calculated. These values are arranged in tables called Tables of Natural Trigonometric Functions. EXERCISE 12 By the use of squared paper, construct the following angles, making use of their natural functions: 1. 30~. (Use sin 30~= -.) 2. 45~. 3. 60~. 4. If tan 61~ 37' = 1.85, construct the angle 61~ 37' on squared paper. By use of the table of natural tangents, construct: 5. 420 30'. 6. 56 37'. 7. 47.24~. 8. 72.37~. By use of the table of natural sines, construct: 9. 61~ 23'. lo. 470 15'. 11. 52.35~. 12. 63.84~. Find the numerical value of: 13. 2 sin 30~ + cos 60~ + sin 90~. 14. b tan 30~ + c cot 60~ + a tan 0~. 15. 4 tan 0~ + 4 sin2 45~ + 2 cos 45~. 16. tan 30~ cos 90~ - 4 sin 60~ + cos2 0~. 17. tan 30~ cot 30 ~- 2 sin 45~ tan 45 ~- 6 cos 60~ cot 45~ + sin 90~. 18. sec 60~ cos 60~ - tan 30~ cot 60~ + tan 60~ cot 30 ~- 20 sin 30~. 19. Show that (sin 60~ - sin 45~) (cos 30~ + cos 45~) =. If P = 00,'Q = 300, R = 450, S = 60, T= 90~, find the value of each of the following expressions: 20. sin Q + cos R- 1. 21. tan2 P + tan2 Q + tan2 R. 22. cos P cos Q cos R + sin R sin S sin T. 23. sec P + 2 sin Q + 2 cos2 R + - tan2 S + cosec T. 24. Does twice the tangent of 45~ = the tan of 90~? Why? 25. Does sin 30~ + sin 45~ = sin 75~? 26. Does cot 30 +-cot 45~ = cot 75~? 27. Draw a diagram showing the trigonometric functions as lines when Z AOP is less than 45~. 28. Also when Z AOP is greater than 45~. 29. Also when Z AOP equals 45.~ TRIGONOMETRIC FUNCTIONS 45 30. Given that x is greater than 45Q and less than 90~, show on a diagram similar to Fig. 17 that tan x is greater than cotx. 31. Given that x is less than 45~, show that sec x is less than CSC x. 32. Show that cos x is always less than cot x. 33. Show that sin x < tan x < sec x. 34. Show that cot x < csc x. 35. If a flagstaff is at a distance of 150 ft. and the angle of elevation (see Art. 88) of the top of the flagstaff is 30~, find the height of the flagstaff. 36. Find its height if the angle of elevation of the top (at the same distance) is 45~. Is 60~. 37. Make up two examples similar to Ex. 35. 38. The Washington Monument is 555 ft. high. At a certain place the angle of elevation of its top is 30~. Find the distance of the monument from this place. 39. At a certain spot 165 ft. from the top of a particular part of Niagara Falls the angle of depression (see Art. 88) of the bottom of the falls is 45~. What is the perpendicular extent of the falls? 40. How many of the examples in this exercise can you work at sight? 38. Many trigonometric equations involving only acute angles may now be solved. Ex. 1. Find the value of x which satisfies the equation sin x=-. Since sin 30~ -, in the given equation x = 30~, Ans. Ex. 2. Solve sin x = cosx. Dividing each member by cos x, tan x = 1..'.x = 45~, Ans. Ex. 3. Solve tan x-1 = 2 sin x- 2 cos x. Substituting for tan x, sin 1 = =2 sinux;- 2 cos x. COS X Hence, sin x - cos x= 2 sin x cos x - 2 cos2 x. Factoring, (sin x - cos x)(1 - 2 cos x) = 0. Hence, sin x - cos x = 0... tan x = 1, x = 45~. Also 1- 2 cos x = 0... cos x=, x= 60~. Hence, x - 45~ 60~, Ans. 46 TRIGONOMETRY Ex. 4. Given sin x = cos 4 x, find x. By Art. 26 we may substitute for sin x its equal, cos (90 ~- x). Then cos (90~ - x) = cos 4 x..-. 90- x= 4 x. 5x = 90~. x = 18~, Ans. EXERCISE 13 Solve each of the following equations: 1. tan x =3. 12. 2. sin2x-. 13. 4. 3. cot x =3 tan x. 14. 4. cot2 x =. 15. 5. V/1- sin2 x = 1 + sin x. 16. 6. sec x 2. 17. 7. tan x + cot x = 2. 18. 8. sec x= V2 tan x. 19. 9. cos2 x - sin2 x = sin x. 20. 10. tan2 x + 2 sec x = 11. 21. 11. 3 cot2 x + cot x = 4. 22. 2 sin y + csc y = 3. 2 sin x V3 + 4 cos x = 5. sec x = 2 tan x. 4 sin2 x- tan' x = cot2 x. cot x + 2 tan x = 5-s 2 3 cos x + tanx = 1 +3 sinx tan x = 2 cot x - 1. csc y = 2 cot y. 2 sin x + cos x = 2. 2 sec x- cos x = 1. sin x sin x = 2 Solve: 23. sin x = cos 5 x. 26. see (45~ + x) = cse x. 24. tan y = cot 8 y. 27. sin y = cos ny. 25. cos ~ x = sin x. 28. sin 3 x = cos 2 x. 29. If a church steeple is at a distance of 80 ft., and the steeple is 80 ft. high, find the angle of elevation of the top of the steeple. 30. If the height of the steeple is 80.5 ft. and the distance of the base is 100 ft., see if you can find the angle of elevation of the top of the steeple by use of the table of natural tangents (pp. 91-96 of the tables). 31. Make up an example similar to Ex. 29. 32. Make up an example similar to Ex. 30. 33. In a right triangle given c = 62, a = 31, find A. 34. Given c = 150, a 75, find B. 35. Given c = 120, b = 60 V3, find A. 36. How many of the examples in this exercise can you work at sight? TRIGONOMETRIC FUNCTIONS 47 39. Tables of Logarithms of the Trigonometric Functions from 0~ to 90~. In performing numerical work involving trigonometric functions, it is usually more expeditious to proceed by the use of logarithms. Hence the logarithms of the natural trigonometric functions have been obtained once for all and arranged in tables called Tables of Logarithmic Trigonometric Functions. The use of these tables is explained in the Introduction to the Tables (Artso 7-11). EXERCISE 14 By the use of five-place tables, find: 1. log sin 26~ 18'. 9. log sin 4~ 6' 55". 2. log cos 12~ 16'. 10. log cos 17~ 17' 30". 3. log tan 36~ 18'. 11. log cot 37~ 28' 50". 4. log cot 76~ 18'. 12. log sin 78~ 59' 30". 5. log tan 55~ 16'. 13. log tan 86~ 46' 5". 6. log tan 15~ 18'. 14. log tan 4~ 44' 50". 7. log cos 86~ 52'. 15. log cos 45~ 48' 48". 8. log tan 36~. 16. log cot 60~ 52' 6". 17. We have proved (see Art. 33) that sin 30~=.5. Obtain log.5 and thus show that the value of log sin 30~ as given in the table is correct. 18. Similarly verify the value of log sin 45~, and of log tan 60~, as given in the table. 19. In the rt. A ABC, a = b tan A. (Why?) If A = 18~ 16' and b= 18.63, find a. 20. In the rt. A ABC, b= c cos A. (Why?) Find b if c = 18.675 and A = 36 36' 36". By the use of four-place * tables, find: 21. log sin 15.3~. 24. log tan 78.8~. 22. log cos 47.5~. 25. log sin 27.35~. 23. log cot 33.7~. 26. log cos 26.36~. * When the term " four-place tables " is used in connection with angles, the four-place logarithmic tables for the decimally divided degree are meant. See Arts. 18-19 of the tables. 48 TRIGONOMETRY 27. log tan 63.78~. 28. log cot 12.65~. 31. In the rt. A BAC, 29. log cos 40.16~. 30. log cot 29.23~. b = a cot A. (Why?) If A = 18.67~ and a =.2167 feet, find b. 32. In the rt. A ABC, a=c sin A. A=59.72~, find a. Also find b, if b = c (Why?) If c= 17.65 and cos A. EXERCISE 15 Using five-place tables, find A, give 1. log sin A = 9.59632 - 10. 2. log tan A = 9.73777 - 10. 3. log cos A = 9.90951 - 10. 4. log cot A = 10.07029 -10. 5. log sin A = 9.96159 - 10. 6. log tan A = 0.44540. 7. log cos A = 9.53390 -10. 8. log tan A = 1.06575. 9. log sin A = 9.95788 - 10. 10. log cot A = 1.02921. 11. log sin A = 8.84501 - 10. 12. log cos A = 8.84501 - 10. By use of four-place tables, find A, given: 13. log sin A = 9.6495 - 10. 20. log cos A = 9.8409 - 10. 14. log cos A = 9.8063 - 10. 21. log tan A = 0.2575. 15. log tan A = 9.7384-10. 22. log cot A = 2.0248. 16. log cot A = 0.4755. 23. log tan A = 1.5718. 17. log cot A = 9.8248 - 10. 24. log sin A = 9.9596 - 10. 18. log tan A = 0.4422. 25. log cos A = 9.3129 - 10. 19. log cos A =9.6351-10. 26. log cot A = 0.5881. EXERCISE 16 By use of five-place tables find: 1. log sin 0~ 56' 18". 2. log tan 1~ 16' 37". 3. log cos 88~ 13' 26". 4. log tan 88~ 54' 50". Find the angle A if: 9. log tan A = 7.88154 - 10. 10. log cos A = 8.28910 - 10. 11. log sin A = 8.09600 - 10. 12. log cot A = 7.90390 -10. 5. log cot 1~ 18' 36". 6. log cos 89~ 7' 19". 7. log sin 1~ 6' 12". 8. log cot 88~ 16' 32". 13. 14. 15. 16. log tan A = 3.05992. log cot A = 2.88206. log sin A = 6.88800 -10. log cos A = 7.63702 -10. TRIGONOMETRIC FUNCTIONS 49 For " angle whose log sin is" we may write " Z log sin," or " antilog sin," hence find: 17. Z log sin 9.82627-10. 20. L log cot 8.09599-10. 18. Z log tan 10.90261 - 10. 21. Z log cos 8.09599-10. 19. Z log cos 9.06000- 10. 22. Z log tan = 2.77651. 23. In the A ABC, a =c sinA. Find a if c= 18.6 and A= 26~ 18' 48." Find the value of the following: 2 528.7 x cos 83~ 16' 24" x tan2 75~ 18' 24"' 672 cot218~ 32' 54" x sin 69~ cos2 15~ 16' 34" 2 265 x tan 65~ 18' X cos2 14~ 28' 12" 19 cot2 11~ 16' 24" x sin 75~ 15' 45" x.7 By use of four-place tables, find: 26. log cos 88.76~. 27. log sin 0.762~. 28. log cot 89.267~. 29. log tan 1.067~. 30. log tan 88.763~. 31. log cot 0.765~. 32. log sin 1.267~. 33. log cos 89.467~. Find angle A if: 34. log cot A = 8.1067 - 10. 35. log tan A = 8.2574 -10. 36. log cos A = 8.1360 - 10. 37. log sin A = 8.0440 - 10. 38. log tan A = 2.1080. 39. log cot A = 2.0532. 40. log sin A = 7.9100 - 10. 41. log cos A = 7.9932 - 10. 49. In the rt. A ABC, a = c sin A. and A - 1.267~. 50. In the rt. AABC, b = a cot A. and A = 2.166~. Find: 42. log cot 88.676~. 43. log tan 88.676~. 44. Z log cot 8.1078 - 10. 45. Z log tan 8.0295 - 10. 46. Z log cos 8.0959 - 10. 47. Z log sin 8.0371 - 10. 48. log tan 88.68~. (Why?) Find a if c = 126.27, (Why?) Find b if a = 0.4267, 51. Find the value of 632.7 x cos 78.16~ X tan' 71.62~ 426.8 x sin 13.25~ X cot2 12.47~ x.8 52. Find the vale of 326 x tan 38.25 X cos2 88.627 43 x cot 0.826~ X sin2 2.467~ 50 TRIGONOMETRY EXERCISE 17. REVIEW 1. In the right A ABC, given tan A = A- and a = 16, find b, c, and the other functions of A. 2. If cos A = 8, find the value of sin A+tan A 17 cos A - cot A 3. Show that cos 60~ cos 30~ + sin 60~ sin 30~ = cos 30~. cot 45' + cot 90~ 4. Show that cot 45-cot 90~ 1. 1- cot 45~ cot 90~ (Work Exs. 5-12 without the use of tables.) 5. Which is greater, sin 49~ or cos 49~? 6. If sin A = 3, is A greater or less than 45? 7. If tan A = 2, is A greater or less than 60? 8. Which is the greater, tan 37~ or cot 37? 9. If A = 60~, show that sin A = /1 -cos A 2 10. If A = 60~, show that cot I A = / + cos A 1 - cos A 11. Which is greater, sin 45~ or 1 sin 90? sin 60~ or 2 sin 30? tan 30~ or - tan 60~? 12. If x= 30~ and y 60~, show that sin x cos y + cos x sin y= si (x + y). 13. + cot A see A + csc A 13. Prove = 1-cot A secA-csc A Pov + tan2 A sin2 A 14. Prove -- 1 + cot2 A cos2 A P + cos A 15. Prove cos A= (csc A + cot A)2. 1 - cos A 16. If x =30~, show that tan 2 x = 2 tan x 1 - tan2 x 17. If x = 30~, show that sin 3 x = 3 sin x - 4 sin x. 18. If x = 30~, show that cos 3 x = 4 cos x - 3 cos x. Solve the following trigonometric equations:19. tan x + 3 cot x = 4. 20. 2 sec2x- tan2 x = 5. 21. 3 csc2x -2 cot x =4. TRIGONOMETRIC FUNCTIONS 51 If P = 0~, Q = 30~, R = 45~, S= 60~, T = 90~, find the value of: 22. cos2 Q + cosS + cos2 T + 2 cos Q cos Scos T. 23. sec Q ( + tan R) - sin3 T(cos R + sin S cos Q). 24. 1 + tan2 S + 3(cos P sin2 R - sin S). 2- tan2 R 25. If 25 sin A = 7, find cot A and csc A. 26. If p cot 0 = Vr2 —p2, find sin 0. 27. If i denotes the angle of incidence of a ray of light falling on a piece of glass, and r the angle of refraction, then sin i = 3 sin r. Find r when i =27~ 17'. 28. If at a distance of 300 ft. the angle of elevation of the top of one of the big trees of California is 45~, how tall is the tree? 29. If at a distance of 300 ft. the angle of elevation of the top of a tree were 42~, see if you can find out how tall the tree would be. (Why are we able to determine this height by trigonometry and not by geometry?) 30. Who first, and at what date, defined the sine of an angle as the ratio between two lines (see p. 165)? Give the different substitutes for this idea of the sine that had been used before this time. Why is the ratio definition of the sine superior to each of these? 31. Explain the origin and literal meaning of the word sine (see p. 166). 32. Who first invented each of the other trigonometric ratios, and at what time (see pp. 162; 164)? 33. Give some of the various names used for these ratios, with the names of the inventors of these names. 34. What nation first used the trigonometrical identity sin2 A + cos2 A = 1 (see p. 172)? tan x = si? cos x 35. Give an account of the computation of trigonometric tables (see pp. 168-170). CHAPTER III RIGHT TRIANGLES 40. Two Cases arise in the trigonometrical solution of right triangles. CASE I. Given one side and an acute angle. CASE II. Given two sides. In each of these cases it will be observed that three parts are really given, since the right angle is known. CASE I 41. The solution of Case I is effected as follows. Subtract the given angle from 90~. This will give the unknown angle. The unknozon sides may then be found by means of the fbllowing: 1. Either leg = (sine of Z opposite) x hypotenuse. 2. Either leg = (cosine of Z aljcacent) x hypotenuse. 3. Either leg = (tangent of Z opposite) x other leg. 4. eypotenuse = (secant of either acute Z) x (leg adjacent to that/ ). Also (either leg) = (cot of Z adjacent) x (other leg); hyp. = (csc of either acute Z) X (leg opposite that L)., Proof By def., sin A= a.'.a= c sin A. c / a Also cosB=...a= c cos B. c tan A =a-..a = b tan A. b A b C c Also sec B=-..c= a sec B. FIG. 18. a 52 RIGHT TRIANGLES Similarly it may be proved that: b=c sinB, b=ccosA, b=a tanB, andc=b secA. Ex. 1. Given A = 55~ 43' 29", c = 415.18, find the remaining parts of the right triangle. We first draw a diagram (Fig. 19) of the triangle to be solved, and on this diagram write the known magnitudes (415.18 for c, and 55~ 43' 29" for A). We also indicate the parts to be computed (a, b, B) by annexing the = mark to each of these. During the numerical computation, as soon as the result for any part is ascertained, this result should be entered on the diagram after the proper = mark. Z B = 90~ - 55~ 43' 29" = 34~ 16' 31". B a = 415.18 sin 55~ 43' 29". (Art. 41, 1).'. log a = log 415.18 + log sin 55~ 43' 29". / 415.18 log 2.61824 55~ 43' 29" log sin 9.91716 -10 a = 343.085 log 2.53540; Also b = 415.18 cos 55~ 43' 29". (Art. 41, 2).. log b = log 415.18 + log cos 55~ 43' 29". 415.18 log 2.61824 55' 43' 29" log cos 9.75064 - 10 - A -. _ b = 233.821 log 2.36888 (As a check use a = b tan A.) FIG. 19. Ex. 2. 1A b= FIG. 2( Given a=.0723, B = 3~ 47' 7", find the re B ing parts of the right triangle. / A = 90~ - 31~ 47' 7" = 58 12' 53". b =.0723 tan 31~ 47' 7't.0723 log 8.85914 - 10 31~ 47' 7" log tan 9.79216 - 10 b =.448022 log 8.65130 — 10 c=.0723 sec 31~ 47 7" -.0723 cos 31~ 47" 7' 'main J..9723 log 8.85914 - 10 31~ 47' 7" log cos 9.92943 - 10 colog cos 0.07057 c=.0850567 log 8.92971 - 10 (As a check use b= c cos A.) 54 TRIGONOMETRY Ex. 3. By use of four-place tables solve the right triangle in which b 21.635, A =47.23~. Z B = 90~ - 47.23~ = 42.77~. Also c = 21.635 tan 47.23~. (Art. 41, 3) B.'. log a = log 21.635 + log tan 47.23~. 1/ 21.635 log 1.3352 47.23~ log tan 0.0339 ~/ I a =23.394 log 1.3691 By Art. 41, 4, c = 21.635 sec 47.23~ = 21.35 cos 47.23~ 47.23~.'. log c = log 21.635 +colog cos 47.23~ A 21.6335 21.635 log 1.3352 FIG. 21. 47.23~ colog cos 0.1681 c= 31.864 log 1.5033 (As a check use a = c cos B.) 42. First Estimates. Graphical Solutions. In the solutions of triangles fully one half the mistakes commonly made, and those the most important ones, are eliminated by making a rough mental forecast of the results before proceeding with the exact numerical work. Thus in solving Ex. 1 of Art. 41, the pupil should first of all observe that, the hypotenuse being 415.18, each of the legs will be less than 415.18; and also that, since angle B is less than angle A, side b must be less than side a. If then as a result of his exact numerical calculation, the pupil finds a leg greater than 415.18, or a less than b, he knows at once that a mistake has been made. Similarly it is useful, by means of the rule and protractor, to make a drawing according to scale of the triangle to be solved, and from the figure to determine as accurately as possible the dimensions of the unknown parts by measuring them according to scale. Such results should be accurate enough to aid in eliminating any large errors in the numerical work. (Indeed, if the work be neatly done, the results obtained from the diagram will be accurate enough for many practical purposes.) RIGHT TRIANGLES 55 43. Exact checks of the numerical accuracy of the work of solving right triangles are obtained by calculating some side or angle of the triangle by a formula different from those already used in the computation, and observing whether the results thus obtained accord with those obtained in the first solution. Thus, to check the accuracy of the solution given for Ex. 1, Art. 41, determine whether tan A=-; that is, compute the value of the fracb tion 343.085 and also obtain from the table the value of tan 55~ 43' 29" 233.821 and observe whether these two values accord. EXERCISE 18 State at sight the formula value of x (or of x and y) in each of the following triangles: Thus in Ex. 1, (1), x = 208 sin 40~. 1. (1) (2) (3) (4) (5) x I X a ( a Y x x 40 ______ ^ 40 65~ Y 627 y X 2. (1) (2) (3) (4) (5) Y 3 8aX Xc xc a - " y x y 3. Make up an example similar to Ex. 2. By use of five-place tables solve each of the following triangles, given: (In working each example outline all the work carefully before looking up any logs-see Ex. 1, p. 18.) 4. A=28~, b=12. 6. A=460 18', b =48.527. 5. A =78~, c=26.735. 7. A =28~ 17', c =24.16. 56 TRIGONOMETRY 8. B =54~ 43' c = 1123. 10. A = 38 16' 24", c = 3.6289. 9. B = 37~ 19', b = 293.8. 11. B= 72 16' 42", a=22.684. 12. Given c=.52684, B= 63~ 18' 48"; find a. 13. Given A = 37~ 25' 20", c =.356; find b. Find the remaining parts in each of the following right triangles, given: 14. A = 63~ 28' 40", a = 256.43. 15. c = 13.867, A = 87~ 16' 30". 16. A=51 9' 6", c=.19678. 17. a =126.78, A =26~ 18' 36". 18. Given A = 5~ 16' 32", b =.96156; find c. 19. Given A = 37 14' 15", b = 217; find a. 20. If the top of the Statue of Liberty in New York harbor is 301 ft. above the water surface, and a boat in the harbor finds the angle of elevation of the top of the statue to be 12~, how far is the boat from the statue? 21. If a certain point on the brink of the Grand Canon of the Colorado is known to be a horizontal distance of 3 miles from the Colorado River and the angle of depression of the river is 17~, how deep is the cafion at that place and how far from the observer is the river in a straight line? 22. Which of the examples in Exercise 22 are you able to solve by Case I? Solve one of these. 23. Make up a similar practical problem for yourself and solve it, as for instance one concerning the Bunker Hill monument (221 ft. high). Solve the following right triangles, by use of four-place tables, having given: 24. A=32.6~, b =18. 28. A =37.67~, c= 126.7. 25. A=56~, c= 2.678. 29. B=76.25~, a =.926. 26. B=38.2~, c=.7685. 30. A=21.32~, a= 16.256. 27. B= 82.5~, a= 12.56. 31. B= 66.27~, b=.0087. 32. Given c =.6243, B = 51.25~; find a. 33. Given A = 77.26~, c =.5163; find b. 34. Given B = 39.29~, b= 41.67; find a. RIGHT TRIANGLES 57 Find the remaining parts in each of the following right triangles, given: 35. c = 13.13, A = 88.17~. 36. B = 42.16~, a=.5252. 37. Given A = 5.26~, b = 128.6; find c. 38. Given B = 87.267~, c = 22.67; find a. 39. Given A = 4.276~, a = 26.32; find b. 40. Work Exs. 20-23 by four-place tables. Solve without the use of tables, having given: 41. A=30~, b=7. 45. A =60~, a=2000. 42. A=45~, c=12. 46. B=30~, c = 1200. 43. B=60~, b=25. 47. A=45~, b=200. 44. B=30~, a=1000. 48. A=30~, c=20d. 49. Solve Exs. 6 and 7 of this exercise without the use of logarithms (i.e. by the use of the Tables of Natural Sines, etc., pp. 91-96). 50. How many of Exs. 41-48 can you solve at sight without drawing a figure? 51. On the figure if s ADB and DCB are right As, find BD, BC, and DC at sight. 30 52. On Fig. 52, p. 93, if OP= 1, what is the value of OQ? of PQ? of QN? of ON? A,300 A D CASE II TWO SIDES GIVEN 44. The Solution of Case II is effected as follows: Find one of the angles of the given triangle by using that one of the following trigonometric ratios which contains the two given sides: _L opp. 1. sine of either acute Z = hyp. o adj. 2. cosine of either acute L = ad-. hyp. _3 opp. 3. tangent of either acute L =. adj. 58 TRIGONOMETRY Find the remaining parts of the triangle by Case I (but if the hypotenuse and a leg are given, the other leg may be found by one of the formulas, a = V(c + b)(c- b), b = V( + a)(c-a)). Ex. 1. Given a = 317, c = 438, find the remaining parts of the right triangle ABC. 317 sin A = 38 (Art. 44, 1) B Hence log sin A = log 317 + colog 438 42~~~~/ ~317 log 2.50106 438 log 2.64147 colog 7.35853 - 10 it5/ IL ~A = 46~ 21' 55" log sin 9.85959 - 10 7 n B = 900 - 460 21' 55" = 43~ 38' 5". b = 438 cos 46~ 21' 55". (Art. 41, 2) 14^~ ~438 log 2.64147 A b= - 46~ 21' 55" log cos 9.83888 - 10 F~iG. 22. b = 302.24 log 2.48035 FIG. 22. (As a check use tan A= a~\ by ~b Ex. 2. By use of four-place tables, solve the right triangle in which a=3.104, b 2.965. B 3.104 tan A 3.104 (Art. 44) 2.965 B = 90~ - A. 3.104 c= - = - (Art. 41) cos B A 2.965 0 FIG. 23. 3.104 log 0.4920 3.104 log 0.4920 2.965 colog 9.5279 - 10 43.69~ colog cos 0.1408 A = 46.31~ log tan 0.0199 c = 4.293 log 0.6328 B = 90~ - 46.31~ = 43.69~. 45. Sources of Power in Trigonometrical Solution of Triangles. There is danger that the pupil form mechanical habits of solving triangles without realizing the nature or RIGHT TRIANGLES 59 meaning of what he is doing. He should constantly realize that he is able to do what he is doing because some one before him has computed the legs of every possible right triangle whose hypotenuse is 1, and the other parts when each leg is 1, and arranged the results in tables (natural sines, etc.,) and that he uses these results (and therefore uses the work done in computing them) by the geometrical principle of similar triangles. Also that some one else has made the pupil's work easier by looking up the logarithms of all the numbers in the natural tables and arranging them in other tables, and that the pupil is using this work also. 46. Special Case. Given the hypotenuse and a leg nearly equal, the angle between them will be very small. If this angle be found directly from the parts given, it will be found in terms of the cosine. Since the cosine of a small angle changes slowly as the angle varies, such a solution will not be accurate in the last figures. A more accurate solution is obtained by first calculating the third side by the use of the formula a= (c + b)(c- b) and finding the angle mentioned in terms of the sine. Ex. Given c= 412, b= 410, solve the triangle. By the formula, a = /(412 + 410) (412 - 410) = \/822 x 2..-. log a-I- (log 822 + log 2). 822 log 2.91487 A 2 log 0.30103 2)3.21590 a = 40.546 log 1.60795 Als( 40.546 log 1.60795 412 colog 7.38510 - 10 A = 5 38' 52" log sin 8.99305 - 10 B = 90~ - 5~ 38' 52" = 84~ 21' 8". B ijj a 410 FIG. 24. U 40.546 o sin A = 412 60 TRIGONOMETRY EXERCISE 19 Using five-place tables, solve in full the following right triangles, given: (In working each example outline all the work carefully before looking up any logs-see Ex. 1, p. 18.) 1. c=18.4, a=10.7. 5. c=.89672, a=.68425. 2. c= 37.266, a 20.46. 6. b = 14.222, c= 21.678. 3. a=26.725, c=39.626. 7 a =.0628, b =.0487. 4. a=5, b=6. 8. a =.1777, c=.25643. 9. Given a= 4 yd., b= 9 ft., find A. 10. Given a = 8.701 yd., b = 21.645 yd., find L A. 11. Given b =.26725, c =.39626, find Z B. 12. Solve in full if a=6, b =6. 13. Find A if a =.02678, b =.05537. 14. Solve in full if c = 117.32, a= 112.67. SUGGESTION. First use b = /c2 - a2 = /(c + a) (c - a). 15. Solve in full if b = 358, c = 362. 16. Solve in full if a = 26.63, c = 27.99. 17. If the Mt. Washington railway at a certain place rises 3596 ft. for 3 mi. of the length of the track, what angle on the average does the track make with the horizon? 18. The carpenter's rule for constructing 4 of a right angle is to construct a right triangle whose legs are 5 and 12 inches and take the greater acute angle in the triangle. How far is this from being correct? 19. Which of the examples in Exercise 22 are you able to solve by the methods of Case II? Solve two of these. 20. Make up a similar practical problem for yourself and solve it. Solve by use of four-place tables, having given: 21. c = 23.7, a-15.7. 25. b=6.7, c=9.7. 22. c=.562, b=.3962. 26. b=.12675, a =.14296. 23. a= 33.29, b =27.28. 27. c= 132.96, b = 100.82. 24. a=5, b=8. 28. a=.07282, c=.11111. 29. a =2367, b=1827.6. RIGHT TRIANGLES 61 30. Given a = 11, c = 16, find A. 31. Given a =27.82, b = 33.67, find B. 32. Given c = 156.7, b = 148.2, solve in full. First use a = V c2- b2 = ac + b)(c- b). 33. Given c = 862, a = 854, solve in full. 34. Given a = 98.6, b = 63.4, find A. 35. Given c=.4367, b=.1967, find B. 36. Work Exs. 17-20 by the four-place tables. Without the use of tables solve in full each of the following right triangles, given: 37. a =13, b =13. 41. c=6, a=3V3. 38. c=18, a=9. 42. c= V2, b=1. 39. c=200, = 100. 43. c =100, a= 50V3. 40. a=V3, b=1. 44. a+c= 18, b =6/3. 45. Solve Exs. 3 and 4 of this Exercise without the use of logarithms. 46. How many of Exs. 37-43 are you able to solve at sight without drawing a figure? 47. Isosceles Triangles. If certain parts of an isosceles triangle be given, the unknown parts may often be determined by dividing the isosceles triangle into two equal right triangles by means of a perpendicular drawn from the vertex to the base, and by solving one of the right triangles thus formed. Ex. 1. If the vertex angle of an isosceles triangle is 42~ 30' and a leg is 47.6, find the base. o Draw the altitude OD. Then ZAOD=21~15'. Hence, in the right A AOD, we have a side and an acute angle given, to find the base AD (Case I). Hence 4230/ AD = 47.6 sin 21~ 15'. 47.6 log 1.67761 21~ 15' log sin 9.55923 - 10 AD =17.252 log 1.23684 A B AB = 2 AD = 34.504, FIG. 25, TRIGONOMETRY Ex. 2. By use of four-place tables, solve the isosceles A triangle whose base is 12.25 and vertex angle 28.22~. /I,\ Draw the altitude AD. Then Z BAD= -(28.22~) = 14.11~. l///~ j \ Z~/ B = 90 ~- 14.11~ = 75.89~. AB= 6.125 sec 75.89~ = 6.125 cos 75.89~ ~ \D 6.125 log 0.7872 B 6125 C 75.89~ colog cos 0.6130 FIG. 26. AB = 25.129 log 1.4002 48. A regular polygon may be divided into equal right triangles by lines drawn from the center to the vertices and by the apothems to the sides. Hence if certain parts of a regular polygon are given, the remaining parts may often be determined by dividing the polygon into right triangles and R r solving one of these triangles. It is to be observed that one of the A D A right triangles, as ACD of Fig. 27, has FIG. 27. the radius of the circle circumscribed about the polygon for its hypotenuse AC, and the radius of the inscribed circle, 360~ CD, for a leg. Hence, Z A CA'- 3, where n denotes the n number of sides of the polygon, and A CD of the right 180~ triangle =18 n EXERCISE 20 Using five-place tables, solve each of the following isosceles triangles, given: 1. Base = 120, base Z = 60~. 2. Leg =216, vertex = 110~. 3. Base / = 56~ 18', leg = 8.7265. 4. Base Z = 38~ 17' 50", altitude = 31.42. RIGHT TRIANGLES 63 5. Base = 55~ 18' 24", altitude = 762.89. 6. Base = 8.2364, altitude = 7.8. 7. Vertex / = 113~ 17', base =.12692. 8. Altitude - 4835, base =9248. 9. One side of a regular pentagon is 12. Find the apothem, radius; perimeter, and area of the pentagon. 10. One side of a regular decagon is 1. Find the apothem, radius, perimeter, and area of the decagon. 11. The radius of a circle is 16 feet. Find the side, apothem, and area of a regular inscribed dodecagon. 12. Find the same magnitudes for a regular dodecagon which is circumscribed about a circle whose radius is 17. 13. The diagonal of a regular pentagon is 14; find the side, apothem, perimeter, and area of the pentagon. 14. The apothem of a regular heptagon is 0.69786; find the perimeter and area of the heptagon. If mn denotes the base, h the altitude, I the leg, C the vertex angle, and D the base angle of an isosceles triangle, find: 15. h, m, and C, in terms of D and 1. 16. D, I, and C, in terms of nm and h. 17. D, C, and nm, in terms of h and 1. 18. C, h, and 1, in terms of D and m. 19. D, h, and I, in terms of C and nm. 20. Solve the isosceles triangle in which a leg 2.62731 and the altitude = 1.76683. 21. If a chord 22.67 ft. in length subtends an arc 127~ 23', what is the radius of the circle? 22. If the radius of a circle is 105.27 ft., what is the length of a chord which subtends an are of 54~ 13'? 23. The side of a regular polygon of fourteen sides inscribed in a circle is 21.6 ft.; find the side of a regular twenty-sided polygon inscribed in the same circle. 24. The radius of a circle is R; show that each side of a regular inscribed polygon of n sides is 2Rsin / —i, and that each side of a regular circumscribed polygon is 2? tan ) regular circumscribed polygon is 2 R tan (1 - )* \,n / 64 TRIGONOMETRY 25. Each side of a regular polygon of n sides is m; show that the radius of the circumscribed circle is equal to - csc ('-80), and the radius 2 \n of the inscribed circle is equal to - cot (180 2 \f,n 26. If the chord of an arc of 36~ is 24, find the chord of an arc of 12~ in the same circle. 27. If the chord of an arc of 48~ is 36, find the chord of an arc of 66~ in the same circle. Using four-place tables, solve the isosceles triangle in which: 28. Leg = 36.72, base Z = 32.6. 29. Base = 1600, base / = 67.4~. 30. Vertex Z = 117.72~, altitude = 17.83. 31. Base =.7368, altitude.4864. 32. Altitude = 112.67, leg =128.7. 33. Leg =67.87, base Z = 32.73~. 34. Altitude =.11683, base Z = 76.18~. 35. Base =31.26, altitude = 21.73. 36. Vertex Z = 151.7~, leg =.4363. 37. One side of a regular octagon is 14. Find the apothem and area of the octagon. 38. The apothem of a regular pentagon is 19.7. Find the perimeter of the pentagon. 39. A regular decagon is inscribed in a circle whose radius is 1.76. Find the side and apothem of the decagon. 40. Find the magnitude of the various parts of a regular heptagon circumscribed about a circle whose radius is 21. 41. The diagonal connecting two alternate vertices of a regular dodecagon is 18. Find the side, apothem, and area of the dodecagon. 42. If a chord of 37.82 ft. subtends an arc of 118.3~, find the radius of the circle. 43. If the radius of a circle is 100, what is the length of a chord which subtends an arc of 67.7? RIGHT TRIANGLES 65 Without the use of the tables, solve the following: 44. The base of an isosceles triangle is 50, and the vertex angle is 120~. Find the base angle and altitude. 45. The leg of an isosceles triangle is 100, and the altitude is 50. Find the base angle and base. 46. The altitude of an isosceles triangle is 10, and the base angle is 60. Find a leg and the base. 47. The leg of an isosceles triangle is 6V2, and the base is 12. Find the base angle, vertex angle, and altitude. 48. The radius of a circle is 2. Find the number of degrees in an arc which subtends a chord whose length is 2V3. 49. The diagonal of a square is 10. Find the side of the square. 50. How many of Exs. 44-49 can you work at sight? AREAS 49. General Method of computing Area of a Right Triangle. If b denote the. base, a the altitude, and K the area of a right triangle, by geometry K=I ab.. log K =log + log b + colog 2. Ex. 1. Given A=37~ 19', b=308, find the area of the right triangle. To find log a and then the area we proceed as follows: B a = 308 tan 37~ 19'. (Art. 41) 308 log 2.48855 37~ 19' log tan 9.88210 - 10 a log 2.37065 K 308 log 2.48855 371 2 colog 9.69897 - 10 A 308 K = 36155 log 4.55817 FIG. 28. Ex. 2. Find the area of a right triangle in which the hypotenuse is 417 and the base 356. a = Vc2 —2 b= /(417)2 - (356)2 = V(417 + 356)(417 -356) = 773 x 61... log a = (log 773 + log 61). 66 TRIGONOMETRY K=- ab..'. log K-= log a + log b + colog 2. 773 log 2.88818 ~ log 1.44409 j^~~3~/ 161 log 1.78533 1 log 0.89267 /.^jrZi~~= 1356 log 2.55145.A 8 -5.6 a 2 colog 9.69897 - 10 FIG. 29. K= 38652.7 log 4.58718 Ex. 3. By use of four-place tables find the area of the right triangle in which A= 37.32~ and b= 308 (see Fig. 28). log K = log a + log 308 + colog 2. To find log a, a = 308 tan 37.32~. 308 log 2.4886 37.32~ log tan 9.8821 a log 2.3707 308 log 2.4886 2 colog 9.6990 - 10 K= 36167 log 4.5583 50. Formulas for Area of a Right Triangle. The area of a right triangle may often be obtained more readily by the use of a formula involving only the particular parts of the triangle given. Denoting the area of a right triangle by K, let the pupil show that When the two legs are given, K= -- ab. When an acute angle and the hypotenuse are given, K = c2 sin A cos A (or = 1 c2 sin B cos B). When the hypotenuse and a leg are given, = a(c + )(c-a) (or = b(c + b)(c-)). When an acute angle and a ley are given, = -a2 tan B (or= 1 b tan A), or K= a2 cotA (or= - b2 cot B). By geometry, what is the method or formula for computing the area of an isosceles triangle? of a regular polygon? The formulas given above for computing the area of a right triangle are sometimes useful in computing the area of an isosceles triangle, or of a regular polygon. RIGHT TRIANGLES 67 EXERCISE 21 Using five-place tables, compute the area of the right triangle in which: 1. A=28 18', b = 216. 5. B =63~ 18', c= 124.72. 2. B=72~, a= 196. 6. a=192.7, b=212.97. 3. A=21 16'30", c = 31.967. 7. a=0.73216, c=.9125. 4. c= 46.72, b= 32.54. 8. c= 927.8 ft., b = 759.8 ft. 9. Given a = 2.5 and K= 4.27, find b, c, and A. 10. Given K= 7.256 and A = 26~ 18', find a, b, and c. 11. Given K= 55.686 and c = 15.67, find a, b, and A. Compute the area of the isosceles triangle in which: 12. Base =12.67, leg= 9.267. 13. Base =.67892, altitude =.26217. 14. Base angle = 68~ 18', leg =.2892. 15. Vertex angle = 105~ 17', altitude = 13.67. 16. Vertex angle = 113~ 18', leg 25.6. 17. Given area = 16.72 and base = 6.37, find altitude, leg, and base angle. 18. Given area =.9273 and base angle = 27~ 18', find leg, base, and altitude. 19. Given area = 22.76 and vertex angle = 117~ 55', find leg, base, and altitude. 20. Find the area of the regular pentagon whose perimeter is 3.35. 21. Find the area of the regular dodecagon whose apothem is 1.7267. 22. Find the area of a regular heptagon inscribed in a circle whose radius is 0.7516. 23. Given a regular octagon whose apothem is 2.27; find the difference between its area and that of the inscribed circle. 24. Given n = 9 and K = 30, find r, c, and R. 25. Given n = 11 and K = 35, find the perimeter. 26. Given n = 5 and K = 37, find p and R. 27. If n denotes the number of sides, I? the radius, and C the central angle of any regular polygon, prove that K= nR2 sin 1 C cos i C. 68 TRIGONOMETRY Using four-place tables, find the area of each of the following right triangles, given: 28. A = 34.6~, a = 67.8. 32. b = 8.42, c = 11.26. 29. B = 84~, a = 100. 33. B = 39.24~, c = 23.68. 30. A = 18.62~, b = 72.36. 34. c = 5000, a = 3000. 31. a =.16376, b =.19762. 35. A= 47~, a =.0087. Solve the following right triangles, given: 36. b = 6.37, K = 26.38. 37. K= 1200, A = 63.18~. 38. K=.4962, c=.1635. Find the area of each of the following isosceles triangles, given: 39. Base =.7262, leg=.5263. 40. Altitude = 12.36, leg = 17.27. 41. Altitude = 86.27, base = 111.63. 42. Base angle = 42.67~, leg = 17.43. 43. Vertex angle = 100.24~, altitude = 8.217. 44. Vertex angle = 78.32~, leg =.6526. In an isosceles triangle: 45. Given area =192.67 and base =43.64, find altitude, leg, and base angle. 46. Given area = 0.7362 and base angle = 37.43~, find leg, base, and altitude. 47. Given area = 1367.8 and vertex angle = 113.28~, find base, leg, and altitude. 48. Given area =.1025, and leg =.4916, find the base, altitude, and angle at the base. 49. Find the area of a regular decagon whose perimeter is 27.63. 50. Find the area of a regular pentagon whose apothem is.4782. 51. Find the area of a regular heptagon inscribed in a circle whose radius is 116.2. 52. Given the side of a regular octagon as 5.33, find the difference between the area of the octagon and that of the circumscribed circle. RIGHT TRIANGLES 69 In a regular polygon: 53. Given n = 7 and K= 14, find c, r, and R. 54. Given n = 11 and K= 1000, find r, c, and R. 55. Given n = 9 and K= 47, find r, c, and R. 56. Given n = 14 and K= 800, find the perimeter. Without the use of the tables, find the area of each of the following right triangles, given: 57. a = 100 and A = 60~. 61. a = 80 and c = 160. 58. b =-600 and c = 1200. * 62. b = 40 and c = 402. 59. c = 26.3 and b = 21.2. 63. c = 4000 and A = 30~. 60. B = 60~ and a = 90. 64. A=45~, b=12,0. Also of each of the following isosceles triangles, given: 65. Vertex = 120~, leg = 100. 67. Leg = 40, altitude = 20. 66. Base Z = 30~, base = 200. 68. Vertex Z = 90~, leg = 400. EXERCISE 22. APPLICATIONS Solve, using either set of tables: 1. The angle of elevation (see Art. 88) of the top of a cliff, measured from a point 225 ft. from the base, is 60~. How high is the cliff? 2. At a point 170 ft. from a tower, and on a level with its base, the angle of elevation of the top of the tower is found to be 70~ 18' [70.3~]. What is the height of the tower? 3. The angle of elevation of the sun is 65~ 30' [65.5~] and the length of a tree's shadow, on a level plane, is 52 ft. Find the height of the tree. 4. If the Eiffel Tower is 984'ft. high, what will be the angle of elevation of its top, when viewed at a distance of a mile? 5. The length of a kite string is 700 ft., and the angle of elevation of the kite is 44~ 36' [44.6~]. Find the height of the kite supposing the kite string to be straight. 6. One of the equal sides of an isosceles triangle is 62.8 ft., and one of the equal angles is 52~ 18' 36" [52.31~]. Find the base, altitude, and area of the triangle. 7. What is the elevation 'of the sun, if a tree 82.6 ft. high casts a shadow 105.8 ft. long on a horizontal plane? 70 TRIGONOMETRY 8. A ladder, 25 ft. long, leans against a house and reaches to a point 21.6 ft. from the ground. Find the angle between the ladder and the house, and the distance the foot of the ladder is from the house. Why are we able to solve an example like this by trigonometry when we are not able to do so by geometry? 9. The Washington Monument is 555 ft. high. How far apart are two observers 555 who from points due west of the monument 5 4 8~0417' observe its angles of elevation to be 25~ and 48~ 17' [48.28~] respectively? 10. If the Grand Canion of the Colorado is 5000 ft. deep, what will be the angle of depression of the river flowing through it when viewed from the brink of the caion at a horizontal distance of 3 mi.? 11. If a hillside has a slope of 7~, a dam 10 ft. high will force the water how far back up the hillside? 12. A tower.125 ft. high stands on the bank of a river. The angle subtended by the tower at the edge of the opposite bank is 23~ 31' [23.52~]. Find the width of the river. 13. What is the height of a hill if its angle of elevation taken at the foot of the hill is 40~ 18' [40.3~] and if this angle taken 150 yd. from the foot of the hill and on a level with the foot is 28~ 42' [28.7~]? 14. From the summit of a hill, there are observed two consecutive milestones on a straight horizontal road running from the base of the hill. The angles of depression (see Art. 88) are found to be 12~ and 7~ respectively. Find the height of the hill. 15. A valley is crossed by a horizontal bridge, whose length is 1. The sides of the valley make angles m and n with the plane of the horizon. Show that the height of the bridge above the bottom of the valley is cot m + cot n 16. Upon a hill overlooking the sea stands a tower 70 ft. high. From a ship the angle of elevation of the base and top of the tower are respectively 15~ 4' [15.07~] and 15040' [15.67~]. What is the height of the hill and the horizontal distance of the ship from the tower? 17. Given: K Z AKF= Z ARK= Z RTF= 90~. T Z KAR = 60~ and AR = 12. Without the use of the tables find the length of all the other lines in the 0 figure. A 12 F RIGHT TRIANGLES 71 18. A boy standing m feet behind and opposite the middle of a football goal, sees that the angle of elevation of the nearer crossbar is A, and the angle of elevation of the crossbar at the other end of the field is C. Prove that the length of the field is m (tan A cot C- 1). 19. A railroad embankment is 7 ft. high. If the top of the embankment is 8 ft. wide and the sides slope at an angle of 43~, what will be the width of the base? 20. If the Metropolitan Life Insurance building of New York City is 700 ft. high, how far from the building is an observer when the angle of elevation of the top of the building is 7~ 36' [7.6~]? 21. A man standing on the bank of a river observes that the angle of elevation of the top of a tree on the opposite bank is 60~; when he retires 50 m. from the edge of the river, the angle of elevation is 30~. Without the use of the tables find the height of the tree and the width of the river. s 22. Given: KP=6 m.; ZK -f=Z F=60~; LSRN= 45~; R N and RNTP a square. Without the use of the tables find the 6O lengths of KR, PR, RS, ST, SF, and TF. K P T F 23. A tower and a monument stand on the same horizontal plane. The height of the tower is 35.6 m. and the angles of depression of the top and base of the monument, as observed from the top of the tower, are respectively 5~ 16' 48" [5.28~] and 8~ 18' 30" [8.3~]. How high is the monument? 24. A flagstaff stands on the roof of a building. From a point A on the ground the angles of elevation of the foot and the top of the flagstaff are 37~ and 46~, respectively. From a point B, 250 ft. farther off and in line with A and the base of the building immediately below the flagstaff, the angle of elevation of the top of the flagstaff is 27~ 30' [27.5~]. Find the length of the flagstaff. 25. From the top of a lighthouse, 150 ft. above the sea level, the angle of depression of a buoy situated between the lighthouse and the shore was 62~ 14' [62.23~] and that of a point on the shore in a straight line with the buoy was 12~ 10' [12.17~]. Find the distance, in feet, of the buoy from the shore. 26. The base of a rectangle is 50.62 and its diagonal is 71.6. Find the altitude of the rectangle and the angle which the diagonal makes with the base. 72 TRIGONOMETRY A 27. Given: OA =, 1i \ Z ABO = BCO = 90~. B<y ^, /Express AB, OB, BC, OC in terms of trigonometric functions of x and y. 0 C 28. The Singer building of New York City is 612 ft. high. Make up some problem concerning this which can be solved by trigonometry. 29. The diagonals of a rhombus are 42.28 and 30.58. Find the sides and angles. 30. Make up (or collect) as many different examples as you can showing the practical uses of the solution of right triangles by trigonometry, each example being distinct from the rest either in principle or in the field of its application. 31. Who first, and at what date, taught the trigonometric solution of triangles in the same general way as is done at present? CHAPTER IV GONIOMETRY TRIGONOMETRIC FUNCTIONS OF ANGLES IN GENERAL 51. Angles greater than 90~. In solving oblique triangles, angles greater than 90~ may occur. Hence it is important to learn what the trigonometric functions of an obtuse angle are. Similarly the radius of a rotating wheel, as in a dynamo, generates angles greater than 360~ and by successive rotations generates angles unlimited in size. In astronomy, the heavenly bodies, by successive rotations about an axis, and by revolutions in an orbit, also generate angles unlimited in size. Hence a general method is needed of determining the trigonometric functions of angles unlimited in size. 52. The Four Quadrants. Definitions. Let AC (Fig. 30) be the horizontal diameter of a circle ABCD, and BD the diameter perpendicular to AC. B Then AOB, BOC, COD, and DOA P2 are termed the first, second, third, and fourth quadrants of the circle. c A On Fig. 31 the four parts into which a plane is divided by the lines XX' and YY' p3\ are also termed quadrants and arejnumbered in the same order as the quadrants of a D circle. FIG. 30. In treating of the properties of angles in general, it is convenient, wherever possible, to let the angles start at the same place, as OA (that is, to have the vertex and a side in common). Let the rotating radius start in the position OA and rotate toward the position OB (in the direction contrary to'that in which the hands of a clock move, or counter-clockwise). 73 74 TRIGONOMETRY The ZsAOP, AOP2, AOP,, AOP4 are called angles in the first, second, third, and fourth quadrants respectively. The initial line of an angle is the rotating radius, which generates the angle, in its first position, as A O. The terminal line of an angle is the rotating radius in its final position, as OP2 for Z AOP2. By continuing the rotation of OA, angles greater than 360~ will be generated. If two angles differ by 360~, or by any exact multiple of 360~, they will have the same terminal line. Coterminal angles are angles which have the same terminal line, as 37~ 397~, and 757~. In general an angle is said to be of or in that quadrant in which its terminal line lies. 53. Negative Angles. In algebra it is shown that negative quantity is quantity exactly opposite in some respect, as, for instance, in direction, from other quantity taken as positive. Hence if the rotating radius move from the position OA (Fig. 30) toward the position OD (that is, in the same direction with the hands of a clock, or clockwise), a negative angle, as the acute Z AOP4, will be generated. If the radius continue to rotate in this direction, a whole series of negative angles will be formed similarly. 54. Rectangular Co'rdinates. In order to define the ~Y ~ trigonometric functions of angles greater than 90~, and of negap2 p/P rtive angles, two straight lines, + / + XX' and YY' (Fig. 31), interM2m < -__o / _. —__-X secting at the point 0 and perpendicular to each other, are PS Pp taken and called axes. The Y signs of other lines used are deFIG. 31. termined by their position with GONIOMETRY 75 reference to these axes Lines drawn from YY' to the right (and II XX') are taken as +; lines drawn from YY' to the left (and I1 XX') are taken as -. Lines drawn from XX' above (and I1 YY') are taken as +; lines drawn from XX' below (and I1 YY') are taken as -. The origin is the point in which the axes intersect, as the point 0 on Fig. 31. The ordinate of a point is the distance of the point above or below the axis XX'. The abscissa of a point is the distance of the point to the right or left of the YY' axis. Thus, the ordinate of P1 is MP,; the abscissa of Pi is OM1. Coordinates is the general term for abscissa and ordinate of a point. The coordinates of a point may be written together in parenthesis with abscissa first and a comma between. Thus if OM1 = a, and MliP = b, the coordinates of P1 are (a, b). The distance of a point is the line drawn from the origin to the point, thus on Fig. 31 the distance of Pi is OP1. The distance of a point is independent of sign. 55. Definitions of Trigonometric Functions of Any Angle. Y Y Y Y PIjlr,< M3I 4D // o xM 0M X 0 X o0 X P4 FIG. 32. FIG. 33. FIG. 34. FIG. 35. If we regard an angle as formed by an initial line and a line drawn from the origin to a point whose abscissa and ordinate are considered, then sine of an angle = ratio of ordinate to distance; cosine of an angle = ratio of abscissa to distance; 76 TRIGONOMETRY tangent of an angle = ratio of ordinate to abscissa; cotangent of an angle = ratio of abscissa to ordinate; secant of an angle = ratio of distance to abscissa; cosecant of an angle = ratio of distance to ordinate. Thus in Figs. 32, 33, 34, 35, sin L XOP1 - M-, OP, =MP2 MsP-3 'sin P M4P4 sin L XOP2 = -, sin Z XOP, =, sin X XOP, 4 2-OOP2 OP4 Let the pupil point out in like manner the other trigonometric functions of the angles XOP1, XOP2, XOP3, XOP4. 56. Trigonometric Functions represented by Lines. If a circle (Fig. 36) be drawn with 0 as a center and a radiis OA, equal to 1, and with MAP1, M2P2, MIP3, M4P4, perpendicular to XX', P1/~P xy y1 -AT4 P3 FIG. 36. FIG. 37. sin / AOP _ M P _ MP1 _ P, OP1 1 Similarly, sin L AOP2 = M2P2; sin / AOP3 = M3P3; and sin Z AOP4 = M4P4. Or, in the circle as described, the sine of an angle is represented by a line drawn from the terminal end of the arc intercepted by the angle, and perpendicular to the horizontal diameter. GONIOMETRY 77 Similarly if (in Fig. 37) NPi, N2P2, N3P3, N4P4 are perpendicular to YY', cos L AOP, - NoP, N1P, = NIP; OP 1 1 cos ZAOP?2= NP2; cos Z AOP = N3P; cos AOP4= N4P4. Or, in the circle as described, the cosine of an angle is represented by a line drawn from the terminal end of the arc intercepted by the angle, and perpendicular to the vertical diameter. Similarly (in Fig. 38), if TT' is tangent to the circle at A, tan z AOP1 = AT, = AT _ AT,; OA 1 tan ZAOP2 = AT2;. tan Z AOP3 = AT3; tan LAOP4 = AT4. Or in the circle as described, the tangent of an angle is represented by a line drawn touching the initial end of the arc intercepted by the angle, and terminated by the radius to the other end of the arc, produced. Y Y' x 0 AX FIG. 38. FIG. 39. Similarly (in Fig. 39), if R1R4 is tangent to the circle at the point B, RB R2o Rs R1 _ cot XAOP? = BR2; cot zAOP3 =BRA; cot ZAOP4 =BR4; or in the circle as described the cotangent of an angle is repre 78 TRIGONOMETRY sented by a line which is the tangent of the complement of the given angle. On Fig. 38 the secants of the four angles used are readily shown to be represented by OT,, OT2, OT3, OT4; or, in general, the secant of an angle is represented by a line drawn from the center through the terminal end of the arc intercepted by the angle, and terminated by the tangent. Similarly on Fig. 39 the cosecants of the four angles used are represented by OR1, OR, OR,, OR4; or, in general, the cosecant of an angle is represented by a line which is the secant of the complement of the angle. It will be convenient to draw a figure for an angle in each quadrant showing the lines which represent the functions of that angle. B _I R B P A',, A' A Ar o 0 MA A' -M 0 B' B' T FIG. 40. FIG. 41. B R R B A X A; A A o B' B' T FIG. 42. FIG. 43. The lines which represent the various trigonometric functions of an angle are not the same as the trigonometric functions which they represent, but they have many of the same properties as the functions or ratios. It is often GONIOMETRY 79 easier to perceive these properties by the use of the lines, than by the use of the ratios which the lines represent. In deriving the properties of the trigonometric functions of angles greater than 90~ we shall derive them from the lines representing the functions; but in such cases we give some specimen proofs showing how these properties may be derived from the ratio definitions (of Art. 55), and in other cases leave it as an exercise for the pupil to derive the proofs from the ratios if the teacher considers it desirable. 57. Signs of the Trigonometric Functions in the Different Quadrants. Of the lines representing the sines of angles in the different quadrants, viz. IVFP,, M2P,, M3P, MP4 (Fig. 36), the first two are above the horizontal axis, and are therefore plus in sign; the last two are below, and therefore minus. Hence the signs of the sines of angles in the four quadrants are respectively +,.+, -, -. The students may obtain the same results from Figs. 32-35 by using the general definitions of trigonometric functions given in Art. 55. Similarly in Fig. 37 the cosine lines NPl, N2P2, NP]3, N4]4P are +, -, -, +, respectively; and in Fig. 38 the tangent lines AT1, AT, AT3, AT4 are +, -, +, -, respectively. Since the sine of a quantity and of its reciprocal must be the same, the sign of the cotangent in the various quadrants must be the same as that of the tangent; that of the secant, the same as the cosine; that of the cosecant, the same as the sine. Or, proceeding geometrically, on Fig. 39, the cotangent lines BR1, BR2, BR3, BR4 are +, -, +, -. The secant is considered as plus when it is drawn in the same direction from the center as the terminal radius (thus OT2, Fig. 38, is opposite in direction from OP2 and is therefore negative). Hence the secant lines OT1, OT2, OT OT4 have the signs +, -, -, +, respec 80 TRIGONOMETRY tively. Similarly the cosecant lines (Fig. 39) OR,, OR2, OR3, OR4 have the signs +, +,-, -. The results thus obtained may be arranged in a table as follows: I. I IIil IV sine and cosecant + +-.cosine and secant + - - tangent and cotangent + - - EXERCISE 23 In which quadrant is each of the following angles? 1. 123~. 6. 4150. 11. 1111~. 2. 1550. 7. — 18~. 12..,222~. 3. 215. 8. - 125~. 13. -1826~. 4. 2850. 9. 612~. 14. 2625~. 5. 338~. 10. - 500. 15. - 1500~. 16. Find the signs of the functions of the angles in Exs. 1, 3, and 5. Give two positive and two negative angles each of which is coterminal with: 17. 25~. 18. -30~. 19. 100~. 20. -100~. Find the smallest possible angle coterminal wvith: 21. 425~. 23. - 300~. 25. -17600~. 22. 780~. 24. 875~. 26. 1493~. In which quadrant does an angle lie: 27. If its sin is positive and cos negative? 28. If its tan is positive and sin negative? 29. If its cot is negative and cos negative? 30. If its csc is negative and cot positive? 31. If its cos is positive and tan negative? 32. If its sec is negative and tan negative? 33. A railroad embankment is 9 ft. high and 43 ft. wide at the base. If each of its sides makes an angle of 27~ 15' [27.25~] with the horizontal, how wide is the top of the embankment? GONIOMETRY 81 34. If a railroad embankment is 7 ft. high and 28 ft. 9 in. wide at the top, and one side has a slope of 23~ 30' [23.5~] and the other a slope of 32~ 45' [32.75~], how wide is the base? 35. Make up a similar example for yourself. 58. Functions of 0~, 90~, 1800, 2700, 360~. In Arts. 34 and 35 it is shown that sin 0~= 0 and sin 90= 1. Similar results are readily perceived for other quadrants by the use of a figure showing the sines as lines in the different quadrants. Thus in Fig. 44 in the first quadrant the sine increases from 0 to 1; in the second quadrant it decreases from 1 to 0; + + + + in the third it decreases from 0 to - 1; in the fourth quadrant it increases from - - -1 to 0. Hence the sines of 0~, 90~, 180~, 270~, 360~, in order, are 0, 1, 0, 4 FIG. 44. - 1, 0. Similarly in the first quadrant (Fig. 45) the cosine decreases from 1 to 0; + in the second quadrant it decreases from 0 to - 1; in the third quadrant it increases + -1 +1 from -1 to 0; in the fourth quadrant it increases from 0 to 1. Hence the cosines of 0~, 90~ 180~, 270~, 360~, in order, are - / 1, 0,- 1, 0 1. FIG. 45. sill x Similarly from Fig. 38, or from the formula tan x si, it is clear Cos X that the tangent in the different quadrants changes from 0 to oo; from - oo to 0; from 0 to o; from - oo to 0. Hence the tangents of 0~, 90~, 180~, 270~, 360~, in order, are 0, ~ o, 0o, oc, 0. The changes in the value of the cotangent, the secant, and the cosecant, and the values of these functions for the above-mentioned angles may be obtained from geometrical figures in like manner, but these values are obtained more readily from the reciprocal formulas 1 1 1 cot = -; sec =-; csc= -. tan cos sin Thus, 1 1 sec 180~ = 1=0 = -1. cos 180 - 1 82 TRIGONOMETRY Obtaining the values of the required functions thus and arranging all the results obtained in a table, we have 0~ 90~ 180~ 270~ 360~ sin 0 1 0 -1 0 cos 1 0 -1 0 1 tan 0 co 0 co 0 cot o 0 o 0 co sec 1 c -1 o 1 SC co 1 0o -1 oo In the above table oo is to be taken as + or - according to the side from which it is approached (see Art. 57). EXERCISE 24 Find the numerical value of: 1. 5 sin 90~ + 7 cos 180~ + 8 sin 30~. 2. m sin 00 +p cos 90~ + c cot 360~. 3. b cos 90~ - c tan 180~ + b cot 270~. 4. (a2 - c2) cos 180~ + 4 ac sin 90~. 5. 2 tan 0~ sin 90~ - 4 sec 0~ sin 270~ + 5 cse 90~ cos 0~ cot 270~. 6. a cos 180~ sec 360~- b tan 180~ sin 270~- a sin 90~ sec 0~ + b sin 90~ cos 270~. 7. m sin 270~ csc 90~ + n cos 180~ cse 270~ cot 270~ - m sec 180~. 8. 6 m csc 90~ cos2 0~ - 17 n sec2 0' cot2 270~ + 3 m sin 270~ sec 360~. 9. Show that 4 cos2 45~ sec 0~ + 6 tan2 30~ sin 270~ + 12 cot2 45~ cos 180~ — 4 tan2 45~ csc 270~ = -8. 59. Trigonometric Functions of Angles greater than 360~. It is evident that the trigonometric functions of angles from 360~ to 720~ are the same in order as those from 0~ to 360~. Similarly for every succeeding 360~, the functions repeat themselves. Hence to find the functions of an angle greater than 360~, Divide the angle by 360~ and find the required trigonometric function of the remainder. GONIOMETRY 83 Ex. Sin 766~ = sin (2 x 360~ + 46~) = sin 46~. 60. Formulas for the Acute Angle extended to any Angle. The equations and formulas proved in Arts. 27-29 concerning the function of an acute angle are true for the functions of any angle. Thus, on each of the Figs. 40-43, MP2 + OM = OP2. That is, sin2 x + cos2 x = 1. Also in each quadrant the A OMP, OAT, OBR are simjlar..'. AT: OA = IMP: OMI, or tan x: = sin x: cos x, sin x or tan — = cos x Let the pupil prove in like manner, 1 1 sin x, cos x csc X sec x Or these results may be proved directly from the ratio definitions of the trigonometric functions of any angle. For if angle XOP of Figs. 32-35 be denoted by x, in any quadrant abs. P2 + ord. P2 = dist. P2, /abs. P 2 /ord. P\2 dlist. P dlist. P= Hence, sixn2 +- cos2 x = 1. Let the pupil prove in a similar manner that tacC2 x + 1 = sec2 x, and cot2 x + 1 = csc2 x. ord. P Also t ord. P dist. P sin x sinx Also tan x = -, or tam x = abs. P abs. P cosx cosx dist. P Al orcl. P dist. P abs. P dist. P ord. P abs. P1 dist. P ord. P ' dist. P abs. P abs. P ord. P or sinx x cscx = 1, cosx x sec x=, tan x cot x= 1. 84 TRIGONOMETRY 61. One function of an angle being given, the other functions may be found in a manner similar to that used in Art. 30. Owing to the fact that for angles less than 360~, two angles correspond to any given function, two sets of answers are found in each example. Ex. 1. Given cos x = - find the other functions of x. By the table of signs (Art. 57) a negative cosine occurs in both the second and third quadrants. 2d quadrant. sin x V= 1- _ (4)2 -= V - 1 = 6v 9_ 9 sill x tan x.-^-., etc. cos X 3d quadrant. sin = V/1 - (4)' = -- =-5 sin x - 3 tan x = = -1 _ -, etc. COS -x Ex. 2. Given tan x = 2, find the remaining functions of x. The positive tangent occurs (see Art. 57) in both the first and third quadrants. 1st quadrant. sec2 x = 1 + tan2 x = 1 + 4 = 5, sec x = V5, 1 1 1, etc. cos = -- = 5 etc. sec x 5 5 3d quadrant. sec2 x = 1 + 4, sec x =- /5, 1 1 cos x -1- - - /5, etc. - V/5 5 In case solutions are sought by the geometrical method, the following figures may be used in Exs. 1 and 2 respectively. (2 Y P11 5 2 FIG. 46. FIG. 47. GONIOMETRY 85 EXERCISE 25 1. Find the numerical value of sin 390~; also of cos 390~, tan 390~, and sec 390~. 2. Find the numerical value of cos 780~; also of tan 780~, sin 780~, and cot 780~. 3. Find the values of sin, cos, tan, and cot of the following angles: 4. 18600. 6. - 675~. 8. - 1740~. 5. -330~. 7. 750~. 9. 2205~. o1. Given cos x = -, find the other functions of x. 11. Given tan x = - -15 find the other functions of x. 12. Given sin x = - — 15, find the other functions of x. 13. Given cot x = 2 and sin x negative, find the other functions of x. 14. Given se x = - m and tan x negative, find the other functions of x. 15. Given tan x = -3, find the other functions of x when x is an angle in the fourth quadrant. 16. Given sec x = -6, find the other functions of x if tan x is positive. 17. Verify geometrically the results obtained in Exs. 10-16. 18. Given cot y = - V5 and cos y negative, find sin y and cse y. 19. Given tan x = —3/3 and cos x positive, find the other functions of x. 20. If 0 is in the second quadrant and if cosec 0 = - find the value of cot 0 + sec 0 tan 0 + cos 0 21. Find the value of cos 0 + cot 0 if 0 is in the fourth quadrant csc 0 + sec 0 and tan 0= - 15 62. Trigonometric Functions of 90~ + in terms of functions of x. The trigonometric functions of 90~+ x may be reduced to functions of x by use of the following formulas: sin (90~ + x) = cos x. cot (90~ + x) = -tan x. cos (90~ + ) = - sin x. sec (90~ + x) = - csc x. tan (90~ + x) = -cot x. csc (90~ + x) = sec x. 86 TRIGONOMETRY Q For, let Z AOP (Fig 48 a) be any angle x in the first quadrant. Let POQ g -~ P 90 be a right angle. Let OP = OQ =1. -R-~ M0A Then Z RQO = zMOP. (sides ) A. RQO = A MOP. (hyp. end acuteZ = ).. sin (90~ + x)= RQ= OM= cos x. FIG. 48 a. cos (90 + x) = OR= O - PM= - sin x. sin (900+x) Cos X COtX tan (90 +) s 00 ) = cot x. cos (90~+ z) -sin x Let the pupil supply the proofs for cot (90~ + x), sec (90~ + x), and csc(90~ + ). The same results may readily be obtained for angles ending in the second, third, and fourth quadrants by use of the following diagrams.,/ jI 4'p ~ M Q Q Q _ P Q FIG. 48 b. FIG. 48 c. FIG. 48 d. Ex. 1. Find the value of sin 300~. sin 300~ = sin (90~ + 210~) = cos 210~ =- sin 120~ - cos 30~ = -1-3. Ex. 2. Reduce tan 923~ to a function of an angle less than 90~. tan 923~ = tan (720~ + 203~) = tan 203~ (Art. 59) - cot 113~ = tan 23~. Ex. 3. Simplify cos (630 + A). cos (630~ + A)'= cos (270~ + A) =- sin (180~ + A) =- cos (90~ + A) = sin A. GONIOMETRY 87 EXERCISE 26 Find the numerical value of: 1. sin 210~. 4. cot 150~. 7. tan 210~. 2. cos 300~. 5. sec 1215~. 8. sin 330~. 3. tan 120~. 6. sec 900~. 9. cos 240~. 10. cos 225~+ 3 sin 330~-tan 225~. 11. cot 840~-3 tan 420~+2 sec 480~. Express each of the following trigonometric ratios in terms of a ratio of some positive angle not greater than 45~: 12. sin 142~. 18. cos 110~. 24. sin (280~ 16'). 13. tan 163~. 19. sin 567~: 25. cot (2100~ 17'). 14. cos 310~. 20. cot 1415~. 26. csc 1325~. 15. sec 185~. 21. csc 1200~. 27. cos 82~. 16. cot 265~. 22. cos 117~. 28. tan 1060~. 17. tan 315~. 23. tan 428~. 29. tan 840~. 30. Prove sin 330~ cos 390~ = cos 570~ sin 510~. 31. Prove tan 45~ sec 1080~ cos 570~ sin 510~ - sin 330~ tan 225~ cos 390' = 0. 32. Find the value of 6 sec2 1080~ tan2 135~ sin 1890~ + 8 cot 45~ cos 1140~ + csc 630~ tan 225~ cos 720~ sin 1830~. Simplify the following expressions: 33. 5 sin (90~ + x)- 6 cos (180~ + x). 34. a sin (90~ + x) + b cos (270~ + x) - c tan (180~ + x). 35. p sin (180~ + x) cos (180~ + x). 36. (a + b) sin (2700 + x) - (a - b) cos (270~ + x). 63. Trigonometric Functions of a Negative Angle. The trigonometric functions of a negative angle may be converted into functions of a positive angle by use of the following formulas: sin (-x) = - sin x. cot (- x) = - cot x. cos (- x) = cos x. sec (- x) = sec x. tan (-x) = - tan x. csc (- ) = - csc x. 88 TRIGONOMETRY For let Z AOP (Fig. 49) be a positive angle, x, and Z AOQ an equal negative angle. Let OP = OQ = 1. Then the right triangles OMP and OMQ are equal. Hence, sin (- x) = MQ = -MP = sin x cos (- x) = OM= cos x 0 -wMx A M sin (- x) - sin x, \ tan ( x) = ( = - cos (- x) cos X Q&^ = ~=-tanx. FIG. 49. Let the pupil supply the proofs for cot (- x), sec (- x), and csc (- x). The same results are readily obtained for angles in the other quadrants by the use of appropriate diagrams. Ex. 1. Find the numerical value of cos (- 225~). cos (- 225) = cos 225~, -- sin 135~ (Art. 62) =- cos 45~ = — 2, Ans. Ex. 2. Simplify cot (180~- A). cot (180~ - A) = - tan (90~ - A), = cot (- A) =- cot A, Ans. 64. Reduction Tables and General Rules. Some of the reductions made by the methods of the preceding articles are used so frequently that it is convenient to collect the results obtained by them, and arrange them in tables for future reference. Thus sin (90~ - x) = cos x. sin (180 - x) = sinx. cos (90~ -x) = sin x. cos (180 - x) = - cosx. tan (90~ - x) = cot x. tan (180~ - x) = -tan x cot (90~ - x) = tan x. cot (180- x) = -cot x sec (90~ - x) = csc x. sec (180~ - x) = -sec x csc (90~ - ) = sec x. csc (180~ - x) = csc x Let the pupil form similar tables for the functions of 270- x, 360- x, 180~ + x, 270~ + x. GONIOMETRY 89 Or the following general rule may be used: Each function of 180~ ~ x or 360~ ~ x is equal in absolute value to the like-named function of x; but each function of 90~ ~ x or 270~ ~ x is equal in absolute value to the co-named function of x.* For example, sin (180~ + x) and sin x by the above rule are equal in absolute value. But it must also be remembered that they are opposite in sign. For if, for instance, x is acute, 180~+ x is an angle in the third quadrant and therefore sin (180~ + x) is negative. But x meantime would be an angle in the first quadrant, hence sin x would be positive. Hence, in general, sin (180~ + x) =- sin x. Let the pupil show in like manner that, by the above rule, sin (360~ - x) =- sin x; also that sin (270~-x) = - cos x. In applying the above general rule to any particular example it will be found that the algebraic sign of the result is the same as the sign of the original function. Thus, sin 330~ = sin (360 ~- 30~) = - sin 30~, the short way of determining the sign of sin 30~ being to note that sin 330~ is-negative since 330~ is in the fourth quadrant and that sin 30~ must have the same sign as sin 330~. If geometrical proofs for the above reduction formulas are, desired, such proofs may be obtained by following the methods of Art. 62. But in such proofs, when constructing an angle like 180~ + x, or 270~ + x on the diagram, it R B is an advantage to construct the 180~, or T 270~ first, beginning with the initial line, and then to annex the angle x to the 180~, or \' 270~, after it has been constructed. A' ' I A Thus, to prove that tan (270~ + x)= -cot x when x is an angle in the second quadrant (i.e. an obtuse angle) we first take (Fig. 50) the positive angle AOB' (270~) and annex B' to it ZB'OP' (=x or LAOP). Then FIG. 50. * At 'this point it is often advantageous to have the class study the solution of Case I of oblique-angled triangles (Arts. 74, 79). This shows the pupil an important application of the preceding principle and introduces variety into the course of study. 90 TRIGONOMETRY (270~+x) =Z AOT (as indicated by the long bent arrow), and tan (270 + x) = AT. Also cot x (or cot AOP) = BR. But ZB' T = Z AOR (construction) Subtracting 90~ from each of these angles we have Z AOT ZBOP... AAOT =/BOP. (leg and acute Z =).'. AT= BR, in absolute magnitude. (horn. sides of = A).. tan (270~ + x) and cot x are equal in absolute magnitude. But AT and BR are opposite in sign. B.-. tan (270~ + x) = - cot x. N, P\ Similarly, to prove sin 270 - x — - cos x when x is an angle in the second quadrant (Fig. 51) we take Z AOB' (270~) and from A' M' — o - jA it deduct Z B'OP' (= Z AOP or x). Hence, sin (270~ - x) = MP', while cos x = NP. Since A OIMP' = A ONP, MP' and NP are equal in absolute magnitude. They are also B opposite in sign. FIG. 51..~. sin (270~ - x) = - cos x. EXERCISE 27 Find the numerical value of: 1. sin (- 225~). 4. cot (- 210~). 2. tan (- 300~). 5. tan ( — 600~). 3. cos (- 1200). 6. sin (- 900~). 7. sec (-240~). 8. tan (- 150~). 9. sin (- 135~). Reduce the' functions of the following negative angles to the functions of positive angles not greater than 45~: o1. — 119. 1.3 - 15. 16. -900~. 11. -81~. 14. -253~. 17. - 216 43'. 12. - 195~. 15 -1000~. 18. - 307.24~. 19. Show that sin 420~ cos 390~ = 1 - cos (- 300~) sin (- 330~). 20. That 3 tan (- 60~) cot (- 210~) + 9 sin (- 240~) cos (- 150) = 4. By the general rule stated in Art. 64 reduce each of the following to a function of x: 21. cos (180~ + x). 23. cos (270~ - ). 25. sec (180~- x). 22. sin (270~ + x). 24. tan (180~ + x). 26. csc (270~ + x). GONIOMETRY 91 Simplify the following expressions: 27. 5 sin (90~ - x) + 8 cos (180~ -x). 28. a sin (270~ - x) - b cos (270~ - x) + c tan (180~ - x). 29. m cos (180~ + A) +p cot (180~ - A) + q tan (270~ + A). 30. sin (270~ + x) cos (270~ - x) sin (180~ - x). 31. sin (x - 90~) + cot (x - 90~) + tan (x - 180~). 65. General Solutions of Trigonometric Equations. If there be no limit to the size of an angle, an indefinite number of angles will satisfy every trigonometric equation (see Art. 38). Ex. 1. Solve sin x=. There are two angles less than 360~ whose sine is ~, viz.: 30~ and 150~. If 360~, or any multiple of 360~, be added to, or subtracted from, each of these angles, the sine is unchanged. Hence, in the above example, x = 30~ ~ n (360~), 150~ ~ n (360~), where n = 0 or any positive integer. Ex. 2. Solve tan x= ~V3. f 60~ ~ n (360~), 120~ ~ n(360), Ans [240~ ~ n(360~), 300~ ~ n (360~). Ex. 3. Solve sin2x=cos2 x. - COS2 X = COS2 X. 2cos2 = 1. cos x = ~ i \/2. X= 45~ n (360~), 315~ ~ n (360~), 135~ ~ n (360~), '25~ ~ n(360~). Or more briefly, x= n (180~) ~ 45~. Ans. The pupil should observe that the values of x in a trigonometric equation differ in an important respect from the values of x in an algebraic equation. Thus, in an algebraic equation the values of x are the roots of the equation and the number of values which x has equals the degree of the given equation. Whereas, for instance in Ex. 3 above, the roots are the values of cos x, while the values of x are inferred from the values of cos x and may be unlimited in number no matter what the degree of the original trigonometric equation. 92 TRIGONOMETRY EXERCISE 28 Solve the following trigonometrical equations, for values of x or 0. 1. sin x =. 2. cos2 = 3 3. tan2x - 1. 4. tan x = cotx. 5. sinx + csc x = 5. 6. tan2x- sec x = 1. 7. 2 cos2 -3 sin x = 0. 8. tan x +cotx =2. 9. cot x + csc2x = 3. 10.2 V/3 cot 0 - csc2 0 1. 11. tan 0 + sec2 0 = 3. 12. cos2 0 + cot2 0 = 3 sin2 0. 13. 1 cot 0- cos 0 + sin 0 =. 14. sec2 0 csc2 + 2 csc20 = 8. 15. 2/3 tan 0 = 3 sec2 - 6. 16. 4 sec2 - 7 tan2 0=3. 17. cot 0 + 2 tan 0 = see 0. 18. sin 0 + -/3 cos 0 = 2. 19. A ship starting from a certain point sailed at the average rate of 9.25 mi. per hour on a course 22~ 15' [22.25~] north of east. At the end of 7 hr. 45 min., how far east of her starting point would she be? How far north? 20. If a railroad embankment is 11 ft. high, 76 ft. wide at the base, and 49 ft. wide at the top, and its two sides have the same slope, find the angle at which each side slopes. 21. In an oblique triangle ABC, A = 127~ 36' [127.6~], AB = 472 ft., AC= 374 ft. By dividing the triangle into right triangles and solving, find BC. 22. P is a spring of water, Q is a house, and R is a barn. If QR = 217 ft., Z PQR = 63~ 40' [63.67~], Z PRQ = 58~ 15' [58.25~], find the distance of the spring from the house and also from the barn, by solving right triangles only. CHAPTER V GONIOMETRY (Continued) 66. Formulas for sin (x + y) and cos (x + ). In Fig. 52 let AOQ be an angle x, and QOP an angle y, the sum of x and y being less than a right angle. P Let OP = 1. Draw PMiI OA, 8 PQ~ OQ, QRB PM.i / Then Z RPQ = Lx (sides I.), PQ = sin y, OQ-= cos y. sin (x + y) P= Ml= QN+- PR. FIG. 52. In rt. A OQN, QN= sin x OQ (Art. 41)= sin x cos y. In rt. A.RPQ, PR = cos x PQ = cos x sin y. Hence, sin (x + y) = sin x cos y + cos o sin y. Also on Fig. 52, cos (x + y)= O = ON-RQ. In rt. A OQN, ON= cos x OQ== cos x cos y. In rt. A RPQ, RQ= sinxPQ= sin x sin y. Hence, cos (x + y) = cos x cos y - sin x sin y. If x and y be acute angles whose sum is an obtuse agle, the above proofs will hold good without any change except that it,P ~^ ~ ~is necessary to notice that in the statement 7 773N\A cos (x + y) = 031= ON- R, OM is a negLi \y ative line and is obtained by subtracting M O N the positive line R Q from the smaller FIG. 53. positive line ON. See Fig. 53. If either x or y is obtuse, the above formulas may be proved as follows: 93 94 TRIGONOMETRY Taking x and y as, still acute, sin (90~ + x + y) = cos ( + ) (Art. 62) = cos x cos y - sin x sin y. But cos x= sin (90~ + x),- sin x = cos (90~ + x). (Art. 62).. sin (90~ + x + y) = sin (90~ + x) cos y + cos (90~ + x) sin-y. Replacing 90~ + x by x', sin (x' + y) = sin x' cos y + cos x' sin y, where x' is an obtuse angle. In like manner the formula can be extended to the case where y is an obtuse angle. The formula for cos (x + y) may also be extended in like manner. By successive additions of 90~ to x and y, these angles may thus be made any angles however large. In like manner the formulas may be shown to be true when x and y are diminished by any integral multiple of 90~. Hence, the above formulas are true when x and y are any angles. Ex. Taking the functions of 30~, 45~, 60~ as known, find sin 75~. sin 75~ = sin (45~ + 30~) = sin 45~ cos 30~ + cos 45~ sill 30~ -= 1Y/.- 1/v + /2.2.2 2 = V2 (V3 + 1), Ans. 67. Formulas for sin (x - y) and cos (x -y). In Fig. 54 let AOQ be a positive acute angle x, and POQ a smaller angle y, subtracted from x. Then Z A OP = x- y. Let OP = 1; draw PMJl OA, PQ~ OQ, QlaI OA, PRI QN. o/ NJ MA Th-en z RQP= x. (sides ~) FIG. 54. Also PQ = siny, OQ=cosy. sin (x-y) = PMJ= QXN- RQ. In rt. A OQT, QN-V= sin x OQ = sin x cos y. GONIOMETRY 95 In rt. A BQP, RQ = cos x PQ = cos x sin y. Hence, sin (x - y) = sin x cos y- cos x sin y. Also on Fig. 54, cos (x - y) = OM= ON+ RP. In rt. A OQNV, ON= cos x OQ = cos x cos y. In rt. A RQP, ]P = sin x PQ = sin x sin y. Hence, cos (x- y) = cos x cos y + sin x sin y. By the same method as that used in Art. 66 these formulas can be proved true when x and y are any angles. Ex. Obtain the numerical value of cos 15~. cos 15~= cos (450- 300), = cos 45~ cos 30~ + sin 450 sin 300 = 5- ~ /-' + ~V\. ~ -V V/2 * 2 V2.* =-16 + 4V2, Ans. 68. Formulas for tan (x +y) and tan (x- y). By Art. 66, tan (x + ) sin (x + y) sin x cos y + cos x sin y tan (x + y) = cos (x + y) cos x cos y - sin x sin y Divide both numerator and denominator of the last fraction by cos x cos y. sin x cos y + cos x sin y -m, cos x cos y cos x cos y Then, tan (x + y)= cos x coy coscos y cosx cos y sin x sin y cos x cos y cosx cos y tan x + tan y or, tan (X + )=-) = tan ( + )=1- tan x tan y Similarly, let the pupil show that tan x - tan y tan {x - y) = tan (-y) =1 + tan x tan y and cot(x+y)= cot x cot y T 1 and cot (x ~ o)= X cot y ~cot 96 TRIGONOMETRY Ex. Find the numerical value of tan 105~. tan 105~ = tan (60~ + 45~) tan 60~ + tan 45~ 1 - tan 60~ tan 45~ -/3 1+ /3_ 2- 3, Result. 1 -3.1 1- /3 EXERCISE 29 1. If sin x = cos x-=, sin y -, cos y = - find the value of sin (x + y). 2. Also of sin (x - y), cos (x + y), and cos (x-y). 3. Find sin (x + 45~), cos (30~ - x), and sin (x - 60~) in terms of sin x and cos x. 4. If tan x = - and tan y =2, find the value of tan (x + y). 5. If cot x = -2, and cot y = i, find the value of cot (x -y). Find the numerical value of: 6. cos 75~. 8. sin 105~. 10. sin 15~. 7. tan 75~. 9. cot 105~. 11. cos 105~. 12. Putting 90~ = 60~ + 30~, find sin 90~; also cos 90~. 13. State in general language the formulas proved thus far in this chapter (thus for sin (x + y) = sin x cos y + cos x sin y, say " the sine of the sum of two angles equals sine of the 1st angle times cosine of the 2d plus cosine of 1st times sine of 2d "). 14. Find tan (45~ + y), and also tan (45~ - y), in terms of tan y. 15. Find cot (60~ + y), and also cot (30~ + y), in terms of cot y. 16. Show that sin (60~ + 45~) + cos (60~ + 45~) = cos 45~. Prove the following identities: 17. cot (45 + A) = cot A-1. 1 + cot A 18. cot (450 -A)= coA+ 1 cot A - 1 19. sin (60~ + A) - sin (60~ - A) = sin A. 20. cos x- sin = -/2 cos (x + 45~). 21. cos x + sin x = -/2 cos (x- 45~). 22. Find the smallest value of x which will satisfy the equation tan (x + 45~) + cot (x - 45~) = 0. GONIOMETRY 97 69. Functions of the Double Angle. In the formula sin (x + y) = sin x cos y + cos x sin y, let y have the value x; then, sin (x + x) = sin x cos x + cos x sin x or, sin x =2sin x cos x. Similarly from the formulas for cos (x + y), tan (x + y), and cot (x + y), let the pupil obtain cos 2 x = cos2 x- sin2 x. 2 tan x tan 2 x = 1 - tan2 x cot2 - 1 cot 2 x = 2 cot x Substituting - sin2 x for cos2 x in the formula for cos 2 x, cos 2 x = 1-2 sin2 x. Substituting 1 - cos2 x for sin2 x in the same formula, cos 2 x = 2 cos2 -1. Ex. Find cos 120~ from the functions of 60~. cos 120~ = cos 2 x 60~ = cos2 60~ - sin2 60~ = (1)2- (1/3)2 1 3 =- 1 Ans. — 4 4 2V EXERCISE 30 1. Given sin 30~ =- and cos 30~=1 -/3, find sin 60~. Also cos 60~. 2. Given tan 30~ = 1 /3, find tan 60~. 3. By the formulas of Art. 69, find the value of sin 120~ and tan 120~. Prove the following identities: 4. sin2= 2tanA 6 sin 2 x cos2x secx. 1 + tan2 A, sin x cos x 5 1-tan2 A 1 + sin 2 0 (tan 0 + l)2 5. cos 1 sin.(tan 1) 1 + tan2 A 1 — sin 2 O (tan 0 - 1)2' 98 TRIGONOMETRY 8. State the formulas for sin 2 x and cos 2 x in general language. 9. Find sin 3 x in terms of sin x. 10. Find cos 3 x in terms of cos x. 11. Find tan 3 x in terms of tan x. 12. Prove sin 4 0 =4 sin 0 cos 0 - 8 sin8 0 cos 0. 13. Given tan 0 = 5, find tan 2 0. 14. Given cos 0 =, find cot 2 0. In a right triangle, C being the right angle, prove: 15. tan B - cot A. 16. tan 2 2 ab 17. sin (A - B) + cos 2 A= 0. b2 a_ ' __ —c_ _ 1 - cos 4 x 18. Show that sin2 x = - cos 2 and sin2 2x,= 1-c 2 2 19. Show that cos x= cos2x and cos 22x= cos 4 2 2 20. Using the results of Exs. 18 and 19, transform sin4 x into 1 cos 4 x - I cos 2 +. 2 ' 8 21. Also transform cos4 x into an expression in terms of cos 2 x and cos 4 x. 22. Also show that cos6 x may be changed to the form 1- (5 + 8 cos 2x -2 sin2 2 x cos 2 x +3 cos 4 x). 70. Functions of the Half Angle. From Art. 69, cos 2 A= 1- 2 sin2 A. Hence, 2 sin A = 1 -cos 2 A. Let A=- x; then 2A=x. Hence, 2 sin2 1 x= 1- cos x... sin x= ~\- cos 2 Similarly, from cos 2 A= 2 cos2 A - 1, we obtain, cos I- X+= ~ C V. ~2 2 GONIOMETRY 99 Also sin I x l-cos X tan — x =2 — i 1~\- COSX.,. tan -2 = + 2 1 +cos x This formula may be reduced to another convenient form, thus: tan 2x= (1-cos x)2 - (1-cosx)2 1-cosX (1 + cos x) (1 - cos x) 1 - cos' x sin x 1 - cos x.'. tan x =1 ---cos sin x Similarly, I1 +cos x cot = -- sin x Ex. Find tan 221~ from the functions of 45~. ta n2 1- 1 cos 45~ 1 —2 2- V2-\ A/ tan22~ sin 45~ = 0- 2 -2 EXERCISE 31 1. State the formulas for sin 2 A, cos 1'A, and tan 1 A in general language. 2. Given cos 30~= -/V3, find sin 15~, tan 15~, cos 15~. 3. Given sin 45~ = -2, find cot 222~, cos 220~, sin 22-~. 4. Given cos 90~=0, find the functions of 45~. 5. Given sin A= 2 and A acute, find cos 2 A, cot I A, tan - A. 6. Given2 ) 2 2 0 0 n0 6. Given cos 0 = a, find cos -, cot, tan 25 2 2 Prove the following identities: 7. tan~A= sinA 2 1 + cos A 9. sec2 2 see0 2 se 0 + 8. cot 'A= sinA 10. csc2 0 _ 2 sece 2 1- cos A 2 see 0 -11. sin -A + cos A = V/1 + sin A. 12. Express cos A, sin A, and cot A, in terms of cos 2 A. 13. Find the value of tan - + sec x if x is in the second quadrant cot I X + cos x anc sin x= 5. 5 ' 100 TRIGONOMETRY 14. If x is in the fourth quadrant and csc x = - _, find the numerical value of sin 1 x + sec x value of 2 cot x + cos x 15. In a right triangle show that tan A -= c —b 16. By use of this formula solve the right triangle in which c = 122 and a==120 (that is, the Ex. of Art. 46). 17. If the diagonal of a rectangle is 171 in. and one side of the rectangle is 13 ft. 7 in., find the angle between the diagonal and side. 18. Make up and solve a similar example for yourself. 71. Sum or Difference of Two Sines or of Two Cosines (Logarithmic Formulas). Adding and subtracting the formulas of Art. 66, and also those of Art 67, sin (x + y) + sin (x- y) = 2 sir x cos... (a) sin (x + y) - sin (x - y) = 2 cos x sin y.. (b) cos (x + y) + cos (x - y) = 2 cos x cosy... (c) cos (x + y)-cos (x-y)=-2 sin x sin y... () If we let x+y=A, and x-y=B, then = (A + B), and y = -(A - B). Hence, by substitution in (a), (b), (c), (d), sinA+sin B=2 sin (A +B) cos -(A-B)... (1) sinA-sinB=2cos (A+B)sin- (A-B).. (2) cos A +co =os =2 (A+ B) cos -(A-B)... (3) cosA-cosB=-2sin (A+B) sin (A-B).. (4) These formulas enable us to convert the sum or difference of two sines, and also of two cosines, into a product of two functions, and hence open the way in certain examples for us to save labor by the use of logarithms. GONIOMETRY 101 Ex. Convert sin 50~+ sin 30~ into a product. By formula (1), sin 50~ + sin 30 = 2 sill 2(50 + 30~) cos (50 - 30~) = 2 sin 40~ cos 10~. EXERCISE 32 Prove 1. sin 40~ + sin 10~ = 2 sin 25~ cos 15~. 2. sin 60~ + sin 30~ = V2 cos 15~. 3. cos 80~ - cos 20~ = - sin 50~. sin 33 - sin 3~ + sn 3~ 6. sin 5 x + sin x 4. =tan 18. 6. = tan 3 x. cos 33~ + cos 3~ cos 5 x + cos x cos 27~ - cos 3~ ot 15~ cos 80~ + cos 20~ sin 270 + sin 30 sin 80~ - sin 20~ 3' 8. sin A +sinB cot (A- B) cos A -- cos B 9. cos 4 x+ cos 2 x cot 3x. sin 2 x + sill 4 x 0 sin A-sin B cot A+B cos A - cos B 2 11. cos 20~ + cos 100~ + cos 140~ = 0. sin2 3x 12. sin x + sin 3 x + sin 5 = sin x 13. Given sin A= 1 and sin B, find sin (A + B), sin (A- B), cos (A + B), cos (A-B), sin 2 A, sin 2 B, cos 2 A, cos 2 B, when A and B are both in the first quadrant. 14. Find the numerical value of sin (60~+ 30~). Also of sin 60~ + sin 30~. Show geometrically why sin (60~ + 30~) does not equal sin 60~ + sin 30~. Reduce each of the following to a form adapted to logarithmic computation (that is, to products or quotients): sin 37~ + sin 22 sin 4 A - sin 2 A 15. 16. cos 38~ - cos 16~ cos 6 A 17. sin2 A - sin2 B. 18. Compute the value of the expression in Ex. 16 when A= 14~. Also of that of Ex. 17 when A = 38~ and B = 24~. 19. Make up for yourself an example similar to Ex. 17. 102 TRIGONOMETRY 72. Complex Trigonometrical Identities. Besides those already arrived at, many other complex relations between the trigonometrical functions may be proved. Usually these relations are proved to the best advantage by reducing the two expressions, which are compared, to some common form, and hence inferring their identity by Ax. 1 (see Art. 31). In most cases it is best to reduce given functions to sine and cosine. Ex. 1. Prove that 1cos 2 A tan A. sin 2 A 1-(1-2 sin2 A) sin A 2 sin A cos A cos A 2 sin2 A sin A 2 sin A cos A cos A sill A sin A cos A cos A Or if the teacher prefers, the proof may be put in the following form: 1 - cos 2 A _ -(1 -2 sin2 A) 2 sin2 A _ sin A sin 2 A 2 sin A cos A 2 sin A cos A cos A Ex. 2. Prove sin (A + B) sin (A - B) = sin2 A - sin2 B. (sin A cos B + cos A sin B)(sin A cos B - cos A sinB) = sin A - sin2 B. sin2 A cos2 B - cos2 A sin2 B = sin. A - sin2 B. sin2 A (1 - sin2 B) - (1 - sin2 A) sin2 B = sin2 A - sin2 B. sin2 A - sin2 A sin2 -sin sinA sin2 B = sin2 A - sin2 B. sin2 A ' sin2 B = sin2 A- sin2 B. 73. Functions of the Angles of a Triangle. If the sum of three angles is 180~, the functions of the angles have important relations. Ex. If A + B +C = 180~, prove that sin A + sin B + sin C =4 cos I A cos ` B cos ~ C. A + B = 180 - C and I (A + B) = 90~- C. GONIOMETRY 103 Hence sin -I (A + B) =sin (90~ - 0 ) = cos I C. 2 2 sin A + sin B + sin C = sin A+sin B + sin [180~ - (A + B)] = in A + sin B + sin (A + B) =2 sin I (A + B) cos 1 (A -B) +2 sin ~ (A + B) cos 1 (A+ B) (Arts. 69, 71) = 2sin 1 (A +B)[cos 1 (A —B) +cos 1 (A+ B)] = 4 cos - C cos - A cos 1 B. EXERCISE 33 Prove the following identities: 1. c eosO + sin 0 sin 2 0 +-1 cos 0- sin 0 cos 2 0 2. 2 cos (45~ + i- A) cos (45~ -- A) = cos A. 3. cos (A + B) cos (A - B) = cos2 B - sin2 A. 4. tan (45 + ) - tan (45~ - x) 2 tan 2 x. 5. (V/1 + sin x - V - sin x)2 = 4 sin2 1 x. 6. cos( + y) + cos (x-y) cos (x -y) -- os ( + y) cos x cos y sin x sin y. tan (450 + - A) + tan (45~0 - A) A tan (45~ + A) - tan (45~ — 1 A) 8. cos3A+sin 3 A 2 ot2A. sin A cos A 9. cosA- sin Ase2A-tan 2 A. cos A + sin A sin 0 + sin 2 0 10. tan= sinO+sin2O 1 + cos 0 - cos 2 0 cot 0 -1 1 -- sin 2 0 cot 0 + 1 cos 2 0 1-tan2' 1 12. - 2 = cos X. 1 + tain2 x If A + B + C = 180~, prove that 13. cosA +cosB+cos C= 1 +4sin A sin B sin C. 14. tan A + tan B +.tan C= tan A tan B tan C. 15. cos (A + B + C) = - cos2 C. 104 TRIGONOMETRY EXERCISE 34. REVIEW 1. Given cos 0 = - and 0 is in the third quadrant, find csc 0, cot 0, sin I 0, tan (180~- 0), sin (- 0). 2. Given tan I x = 2 (and x acute), find sin x. 3. Given sin 2 x = -3 V3, find cot x. 4. Given cos ~ x = -, find sin 2 x and tan 2 x. 5. Given cot 30~= /3, find cos 15~, ese 15~, and tan 15~. 6. Given sin A = 3 and A acute, cos B = I and B acute, find (a) sin (A-B); (b) cos (A+B); (c) cos (A-B); (d) sin 2 B; (e) cos 2 B; (f) tan 2 B; (j) cot 2 A; (h) tan (A - B); (i) cot (A + B); (j) cos I B. 7. Given cot 0 = -2 and 0 is the second quadrant, find (a) sec 0; (b) tan (180~ - ); (c) cot (180~ +'0); (d) cos (- ). 8. Find sin, cos, tan, cot, of: (a) (x-; (b) (7-.0); (c) (-3-T); (d) (r + x); where r= 1800. Prove the following: 9. tan x = — cos 2x sin x + sin 2 x 9. tan x - 12. - - ttan. x. sil 2 1 + cos x + cos 2 x 10. tan A = - sA 13. osin (A + B) tan A + tan B sin A cos A cos B 2 sin A- sin 2A 1- cos A sin 21~ - sin 5~_tan 13. 11. - _- -- -- 14. tan 13'a 2 sin A + sin 2 A 1 + cos cos 21~ + cos 5~ 15. cos90 +cos50 +cos0 cot5. sin 9 0 + sin 5 0 + sin 0 16 COS2 tan Sin2 XCot 2 19. tan x +cotx + 1 2 sin 2 x 16. cos2 x tan2 x q sin2 x cot x~- = 1. 19. -- tanx + cot x-1 2-sin 2 x cos 75 + cos 15 3 20. cos 2 cot sin 75~ - sin 15~ cos 2 x -1 18. sin A + sin BB) 21 sin (x + /) cot x + cot y cos B - cos A 2 sin (x -y) cot y -cot x 2 22. cos = 2 - tan + + tan - 23. sin (x + y) sin (x - ) tan2x - tan2 y. cos2 x cos y 24. cos 5 x +cos 3 x = 2 cos 4 xcosx. s in 2 x 1 2 tan x + tan2 x - 1 sin 2 5. 2 tan tan x sin2x —1 2tanx —tan2x —1 GONIOMETRY 105 26. sin (45~ + x) + sin (45 - x) = V2 cos X. 1 +cot2( +x) 27. - csc 2 x. 1 -cot2 (-+ x) 1 - cot2 s _ 28. 4 = — sin2x. 1 + cot2 7- _ \4 291 cosx + cos 2x sin x - sin 2 x cos x sin x 30. cos 12 x + cos 6x + cos 4 x + cos 2 x = 4 cos 5 x cos 4 x cos 3 x. 31. tan (45~+-) 1 + sinx 2 1 - sill x 32. (sin x cos y -cos x sin y)2 + (cos x cos y + sin x sin y)2 = 1. 33. cos2 -x (tan x - 1)2 =1 - sin x. 34. Find the value of s 0 cos 0 when cot 0 = -,and 0 is in quadsec 0 + sin 0 2 rant II. 35. Find the value of tan0 + cos 0 when sin — 4 and is in the cot 0 + sec 0 5 3d quadrant. 36; Simplify cos 300~ - cot + 60~) +cot 150~ -tan (-). 37. Simplify sin 660~ + tan (37r - 60~) + cot 330~+ cos (-30~). 38. Simplify: (a -- b) sin j-3 a 3 (a-b) sin 7r- (a + b) tan 225~ + (a2 + b2) cot 3- a cos 2 2 \2 39. If tan 2 0 = 2, find tan 0 and sin 0, 0 being in the 3d quadrant. 40. Prove sin (A B) t an A tan B cot B + cot A sin (A - B) tan A - tan B cot B - cot A 41. If A is an angle in the second quadrant and sin A= 3, find the value of sin 2 A + cos 2 A. If A +- B+ C-= 180~, prove: 42. sin A + sin B - sin C= 4 sin sn B cos- C. 43. cot 1A + cot B +cot - C = cot 1 A cot - B cot - C. 44. sin 2 A + sin 2 B + sin 2 C = 4 sin A sin B sin C. 45. cos2A+cos2 B cos2 C= - (4cos A cos B cos C+1). 46. tan A - cot B = sec A csc B cse C. 106 TRIGONOMETRY In a right triangle, C being the right angle, prove 47. sin2 1B _ c 49. tall A= 2 2c 2 a+c I. 1 2 a + OS21 b + c 48. (cosA+ sinl ) 50. cos21A=b. 2 2 j 2 2 c 1 * -- cos 2x 2 1 + os 2 x Using sin x cos x sin 2, sin2 =x -- - cos x I= 2 2, 2 2 2 transform: 51. sin2 x cos2 x into (1 - cos 4 x). 52. sin4 x Cos2 x into -1-(1 - cos 4 x) -8 sin 2 x cos 2 x. 53. sin4 x cos4x into an expression in terms of the cosines of even multiples of x. 54. sin8 x into an expression of the same general kind as in Ex. 53. 55. What nation first used the formula for sin I A? 56. What man discovered the formula for sin 2 A? 57. Who first published the formulas for sin (A - B) and cos (A - B), and at what date? CHAPTER VI OBLIQUE TRIANGLES TRIGONOMETRIC PROPERTIES OF OBLIQUE TRIANGLES 74. Law of Sines in a triangle. In any triangle the sides are to each other as the sines of the angles opposite. C C b a 6 / A B A A/, ID ^ A^^ ^ --- —-— ~DB C c FIG. 55. FIG. 56. In Fig. 55 the angles A and B are both acute. In Fig. 56 the angle A is acute, and angle ABC obtuse. Let CD, denoted by p, be the altitude in each triangle. In Fig. 55, in the rt. A ACD, p = b sin A; (Art. 41) in the rt. A CBD, p = a sin B; (Art. 41).. sin A = a sin B. (Ax. 1) In Fig. 56, in the rt. A CD, p = b sin A; in the rt. A BCD, p = a sin (180~ - ABC) = a sin Z ABC. (Art. 64) Hence in A ABC in both figures, b sin A = a sin B, or a: b = sin A: sin B. In like manner, b: c..= sin B: sill C, and a: c = sin A: sin C. Or, collecting results, a b c sinA sinB sin C 107 108 TRIGONOMETRY 75. Law of Tangents in a triangle. In any triangle the szum of any two sides is to their difference as the tangent of half the sum of the angles opposite the given sides is to the tangent of haf the difference of these angles. In a triangle ABC (Figs. 55 and 56), a: b = sin A: sin B. (Art. 74) By composition and division, a +b_ sinA+sinB a - b sin A - sinB 2 sin - (A + B) cos - (A - B) 2 cos 1 (A + B) sin I (A -B) (Art. 71) Or, a + b tanl (A + B) Or,2 a-b tan (A-B)' In like manner, b + c _tan ( + C) b-c tan2 (B- C)' and c + a tan I(C+A) c-a tan (C- A)' It is also helpful to have a geometric proof of the Law of Tangents. This may be obtained as follows: In a given triangle ABC (CB >AC), rAd produce ACto D, making CD= CB or a. f/,/y\ On CB mark off CE = AC or b. /,' \ Draw the straight line DB. a,/ \Then AD= CD + CA = a + b. J/ \ Also EB CB -CE= a -b. ZDCB, being an exterior angle of ~// a. /AACE,=x-+x=2x. / b \jF Also ZDCB, being an exterior angle b/ E -— \ of / ACB, =A + B (of A ACB). i/ ^-2-X \.'. 2 z= A +B (Ax. 1), or x = L (A +B)..A4-/ — -- - B Also, A B ~FIG. 57. ~ ZFFAB= A -x = A -1 (A + B) Ao (t -B). Also A ADF and EFB are similar (two As equal). OBLIQUE TRIANGLES' 109.-. AFD= EFB..-. AFI-DB. In AAFD and EFB, DF: FB= a + b: a-b. In A AFD and AFB, tan x: tan Z FAB = DF: B = DF: FB. AF AF By Ax. 1, a + b: a- b = tan x: tan L FAB = tan (A + B): tan 1 (A- ). 76. Law of Cosines in a triangle. In the triangle ABC, Fig. 55, by geometry, a2 = b2 + c2 - 2 c x AD. But in the rt. A CD, AD = b cos A... a2 = b2 + c2- 2 bc cos A. If A is an obtuse angle, Fig. 58, by geometry, a2 b2 + c2 + 2 c x AD. But in the rt./ A CD, AD = b cos / CAD = bcos (180~ - A) = - b cos A. a = b2 + c - 2 be cosA. i Hence 2 or in either case, be cos A = b2 + c2 - a, a b2 + c2- a2 b cos A= 2\ - e manner it may be proved A FIG. 58. 2 + c22 _ b 2 +- b2 - c2 cosB== 2 ac Cs C 2ab In lik that 77. Formulas derived from the Cosine Formula. The formula for cos A in Art. 76 has a numerator which is primarily a sum and difference, hence logarithms cannot be used in computing numerical values from it. In order to put this formula in such a shape that its value can be computed by the aid of logarithms, it is necessary to transform the numerator of the fraction into a product. This is done 110 TRIGONOMETRY by the use of the formula for the cosine, or of that for the sine of a half angle (Art. 70). Thus: 2 co2 A=l+cosA= 1b2 + c _a2 2 be _ 2be b2 + C2 c a2 (b + c)2 - a2 2 bc 2 bc F (b + c+- a)(b +c -a) 2 be Let 2 s = a + b c; then, subtracting 2 a from each member, 2s-2a=b+-c —a. Hence, 2 cos2 A 2 s (2 s -2 a 2 be s(s - a) or cos 1 A= (- c. C 2 be In like manner,,1- -B ( - b) C (S - C) cos12 B=\/-(-), cosG=\/ --- —* c)ac~ a2 Also from Art. 70, 2 sin2 A= 1 - cos A = 1 -b2 +2 a2 ~~~2 2 bc 2 be - b2 - + a2 a-2- b2+ 2bc -c2 2 be 2 b a2- _(b - c)2 (a + b - c)( - b + c) 2 be 2 bc _ (2 s - 2 c)(2 s - 2 b) _ 4(s - b)(s - c) 2 bc 2 be Hence, sin 1 A = j(s - b) (s - c) 2 - be In like manner, sin ~B=/(s-a) (s-C) ( (s - -a) b) Ji 2 ac sbJ V a Dividing the formula for sin 1 A by that for cos - A, tan 2- A - \(s - b)(s - c) tan-|A=^ (^ * s(s- a) Similarly, tan B ( ( and tan ( C a) (s- a) (s - ) 2 B=-\jrs - b2 (S - C) s - F c/s( s-c) OBLIQUE TRIANGLES 111 EXERCISE 35 1. Prove that the diameter of a circle circumscribed about a triangle is equal to any side of the triangle divided by the sine of the angle opposite that side. 2. By means of the property of sines, prove that the bisector of an angle of a triangle divides the opposite side into segments which are proportional to the sides forming the given angle. 3. In any triangle ABC, prove that a = b cos C + c cos B. State this property in words. Write the two similar formulas for b and c. What does the above formula become when C= 90~? 4. Prove that the radius of an inscribed circle of a triangle is equal to c sin A sin 2 B where c is one side of the triangle and A and B cos I C are the angles adjacent to c, and C is the angle. opposite c. 5. Prove sin A= -- s (s -a)(s -b)(s -c) if s= a be 2 6. Prove cos A = s(s-a)- (s-b)(s- c) be 7. Find the form to which the formula 2ab =tan (A+B) ca-b tan 1 (A-B) reduces, and describe the nature of the triangle, when (I) C= 90~, (II) A -B=90~, and B =C. 8. What does a2= b2 +c2-2 be cos A become when (I) A=90~, (II) A = 0~, (III) A = 180~? What does the triangle become in each of these cases? a sin A 9. What does a sin= become when A is a right angle? When b sin B B is a right angle? SOLUTION OF OBLIQUE TRIANGLES 78. Cases in the Solution of Oblique Triangles. Four cases occur in the solution of oblique triangles according as the parts given are I. One side and two angles. II. Two sides and the included angle. III. Three sides. IV. Two sides and an angle opposite one of them. 112 TRIGONOMETRY CASE I. ONE SIDE AND Two ANGLES GIVEN 79. To solve Case I use the law of sines (Art. 74), thus: Subtract the sum of the two given angles from 180~; this will give the third angle. The unknown sides may then be found by the following proportion: unknown side: known side = sine of angle opposite the unknown side: sine of angle opposite the known side. In solving oblique triangles by the use of logarithms it is of special importance to make an outline or skeleton of the work before looking up any logarithms, and then to do all the work connected with the use of the tables together. Ex. 1. Given A = 67~ 21', B= 57~ 48', b = 367. Solve the oblique triangle ABC. SOLUTION B 6721 A 367 0 FIG. 59. C = 180~ - (67~ 21' + 57~ 48') = 54~ 51'. Then by the law of sines (Art. 74), (Check) a sin 67~ 21' c sin 54~ 51' a sin 67~ 21' 367 sin 57~ 48' 367 sin 57~ 48' c sin 54~ 21' Before looking up any logarithms in the tables the pupil should outline the work as follows: 367 log.... 367 log.... c. log... 67~ 21' log sin... 54~ 51'log sin.... 67 21' log sin... 57~ 48' colog sin.... 57~ 48' colog sin.... 54~ 51' colog sin... a=..... log.... c=..... log.... a=..... log.. OBLIQUE TRIANGLES 113 The pupil can then look up all the logarithms at once and fill in the above tabulated form. (Any logarithm occurring more than once on being taken from the tables should be entered uniformly wherever it belongs.) Proceeding thus, he should obtain 367 log 2.56467 67~ 21' log sin 9.96541 - 10 57~ 48' colog sin 0.07253 a = 400.227 log 2.60231 367 log 2.56467 54~ 51' log sin 9.91257 - 10 57~ 48' colog sin 0.07253 c = 354.625 log 2.54947 (Check) c log 2.54947 67~ 21' log sin 9.96541 -10 54~ 51' colog sin 0.08743 a log 2.60231 Ex. 2. Solve the triangle ABC, given B = 83.11~, and b = 7641. 1 83.11V 18.29 A 7641 C FIG. 60. A = 18.29~0 C = 180~ - (18.29~ + 83.11~) = 78.6~. Then by the law of sines (Art. 74), a sin 18.29~ 7641 sin 83.11~ 7641 log 3.8832 18.29~ log sin 9.4967 -10 83~ 11t colog sin 0.0032 a = 2416.11 log 3.3831 c sin 78.6~ 7641 sin 83.11~ 7641 log 3.8832 78.6~ log sin 9.9913 - 10 83.11~ colog sin 0.0032 c = 7546 log 3.8777 (Check) a sin 18.29~ c sin 78.6~ c log 3.8777 18.29~ log sin 9.4967 - 10 78.6~ colog sin 0.0087 a. log 3.3831 114 TRIGONOMETRY The accuracy of the work in Exs. 1 and 2 might also have been checked by use of the formula a2 = b2 + c2 - 2 be cos A, or of cos ~-A = (-a). bc In general in solving oblique triangles the accuracy of the work in any one case can be checked by applying to the results obtained one of the rules or formulas of the other cases. EXERCISE 36 Find the remaining parts of the triangle, given: i. a = 12.632, A = 65~ 35', B = 73~ 18'. 2. a = 300, B = 10~ 18', C= 35~ 22'. 3. b = 1000, B = 49~ 18', C= 72~ 50'. 4. c = 1640.22, C= 18~ 25', B = 52~ 16'. 5. A= 66~ 18' 36", B = 43~ 43' 48", c =.87654. 6. = 100~ 18' 42", B = 50 40' 16", c = 114.682. 7. C = 22~ 18' 24", B = 58~ 12' 24", = 1.26984. 8. A= 68~ 15' 20", B = 43~ 18' 36", a = 1.8263. 9. B = 57~ 23' 12", A = 54~ 21' 18", c =.20814. 10. Given a =5.267, A =30, B = 45~, solve without using the tables. 11. Given c= 1000, A =60, B=45~, find a and b without using tables. 12. In a parallelogram given a diagonal d, and the angles m and n which this diagonal makes with the sides, find the sides. Find the sides when d = 14.632, and m = 38~ 18', and n = 12~ 32'. Using four-place tables, find the unknown parts, having given: 13. a = 14.26, A = 52.16~, B = 71.11~. 14. c = 200, C = 18.16~, B = 80.52~. 15. b =.7125, A =116.18~, C = 38.25~. 16. a = 63.28, B 63.28~, C= 36.82~. 17. b =4000, B = 17.28, C = 82.26~. 18. c= 8, A =79.26~, B 99.99~. 19. a = 19.28, B = 42.8~, C = 19.53~. OBLIQUE TRIANGLES 115 20. c =.2265, B = 71.28, A = 52.85. 21. b = 176.8, = 9.82~, = 68.22~. 22. a = 4812, B = 75.6~, C = 48.71. 23. b = 14.267, C = 110.6~, A = 41.63~. 24. c = 712.8, B = 44.18~, A = 79.22. Without the use of tables, solve, having given: 25. a = 100, B = 60~, A = 60~. 27. a = 500, A = 75~, B = 60~. 26. A = 120~, B = 30~, c = 200. 28. b = 200, A = 105~, c = 45~. Solve Exs. 29-31 by either set of tables. 29. A ship S can be seen from two points M and N on the shore. The distance MN is 700 ft., and the angles SMNV and SNM are 57~ 42' [57.7~] and 75~ 18' [75.3~] respectively. Find the distance of the ship from M. 30. A balloon is directly over a straight road, and between two points on the road from which it is observed. The distance between the two points is 2652 yd., and the angles of elevation of the balloon as seen from the two points are 58~ 50' [58.83~] and 47~ 24' [47.4~] respectively. Find the distance of the balloon from each of the given points, and also the height of the balloon from the ground. 31. Which examples in Exercise 41 can be worked by Case I? Work such of these examples as the teacher may direct. 32. Make up some practical problem which can be solved by the method of Case I and solve it. CASE II. Two SIDES AND THE INCLUDED ANGLE GIVEN 80. To solve Case II we have the following method by the use of the law of tangents (Art. 75): Subtract the given angle from 180~; divide the remainder by 2. The result will be half the sum of the unknown angles. One half of their difference may then be found by the following proportion: tan 2 the difference of the unknown angles: tan 1 their sum = difference of the two given sides: their sum. 116 TRIGONOMETRY Then 1 sum of unknown As + - their difference = greater unknown Z. sum of unknown As - their difference = smaller unknown Z. The third side is found by Case I. Ex. 1. Given a=4527, b=3465, C=66~ 6' 28", solve the triangle.* A / a + b = 7792. a - b — =1062. AZxA/ \ A+ B= 180~ -660 6' 28 = 113~ 53' 32". / 6'28\B (A + B) = 56~ 56' 46". 4527 FIG. 61. By the law of tangents (Art. 75), tan (A - B): tan - (A + B) = a - b: a + b, that is, tan 2 (A - B): tan 56~ 56' 46" = 1062: 7992. tan I ( 1062 tan 56~ 56' 46" tan2(-B)= 7992 1062 log 3.02612 56~ 56' 46" log tan 0.18659 7992 log 3.91266 - 10 colog 6.09734 - 10 2 (A -B) = 11~ 32' 28" log tan 9.31005 - 10 1 (A + B) =56~ 56' 46" - (A-B) = 11 32' 28" A = 68~ 29' 14" B = 45~ 24' 18" The side c may now be found by Case I. Thus we ave c sin 66~ 6' 28" Thus wse have 4 =1 3465 sin 45~ 24' 18" * If only the third side, c, is required, and the numbers representing the other sides, a and b, are small, the solution may often be readily effected by the formula of Art. 76 without the use of logs. Thus given a = 5, b = 6, C = 60~, find c. c =/a + b2 - 2 ab cos C' -\/25 + 36 - 60 x ~ = v31 = 5.5775. OBLIQUE TRIANGLES 117 3465 log 3.53970 66~ 6' 28" log sin 9.96109 - 10 45~ 24' 18" log sin 9.85254 - 10 colog sin 0.14746 c = 4448.9 log 3.64825 (What checks can you suggest for the work?) Ex. 2. Given c= 30.15, a= 18.159, B= 54.22~, solve the triangle. B c - a = 48.309. c - a =11.991. 54 C + A = 1800 - 54.22~ = 125.78~. / (C+ A)= 62.890. A. --- — c ---By Art. 75, FIG. 62. tan I (C- A): tan - (C + A) = - c - a: c + a; that is, tan 1 (C- A): tan 62.89~= 11.991: 48.309..tan -) 11.991 tan 62.890.'. tan ~ (C — A) =. 2 48.309 11.991 log 1.0789 62.89~ log tan 0.2908 48.309 log 1.6840 colog 8.3160 - 10 ~ (C — A) 25.870 log tan 9.6857 - 10 (C + A) = 62.89~ (C- A) = 25.87~ C= 88.76~ A = 37.02~ The side b may now be found by Case I. b sin 54.22~ 18.591 sin 37.02~ 18.159 log 1.2591 54.22~ log sin 9.9092 - 10 37.02~ log sin 9.7797 - 10 colog sin 0.2203 b = 24.467 log 1.3886 (What checks can you suggest for the work?) 118 TRIGONOMETRY EXERCISE 37 Using five-place tables, solve the following triangles, having given: 1. a - 27.7, b - 18.6, C= 68~. 2. b = 400, c = 250, A == 68~ 18'. 3. A = 30~ 12' 20", b ==.24135, c =.35627. 4. B= 63~ 35' 30", a =.062788, c =.077325. 5. A = 123~ 16' 30", b = 2.1625, c = 3.1536. 6. A =52 6', b = 420, c = 200. 7. C= 60~, b = 9, a = 7. Find c only. SUGGESTION. C= Va2 - b2 —2 ab cos C. 8. c = 26.369, b = 17.268, A = 32~ 18' 30". 9. B = 168~ 18' 39", c = 186.27, a = 132.91. Using four-place tables, solve the following triangles, having given: 10. a = 200, b = 260, C = 51.82~. 11. b = 1.763, c= — 1.112, A = 28.16~. 12. a =.3782, c =.412, = 112.18~. 13. b = 11.65, a = 8.26, = 12.12~. 14. a = 1720, c = 642, B = 78.63~. 15. b = 9, c = 6, A =60~. Find a only. SUGGESTION. a = Vb2 + c2 -2 bc cos A. 16. c = /7, b = V/11, A = 1688~. Find C, B, and a. 17. b = 79.23, a = 100.6, C = 68.25~. 18. a = 1.200, b = 2100, C = 43.18~. 19. a = 12, c = 15, B = 45~. Find b without the use of tables. Solve the following, using either set of tables: 20. Two trees M and P are on opposite sides of a pond. The distance of 3M from a point K is 159.6 ft., the distance of P from K is 216.8 ft., and the angle MKP is 75~ 18' [75.3~]. Find the distance between the trees. OBLIQUE TRIANGLES 119 21. The length of a lake subtends at a certain point an angle of 120~, and the distances of this point from the two extremities of the lake are 2 and 3 miles respectively. Find the length of the lake. 22. The point 0 is acted on by a force OA of 12 pounds and a force OB of 17 \ pounds, and the angle between the lines A of direction of the two forces is 120~ 43' [120.72~]. What will be the resultant force B and what angle will it make with each of -- the original forces? (Use the principle 0 of the parallelogram of forces.) 23. Two trains leave the same station at the same time on straight tracks intersecting at an angle of 21~ 12' [21.2~]. If the trains travel at the rate of 40 and 50 miles an hour respectively, how far apart will they be in 10 minutes? 24. The sides of a parallelogram are 172.43 and 101.31 and the angle included by them is 61~ 16' [61.27~]. Find the two diagonals. 25. In Exercise 41 which examples can be worked by the methods of Case II? Work such of these as the teacher may direct. 26. Make up some practical problem which can be solved by the method of Case II and solve it. CASE III. THREE SIDES GIVEN 81. The Solution of Case III is effected by the use of the formulas proved in Art. 77. In case it is desired to find only one of the angles of a given triangle it will be best to use that one of the formulas of Art. 77 which will give the required angle most accurately. The cosine formula may be stated in general language thus: The cosine of one half of any angle of a triangle is equal to the square root of one half the sum of the three sides multiplied by one-half the sum minus the side opposite, divided by the product of the other two sides. Thus cos2A- = ss-a)co Rs,- b,) os C- ss - c), co~s Ab S,cos~b B= S cos I = ab 2be, 2ac ab 120 TRIGONOMETRY Ex. 1. If in the triangle ABC, a =123, b=113, c= 103, find the angle A. s = 1(123 + 113 + 103) = 169.5. 2 = 169.5. coslA ---./169.5 x 46.5 s - a 169.5 -123 =46.5. 2 113 x103 B 169.5 log 2.22917 46.5 log 1.66745 113 colog 7.94692-10 103 colog 7.98716-10 2)19.83070-20 - A = 34~ 37' 22" log cos 9.91535- 10 A 113 C.'. A = 69~ 14' 44". 'FIG. 63. In case the half angle (T A) to be computed is small, it is best not to use the formula for cos I A. Why? In case the half angle to be computed is close to 90~, it is best not to use the formula for sin I A. Why? In case it is desired to find all three angles of a triangle, it is best to use the tangent formula of Art. 77. For it will be found that by that method it is necessary to employ the logarithms of but four different numbers, whereas by either of the other formulas it is necessary to use the logarithms of seven different numbers. It is a further advantage to transform the tangent formula thus: tan A = (s a)(s - b)(s-c) 1 (s a)(- b)(s-c) 2 s(s-a)2 s-a s s Let (s ac) = r. Then r r r tan21 A=, tan- B=, tan-1 C r s-a s-b s-c To test the accuracy of the work add the angles obtained. Their sum should differ very slightly from 180~. OBLIQUE TRIANGLES 121 Ex. 2. If in the triangle ABC, a =123, b= 113, c= 103, find the three angles of the triangle. s = 169.5. s - b = 56.5. 46.5 log 1.66745 s - a = 46.5. s - c = 66.5. 56.5 log 1.75205 46.5 x 56.5 x 66.5 169.5 66.5 log 1.82282 169.5 colog 7.77083-10 2)3.01315 r log 1.50658 r log 1.50658 56.5 colog 8.24795-10 r log 1.50658 46.5 colog 8.33255 -10 A = 34037'22"logtan9.83913 -10 - B=29~ 36' 25" log tan 9.75453-10 r log 1.50658 Hence A= 69~ 14'44" 66.5 colog 8.17718-10 B = 59~ 12' 50" ~ C=25~ 46' 15" log tan 9.68376- 10 C = 51~ 32' 30" 180~ 0' 4"1 (check) The fact that the suim of the angles of the triangle as computed differs from 180~ by four seconds is due to the fact that the logarithms used are only approximately correct in the last figure. When five-place tables are used, as in the above solution, the sum of the angles obtained should not differ from 180~ by more than six or seven seconds. Ex. 3. Find the three angles a=26.16, b = 29.15, c= 32.24. s= 43.775 s- b = 14.625 s - a = 17.615 s - c = 11.535 17.615 x 14.625 x 11.535.. r = 43.775 r log 0.9159 17.615 colog 8.7541-10 A=25.07~ log tan 9.6700-10 of the triangle in which 17.615 log 1.2459 14.625 log 1.1651 11.535 log 1.1620 43.775 colog 8.3587-10 2)1.8317 r log 0.9159 r log 0.9159 11.535 colog 8.9280-10 - C=35.54~ log tan 9.8539 - 10 A=50.140 B=58.780 C=71.08~ 180~ (check) r log 0.9159 14.625 colog 8.8349-10 - B=29.39~ log tan 9.7508-10 122 TRIGONOMETRY EXERCISE 38 By use of five-place tables solve each of the following triangles, having given: a = 54, 1. b=47, c =38. ( a = 2.6, 2. b=3.7, c =2.8. a =.117, 3. b =.261, c =.217. a = 122.6, 4. b =169.4, ic = 95.2. r a= 79.38, 13. b = 48.16, [ c =50. r a= 100, 5. [b = 125, c = 140. f a = 1.57, 6. b = 1.7, c = 1.266. f a = 17.03, 7. b = 12.585, c = 11.085. a = 113, 8. b = 147, c =48. a =2, 14. b b = 3, c=4. [a=V /14, 9. qb= /19, [ c = V33. f a = 4.1409, 1o. b= 4.9935, l c = 1.8181. ( a= 2.6, 11. b = 5.7, c = 7.8. f a =17.51, 12. i b = 12.575, [ c = 23.645. Find the largest angle. 15. The sides of a triangle are 10, 17, and 25. Find the smallest angle in the triangle. 16. The sides of a triangle are 3, 4, and 5.5. Find the sine of the smallest angle. 17. The sides of a triangle are 1.1, 1.3, 1.6. Find the cosine of the largest angle. 18. The sides of a triangle are 18, 21, and 25 ft. Find the length of the perpendicular from the vertex of the largest angle to the opposite side. 19. By use of four-place tables solve Exs. 1-18. 20. The distances between three towns, P, Q, R, are as follows: PQ= 51, QR=65, PR=20. If R is due east from P, what is the direction of each place from every other place? If R is N.E. from P, what would each of these directions be? 21. What angle is subtended by an island 2 miles long as viewed from a point 3 miles distant from one end of the island and 4 miles from the other end? 22. MDake up two practical problems which can be solved by the method of Case III and solve them, OBLIQUE TRIANGLES 128 CASE IV. GIVEN TWO SIDES AND AN ANGLE OPPOSITE ONE OF THEM 82. The Solution of Case IV, like that of Case I, is effected by the use of the law of sines (Art. 74). But it has been shown in geometry that when two sides and an angle opposite one of them are given, sev- c eral special cases arise in the construction of the triangle. b a Thus in the triangle ABC (Fig. \ 64) let the given parts be the A c FIG. 34. angle A and the sides a and b. Then under the following conditions the following triangles may be constructed: I. If given z A is obtuse and 1. side opp. A > side adj..... one A. 2. side opp. A < side adj..... no A. II. If given L A is right (same results as in I). III. If given Z A is acute and 1. side opp. > side adj...... one A. 2. side opp. = side adj... one isosceles A. 3. side opp. <side adj. The case last mentioned (3) subdivides into three special cases as follows: (1) side opp. > (side adj.) x (sin given z)... two A. (2) side opp. = (side adj.) x (sin given Z).. one right A. (3) side opp. < (side adj.) x (sin given Z).. o A. In practice, the cases of no solution and of one right triangle or one isosceles triangle as the solution do not often occur. Hence we usually need merely a method of discriminating between the cases where one oblique triangle or two 124 TRIGONOMETRY oblique triangles form the solution. We may state this test in the form of question and answer thus: Q. In general, when are there two solutions in Case IV? Ans. When the side opposite the given angle is less than the other given side. Q. In this case, how may the two triangles be constructed? Ans. Take the vertex between the two given sides as a center, and describe an arc, using the smaller side as radius. It is usual so to letter the figure A that the vertex of the given angle comes at the left end of ^b \l ~the unknown base. Thus given /\ C= 38~, b=152, c= 103, we have c/ / B Fig. 65. FIG. 65. Hence, in solving examples in Case IV, Observe whether the side opposite the given angle is less than the other given side; if it is, there are, in general, two solutions, which construct by taking the vertex between the given sides as a center and describing an arc with the smaller side as radius. In either case find the unknown angle opposite *the known side by the use of the following proportion: sine of unknown Z opp. known side:sine of known Z side opp. unknown Z:side opp. known Z. In case there are two solutions, use in one triangle the angle obtained from the table, and in the other triangle the supplement of this angle. Find the third angle and third side by Case I. Ex. 1. Given a= 84, b=48.5, A=21~ 31', solve the triangle. OBLIQUE TRIANGLES 125 Since the side opposite the given angle, 84, is greater than the other given side, 48.5, there is but one solution. sill B 48.5 sin 21~ 31' -84, sinB=48.5 sin 21~ 31'.'. sin B = 84 48.5 log 1.68574 21~ 31' log sin 9.56440 -10 84 log 1.92428 colog 8.07572 -10 B = 12~ 13' 33" log sin 9.32586 - 10. CFIG. 66. C= 180~- (A + B) = 1460 15' 27". By Case I we find c= 127.211. Ex. 2. a = 22, b = 34, A = 30~ 20', solve the triangle. Since the side a opposite the given angle A is less than the other given side (A being acute, and 22 > 34 sin 30~ 20') there are two solutions to the given triangle. In this case it is well to draw the smaller triangle separately as well as the general figure. FIG. 67. FIG. 67a. By the law of sines (Art. 74), sin B 34 sin 30~ 20' 22.34 sin 30~ 20'.'*. sin B =22 22 34 log 1.53148 To complete the solution of AACB, 30~ 20' log sin 9.70332 -10 Z ACB = 180~ - ( A + ZABO) 22 log 1.34242 colog 8.65758 - 10 = 180~ - 81~ 38' 27" B = 51~ 18'27" log sin 9.89238-10 = 98~ 21' 33".. onFig. 67a, B'=180~ —51 18' 27" Hence by Case I we find = 128~ 41' 33". c = 43.098. To complete the solution of A AC'B' (Fig. 67a). C'= 180~- (A+ B') = 180~ - 159~ 1' 33" = 20~ 58' 27". Then by Case I we find c' = 15.5926. (What checks can be used in the case of each of the two triangles?) 126 TRIGONOMETRY Ex. 3. Given a = 22, b= 34, A = 30.33~, solve the triangle. Since the side a opposite the given angle A is less than the other given side (A being acute and 22>34 sin 30.33~), there are two solutions. In this case it is well to draw the smaller triangle separately as well as the general figure. C= C-= FIG. 68. By the law of sines (Art. 74), sin B 34 sin 30.33~ 22 34 log 1.5315 30.33~ log sin 9.7033 - 10 22 log 1.3424 colog 8.6576 - 10 B = 51.32~ log sin 9.8924 - 10 sin B = 34 sin 30.33~ 22 To complete the solution of A ACB, ZACB= 180~ - (30.33~ + 51.32~) = 98.35~. Hence by Case I, obtain c= 43.1... Z B' = 180~ - 51.32~ = 128.68~. To complete the solution of AAC'B' (Fig. 68a), we have C' = 180~ - (30.33~ + 128.68~) = 20.99~. Hence, by Case I, find c' = 15.6. EXERCISE 39 State the number of solutions for each of the following and construct a figure for each example, lettering it according to the method specified in Art. 82: 1. A =30~, 2. B = 30~, 3. C=45~, 4. A=60~, b = 50, a=60. a = 100, b = 70. a =60, c = 60. b=12, a=10. 9. A= 75.16~, 5. 6. 7. 8. c=18, C= 80~, B = 54~, C= 30~, B = 50~, a=17.6. b = 16, c = 15.5. a = 23, b=36. a=18, c=9. a = 50, b=37. Using five-place tables, solve the following triangles, having given: o1. A = 38 18', b = 120.6, a =138.7. 11. A= 61 18', c=23.7, a=21.25. OBLIQUE TRIANGLES 127 12. C= 104 13' 48', b = 115.72, c= 165.28. 13. B=22~22', a =.6728, b =.81434. 14. A =47~ 19', a=100, c =120. 15. B = 15~ 30' 12", a = 1200 b = 590. 16. C =78~ 818'", a =.26725, c=.37926. 17. B = 26 18' 36", a = 28.604, b = 12.678. 18. A = 131~ 18' 24", a =.8888, c =.4128. 19. C= 31311' 15", b = 11.11, c= 8.267. Using four-place tables, solve the following triangles, having given: 20. B = 32.37~, b = 126.6, a= 138.7. 21. A = 57.366~, c = 22.7, a = 20.672. 22. B= 105.273~, b =306.72, c= 241.8. 23. C=26.223~, a=66.35, c=82.59. 24. B =14.3~, a — 20.17, b = 17.8. 25. A =22.37~, c = 300, a=200. 26. B =63.31~, c =7.67, b=9.54. 27. C=49.31~, b=.17634, c=.15678. 28. In a parallelogram, one side is 167, one diagonal is 295.6, and the angle included by the diagonals is 24~ 18' [24.3~]. Find the other side and other diagonal, and also the angles of the parallelogram. 29. If the angle between two forces is 154~ 20' [154.33~], one of the forces is 960 pounds, and the resultant of the two forces is 440.46 pounds, find the other force. AREA OF AN OBLIQUE TRIANGLE 83. I. Given two sides and the included angle, to find the area of a triangle, use the rule: The area of a triangle equals one half the product of any two sides multiplied by the sine of the angle included by these sides. For let the given sides be a and c. 128 TRIGONOMETRY In Fig. 69a, let Z B be acute; in Fig. 69b, let Z ABC be obtuse. C Cb ab A B A C -D-B A -— JD c D c B FIG. 69a. FIG. 69b. Let p be the perpendicular from C to AB or AB produced. In each figure, the area of A ABC = c x p. In Fig. 69a, in the rt. A CBD, p = a sin B. (Art. 41) In Fig. 69b in the rt. A CBD, p=a sin (180 ~-z ABC) a sin ABC. (Art 64) Hence, in each figure, if we denote area of A ABC by K, K= lac sin B. In case the given parts are a, b, C, or b, c, A, let the pupil state what the formula becomes. Let the pupil also state these formulas in general language. Ex. 1. A =66~ 4' 19", b= 21.66, c=36.94, find the area of the triangle ABC. B By the formula K= ~ bc sin A, KA= 1(21.66 x 36.94 x sin 66~ 4' 19")..-. log K = log 21.66 + log 36.94 + log sin 66~ 4' 19" + colog 2. ' \ X21.66 log 1.33566 36.94 log 1.56750 /64'19". \66~ 4' 19" log sin 9.96097 - 10 A 21.66 C 2 colog 9.69897 - 10 FIG. 70. Area = 365.682 log 2.56310 Ex. 2. Given A = 66.07~, b = 21.66, c = 36.94,.find the area of the triangle ABC. OBLIQUE TRIANGLES 129 By the above rule, = 1 (21.66 x 36.94 x sin 66.07~)... log K= log 21.66 + log 36.94 + log sin 66.07~ + colog 2. 21.66 log 1.3357 36.94 log 1.5675 66.07~ log sin 9.9610-10 2 colog 9.6990-10 Area= 365.75 log 2.5632 84. II. Given two angles and a side, find the third angle as usual. Let the given side be a, then a second side c may be determined as follows: c: a= sin C: sin A. a sina sin C a sin C sin A sin [180~- (B + C)] sin (B + C) Substituting this result in the formula for K in Art. 83, a2 sin B sin C 2 sin (B + C) Hence the area may be found by substituting directly in this last formula. 85. III. Given three sides. In this case we know from plane geometry that K = Vs(s - a)(s- b)(s - c). 86. IV. In case two sides and an angle opposite one of them are given, to find the area it is necessary to find the log sin of the angle included between the two given sides by the method of Case IV (Art. 82), and then proceed as in Art. 83, In some cases two answers may occur (see Art. 82). EXERCISE 40 Using either five-place or four-place tables, find the area of the following triangles, having given: 1. a= 16.7, b = 21.6, C=36~ 18 24" [36.31~]. 2. a =.86, B = 52~ 18' [52.3~], C = 66~ 42' [66.7~]. 130 TRIGONOMETRY 3. a=18, b=14, c=24. 4. b = 200, c = 150, A = 72 18' 30" [72.31~]. 5. b = 600, A = 18~ 26' [18.43~], C= 31~ 44' [31.73~]. 6. b = 14.7, = 18.6, A = 74~ 18' [74.3~]. 7. a =.8167, b =.68256, c =.72623. 8. a = 100, c= 125, B = 170~ 16' [170.27~]. 9. b = 62.8, c = 47.2, A = 60~. 10. Given A=29~ 32' 16" [29.54~], b=500, and a=300, find the difference in area between the two triangles which contain these parts. 11. In a parallelogram, given two adjacent sides, c and d, and the included angle A, obtain a formula for the area of the parallelogram in terms of the given parts. 12. Prove that the area of any quadrilateral is equal to one half the product of its diagonals and the sine of their included angle. 13. Two sides of a parallelogram are 30 and 40 respectively, and their included angle is 60~. Find the area of the parallelogram without the use of tables. 14. The diagonals of a quadrilateral are 17.6 and 20.5, intersecting at an angle of 36~ 18' [36.3~]. Find the area of the quadrilateral. CHAPTER VII PRACTICAL APPLICATIONS 87. Instruments for Measuring Angles. In order to determine unknown heights or distances it is important to have an instrument for measuring angles either in the horizontal or in the vertical plane. Horizontal angles can be measured by the Surveyor's Compass. Both horizontal and vertical angles can be measured by the Transit Instrument. 88. An angle of elevation is the angle between a line drawn from the eye of the observer to the point observed and the horizontal plane through the eye of the observer, when this angle is above the horizontal plane. Thus, on Fig. 71, ACB is the angle of elevation of A as viewed from C. An angle of depression is the angle between a line drawn from the eye of' the observer to the point observed and the horizontal plane through the eye of the observer, when this angle is below the horizontal plane. Thus, on Fig. 71, DA C is the angle of depression of C as viewed from A. 89. I. To determine the Height of above a Horizontal Plane. In Fig. 71 let AB be the object whose altitude is sought, and EF the horizontal plane, and C the point of observation. In the right triangle ABC, what line shall we measure? What angle? How then can AB be computed? 131 an Accessible Object D A A E ( TT (; B F FIG. 71. 132 TRIGONOMETRY 90. II. To find the Distance on a Horizontal Plane to an Inaccessible Object whose Height is Known. In Fig. 71, let AB be the inaccessible object whose height is known; let EF be the horizontal plane and C the position of the observer. In the right triangle ABC, what side is known? What angle can be measured? How then can BC be computed? 91. III. To determine the Height of an Inaccessible Object above a Horizontal Plane. A Let AB, Fig. 72, be the altitude which is to be meas/ /^I A ured, and EF the horizontal //r/ / plane. Place the transit inE D C B F strument at D and measure FIG. 72. the angle of elevation ADB. Measure the distance DC toward B, and measure the angle A CB. By solving the triangle ACD the line A C is found. By solving the right triangle ACB, AB is found. In case it is desired to compute AB by means of right triangles alone, the solution may be effected by dropping a perpendicular CP from C to AD and solving the right triangles DCP, CPA, and CAB (let the pupil supply the exact steps in this process). Or we may proceed by the use of natural tangents thus: On Fig. 72, in A DAB, DB = AB tan Z DAB, in A CAB, CB = AB tan Z CAB. Subtracting, DB-CB, or DC = AB (tan / DAB- tan Z CAB). Hence AB = DC tan Z DAB- tan / CAB' In case it is not possible to move directly from D toward B, we may proceed as follows: Measure Z ADB (Fig. 73). PRACTICAL APPLICATIONS 133 Measure the line DC in the horizontal plane in any convenient direction from D. Measure Zs BDC and DCB. Then in the triangle DOB, D, DB may be computed (How?). Afterward in the triangle ADB compute AB (How?). C FIG. 73. 92. IV. To determine the Height of an Inaccessible Object on an Inclined Plane. Let DF (Fig. 74) be the horizontal plane, DB the inclined plane, and AB the object whose height is sought. If we measure the as ADC and A CB, and the dis/ B tance DC, we may then compute AC (How?). If we then measure c/ / X BDF, we may compute X CAB _D F ------- ---—, (How?). Then AB may be comFIG. 74. puted (How?). 93. V. To find the Distance of an Inaccessible Object. Let A (Fig. 75) be the position of the observer and let it be required to determine the distance from A to B. Let the pupil determine what measurements and computations are necessary in accordance with the figure. A\.. FIG. 75. 94. VI. To find the Distance between two Objects separated by an Impassable Barrier (and possibly invisible to each other). 134 TRIGONOMETRY 4A,',,. Let it be required to find the disA, 7 tance between A and B (Fig. 76), which are separated by a swamp or a mountain for instance. Take a station C from which both A and 2B are visible. Measure the angle C and the a lines CA and CB. In the triangle FIG. 76. ABC, compute AB (How?). 95. VII. To find the Distance between two Objects, both Inaccessible and lying in the Horizontal Plane. Let A and B (Fig. 77) be two inaccessible objects (as B two islands off the shore CD). Measure the line CD and the AACD, B CD, AD C, BDC. In the triangle ACD, compute AC; in the triangle BCD, compute BC; in the FIG. 77. triangle ABC, compute AB. 96. Range Finders. In war, both on land and sea, the use of a range finder to determine the distance of an enemy is becoming general. The essential principle of such an instrument is the finding of the distance of an inaccessible object by the solution of a triangle in which a side (called a base line) and the two angles which include the side are known (see Art. 93). On land a convenient base line is taken and measured. In naval warfare, the distance between two points on the vessel is utilized as a base line. In the range finder the triangle employed is not usually solved by numerical computation, but by some mechanical method, which gives the result sought much more expeditiously. 97. Coast and Geodetic Survey. The essential parts of the work of the coast and geodetic survey are as follows: PRACTICAL APPLICATIONS 135 1. The measurement of a base line AB (Fig. 78) at least 4 or 5 miles long, so accurately that the error shall not exceed -0 of an inch per mile. 2. The choice of a convenient station A P and the measurement of the angles \ PAB and PBA, and the computation of PA and PB in the triangle PAB. --- // 3. The choice of another station Q, the measurement of the angles QBP and QPB, and hence the computation of PQ and / QB. \ I 4. Proceeding in like manner from \ / / station to station till convenient points, C f % and D, are reached, and the length of the line CD computed.. /) 5. The careful measurement of CD and the comparison of its computed length D, with the result of the measurement. This final measurement of CD serves as a test of the accuracy of all the inter- FIG. 78. vening work. By carrying these measurements far enough, a considerable arc of a great circle of the earth may be measured, and from this arc the radius or diameter of the earth computed. 98. Distance of the Sun and Stars. The usual method of determining the distance of the sun from the earth consists essentially in taking a line (AB, Fig. 79) nearly equal to the diameter of the earth as a base A _p line, and observing from each end of AB the angle made by a line 13~B ~drawn to some convenient planet FIG. 79. P. The distance of the planet may then be computed by Art. 93. The ratio of the distance of the sun to that of the planet from the earth being 136 TRIGONOMETRY known by an astronomical law, the distance of the sun is readily determined. The distance of the sun from the earth is thus found to be approximately 93,800,000 miles. The distances of the fixed stars are found by taking the diameter of the earth's orbit as a base line, measuring the angles made by this line with lines drawn from its ends to a fixed star, and making the necessary computations. Thus the trigonometrical solution of a triangle in which a side and the two angles adjacent to it are known is seen to have very wide practical applications. 99. Application to Navigation. Trigonometry also has many applications to different departments of applied science. As an illustration of these ^^"^ Aapplications we will briefly indicate its C/ v BA\ method of use in navigation. "/, \ - If a ship should sail from R to B on E oe the diagram (Fig 80), crossing each meridian at the same angle, for certain _I pnrp purposes the AARB (AB being the arc P: of a parallel of latitude) could be reFIG. 0s. garded as a plane triangle and solved, when necessary, by the methods of plane trigonometry. This form of navigation is called Plane Sailing. The departure between two meridians is the arc of a parallel of latitude comprehended between the two meridians. Thus, AB is a departure between PAP' and PBP'. Evidently the departure between two given meridians diminishes with the distance from the equator. The difference of longitude between two places is the angle at the pole (or the arc on the equator) included between the meridians of the two given places. Thus the difference of longitude for A and D is the angle RPS, or arc RS. In Parallel Sailing a vessel sails due east or west (i.e. on a parallel of latitude) as from A to B. The difference of PRACTICAL APPLICATIONS 137 longitude corresponding to the course sailed may be found by the formula diff. of longitude= departure x sec. latitude. For on Fig. 80, diff. long.:dep. = arc RS: arc AB= OR: CA = OA: CA = OA: 1 CA = sec. lat: 1.. diff. long.: departure = sec. lat.: 1. In Middle Latitude Sailing a ship sails between two places in a course oblique to a parallel of latitude. For short distances (especially near the equator) sufficient accuracy is obtained by regarding the departure as measured on the parallel of latitude midway between the parallels of the two places, and computing the difference of longitude by the formula diff. long. = departure x sec. mid. lat. EXERCISE 41 1. In Exercise 22 point out the examples which are solved by the method of Art. 89. 2. Also those which are solved by the method of Art. 90. 3. Also those solved by principles contained or implied in Art. 91. 4. The angle of elevation of the top of a tree measured from a point 213.5 ft. from its foot is observed to be 18~. Find the height of the tree. 5. A water tower 92.5 ft. high stands on a horizontal plane. An observer finds the angle of elevation of the top of the tower to be 52~. Find the distance of the observer from the base of the tower. 6. Pike's Peak when viewed from a certain point on the Colorado plain has an angle of elevation of 15~ 48' [15.8~]. Two miles farther off the angle of elevation is 11~ 59' [11.98~]. What is the altitude of the mountain above the Colorado plain? If the Colorado plain is 5176 ft. above sea level, what is the altitude of Pike's Peak above sea level? 7. From the top of a hill 350 ft. high the angle of depression of the top of a tower which is known to be 150 ft. high is 57~. What is the distance from the foot of the tower to the top of the hill? 138 TRIGONOMETRY 8. A man standing west of a tree, on the same horizontal plane, observes its angle of elevation to be 48~; he goes north 50 yd. and finds its angle of elevation to be 41~. Find the height of the tree. 9. The angle subtended by a tower on an inclined plane, is at a certain point on the plane 56~; 200 ft. further down it is 28~. The inclination of the plane is 7~. Find the height of the tower. 10. From the top and bottom of a castle which is 75 ft. high the angles of depression of a ship at sea are 19~ and 15~ respectively. Find the distance of the ship from the bottom of the castle. 11. A monument 70 ft. high and a tower stand on the same horizontal plane. The angle of elevation of the top of the tower at the top of the monument is 20~ 40' 12" [20.67~], at the base of the monument it is 53~ 31' 12" [53.52~]. Find the height of the tower and its distance from the monument. 12. The three angles of a triangle are to each other as 11: 13: 6 and the longest side is 11. Find the other two sides. 13. Two mountains, A and B, are respectively 12 and 16 mi. from a point C, and the angle ACB is 72~ 18' [72.3~]. Find the distance between the mountains. 14. In a parallelogram one side is 16.9 and a diagonal is 30.72, and the angle included by the diagonals is 26~ 36' [26.6~]. Find the other side and the other diagonal, also the angles of the parallelogram. 15. A flagstaff 50 ft. in height stands on a tower. From a position near the base of the tower, and on the same horizontal plane, the angles of elevation of the top and bottom of the flagstaff are 41~ 36' [41.6~] and 22~ 18' [22.3~], respectively. Find the distance and height of the tower. 16. The diagonals of a parallelogram are 12.5 and 12.8 ft. respectively, and their included angle is 52~ 16' [52.27~]. Find the sides of the parallelogram. 17. The sides of a triangle are 11, 13, and 16. Find the cosine of the largest angle. 18. From a point 4 mi. from one end of an island and 7 mi. from the other, the island subtends an angle of 33~ 33' 33" [33.56~]. Find the length of the island. 19. Two buoys are 1500 yd. apart. The angles formed by lines from a boat to each buoy form angles with the line between the buoys of 77~ 18' [77.3~] and 51~ 16' [51.27~], respectively. Find the distance of the boat from the nearer buoy. PRACTICAL APPLICATIONS 139 20. Two straight roads cross each other at an angle of 48~ 24' [48.4~] at the point M. Four miles from M on one road is the town of P, and 6 miles from M on the other road is the town of K. How far apart are P and K? (Two answers.) 21. The diagonals of a quadrilateral are 47.6 and 61.23 rd., respectively, and the angle included by the diagonals is 43~ 10' [43.17~]. Find the area of the quadrilateral. 22. To find the distance between two trees T and T', on opposite sides of a river, a line TIC and the angles T'TK and T'KT are measured and found to be 412 ft., 62~ 30' [62.5~], and 57~ 32' [57.53~], respectively. Find the distance TT. 23. Two objects which are invisible from each other on account of a hill are visible from a station whose distances from the objects are 367 yd. and 514 yd., respectively, and the angle at the station subtended by the distance between the objects is 57~ 36' [57.6~]. Find the distance between the objects. 24. Given a circle with radius 19.8 ft. Find the area inclosed between two parallel chords on opposite sides of the center whose lengths are 25.6 and 31.7. 25. Wishing to find the distance between two trees T and T', separated by a marsh, I take TIC on the prolongation of TT' through T, 89 yd. in length, and then take IP, 165 yd. in length, at right angles to KT. The angle T'PT is found to be 33~ 36' 36" [33.61~]. Find the distance from T to T'. 26. Two yachts start at the same time from the same point, and sail one due west at the rate of 9.75 mi. per hour, and the other due northwest at the rate of 11.5 mi. per hour. How far apart will they be at the end of 2 hr. sail? 27. In order to find the distance from a rock R to a buoy B, distances RIK and lKP are measured to points IC and P from which both rock and buoy can be seen, the distance RK being 2500 m., and KP being 3600 m. The following angles are then measured: ZBBKR =38~48' [38.8~], ZBKP = 75~ 54' [75.9~], and ZBPKT= 79~ 30' [79.5~]. Find the distance from the rock to the buoy. 28. A ship sails due east 416 mi. in latitude 40~ 23'. Find the difference in longitude which she makes. 29. A ship leaves latitude 30~ 16' N., longitude 43~ 17' W., and sails N.E. 350 mi. Find the difference of latitude and departure which she makes. Hence find her new latitude and longitude. 140 TRIGONOMETRY 30. A flagstaff 30 ft. high stands on the top of a building. From a point on the ground, the angles of elevation of the top and bottom of the flagstaff are observed to be 41~ and 36~ respectively. Assuming the ground to be level, find the height of the building. 31. A tower stands on a hillside whose inclination to the horizon is 11~; a line is measured straight up the hill from the base of the tower 110 ft. in length and, at the upper extremity 'of the line, the tower subtends an angle of 52~. Find the height of the tower. 32. A rock 60 ft. high stands on the top of a hill whose side is inclined 21~ to the horizon. An observer standing on the hillside below the rock finds the angle of elevation of the top of the rock to be 64~, and a second observer, farther down the slope, and in direct line with the first observer, finds the angle of elevation of the top of the rock to be 42~. Find the distance between the observers, and the distance from the first observer to the base of the rock. 33. A point at 0 is acted on B by a force which gives a velocity of 1376 ft. per second along OA, \ ^A and by another force which gives 113 0 a velocity of 1135 ft. per 1 /1376 second along OB. Z AOX= 30~, \ /BOX = 101~. What will be X'- o \< - X the magnitude and direction of the resultant velocity? 34. Show that the projection of OA plus the projection of OB on X'OX equals the projection of the resultant of OA and OB on X'OX. 35. If, in the figure of Ex. 33, OA = 200 and the resultant = 300, find OB, the angles being unchanged. 36. A tower 190 ft. high stands on the seashore. From its top the angle of depression of two boats are 8~ and 11~ respectively. From the bottom of the tower the angle subtended by the distance between the boats is 101~. Find the distance between the boats. 37. A man on the opposite side of a river from two trees P and Q wishes to determine the distance between the trees. He measures a distance A B, 287 ft. He also measures the angles PAB, QAB, PBA, and PBQ and finds them 31~, 36~, 51~, and 42~, respectively. Find the distance between the trees. 38. Two straight paths cross each other at an angle of 68~. A line is drawn so as to inclose, with the two paths, an acre of ground. This line cuts one of the paths at a distance of 52 yd. from the point of PRACTICAL APPLICATIONS 141 intersection of the two paths. What angle does this line make with each path? 39. A tower 135 ft. high stands at one corner of a triangular garden. From the top of the tower the angles of depression of the other two corners of the garden are 56~ 18' [56.3~] and 19~36' [19.6~], respectively. The side of the garden opposite the tower subtends, from the top of the tower, an angle of 66~. Find the length of the sides of the garden. 40. Two towers are 144 ft. apart. The angle of elevation of one observed from the base of the other is twice that of the first observed from the base of the second; but from a point midway between the towers, the angles of elevation of the tops of the towers are complementary. Find the height of the towers. (Do not use logarithms.) 41. A railroad embankment is 9 ft. high. The length of the slope of the embankment on each side is 14 ft. Find the angle which the slope makes with the horizontal, and also find the width of the embankment at the base if the top is 8 ft. wide. 42. Given the triangle ABC, whose sides are AB =-87.6 yd., AC= 112.7 yd., and BC 121.6 yd. A point, D is taken on the line AC produced through C, so that the angle BDC is 18~ 37' 48" [18.63~]. Find the distance DC. 43. The area of a triangle is 3 acres and two of its sides are 92.6 and 26.72 rd. Find the angle between these sides. 44. A shooting star is observed at two places 200 mi. apart on the earth's surface; the angle of elevation of the star at one station is 27~ and at the other is 63~, the star being in the same plane with the two stations and the center of the earth. Taking the radius of the earth as 3956 mi. find the height of the shooting star above the earth's surface and hence the height of the earth's atmosphere. (What is a shooting star? What causes its light?) 45. Show how to solve each of the cases in oblique triangles by dividing the oblique triangle into right triangles and using the methods of solving right triangles given in Chapter III. Why do we not ordinarily use this method of solving oblique triangles? 46. Make up (or collect) all the different examples you can showing practical applications of trigonometry, each example being distinct in principle or in field of application from the other examples. CHAPTER VIII CIRCULAR MEASURE. GRAPHS OF TRIGONOMETRIC FUNCTIONS 100. Radians, or the Circular Measure of Angles. The method of measuring angles by taking a right angle as the unit, dividing the right angle into 90 degrees, dividing each degree into 60 minutes, etc., is called the sexagesimal method and originated in Babylonia (see Art. 127) in very early times. It continues to be generally used in spite of its awkwardness because of the extensive tables and large number of results stated in terms of it which have been accumulated. However, the advantages of the decimal division of any unit are so great that it is a growing custom to divide the degree of angle into tenths and hundredths instead of minutes and seconds (see many examples in this book). Also within the past century it has become customary in many kinds of work (especially algebraic or theoretic work) to use a unit of angle different from the right angle, called the radian, and to divide this unit decimally. l / 0 ~A radian is the angle which, when B r A its vertex is placed at the center of a circle, intercepts an arc equal to tile radius of the circle. FIG. 81. Thus if the arc AC (Fig. 8) equals the radius AB, the angle ABC is a radian, or the unit angle in the so-called circular method of measuring angles. 142 CIRCULAR MEASURE 143 Hence, to determine the number of radians in an angle whose arc and radius are given, we have the relation no. of radians in an angle= arc, or, B radius\ denoting the number of radians in an o 5 A angle by p, the subtended arc by a, and the radius of the circle by R, p= aR Ex. 1. Find the number of radians FIG. 82. in an angle AOB whose arc is 13 and radius 5. We have, Z AOB = -1 = 2.6 radians, Ans. From the above relation it follows that Any two of the three quantities, number of radians in an angle, arc, and radius, being given, the other may be found. Ex. 2. An angle containing 2.4 radians subtends an arc 14 in. long. Find the radius. Substituting for p and a in the formula p = -, 14 in. 14 in. 2.4= i.. R in 5.83+ in., Ans. R 2.4 101. I. Converting Degrees into Radians. The number of radians about a point in a plane circumference radius 2 rR = --— 27= 7T. R.. 360~= 2, or 6.2832 radians. 45 = or 0.7854 radians. 180~ = T, or 3.1416 radians. 4 90~=_, or 1.5708 radians. 30~= or 0.5236 radians. 260~=' 6'or 1.0472 radians. or.01745 radians. 60~ =, or 1.0472 radians. 1 =j or.01745 radians. 3 180 ( E\ 7/ /) /) 144 TRIGONOMETRY Hence to convert degrees into radians Multiply the given number of degrees by 8 (or by.01745+). Ex. 1. How many radians in 26~ 17' 36"? 26~ 171 361= 26.293+0 =(26.293+)(.01745) radians. 0.45882+ radians, Ans. Ex. 2. Simplify sin (T +. sin (7 + = sinl cos x + cos 6sin x (Art. 66) 6G 6 6 = ~ cos x + 1 -3 sin xl, Ans. (Art. 33) Where the meaning is evident from the context, it is customary to abbreviate "- 7 radians" into "'r." Thus also we abbreviate "sinl radians" into '" sin " and similarly for other expressions. 6 102. II. Converting Radians into Degrees. Since 2 7T radians = 360~ 180~ 1 radian = or 1 radian = 57.29579+0 = 570 17' 45" - 206265". Hence to convert radians into degrees 180~ Multiply the given number of radians by - (or 57.3~-). Ex. Convert 2.5 radians into degrees, minutes, and seconds. 2.5 radians = 2.5 x (57.2958~-) = 143.2395~ = 143 1414' 22", Ans. Hence, if the number of degrees in an angle be denoted by A, the number of radians in it by p, etc., any two of the 0 ^ - ') /'" / /! ~ I\ <,,,,/ Q. 1,K! Yc G — A,^,/ CIRCULAR MEASURE 145 four quantities A, p, a, R being given (provided one of them is a or R), the other two may be found by substitution of the two given quantities in the two equations / a /180~. P = and A=p( ). - 103. The solution of a right triangle containing an angle less than 2~ may often be conveniently effected by the use of radians. For the sine or tangent of a small angle may be taken as equivalent to the number of radians in the angle (i.e. the circular measure of the angle) without appreciable error (see Art. 115). Thus sin A = (in radians) when A is a small angle, is an approximation frequently used in Physics, and the result is accurate to within the probable degree of error in measurement. Ex. If a railroad track has a rise of 1 ft. in every 2000 ft. in its length, what angle does it make with the horizontal? Denoting the required angle by A, sin A = 1 = no. radians in A approximately. 2000.. A 2000 x 206265" = 103+" = 1' 43", Ans. EXERCISE 42 1. Reduce the following angles to circular measure, expressing the results as fractions of 7r: 30~, 135~, 60~, 90~, 210~, 270~, 225~, 72~, 315~. 2. Express the following angles in degrees: 7 7r 7r 2 r 4 7r 3 7r 77r 87r 6' 4' 3' 3' 5 5 5 ' 15' 3. What decimal part of a radian is 1~? 16"? 2' 15"? 5~ 14'? 4. HIow many degrees (minutes and seconds) in 2 radians? 3.2 radians?.003 radians? 5. A circle has a radius of 14 inches. 'How many radians are there in an angle at the center subtended by an arc 21 in. long? By an arc 7 in. long? 146 TRIGONOMETRY 6. In a circle of radius R, an arc 3 ft. 6 in. subtends an angle of 1.5 radians. Find R. 7. One angle of a triangle is 30~, and the circular measure of another angle is 1.5 radians. Find the third angle in degrees. Also in radians. 8. The difference between two angles is 7 and their sum is 110~. Find the angles in degrees; in radians. 9. Find both in radians and degrees the complement and supplement of the following angles: 7T 7T 7T 7r 5 7 6' 3' 4' 9' 18' 10. Write out the trigonometric ratios of the following angles: r 7r 7r 7r 3 r 77r 77 6' 3' 4' 2' 4 ' 6 4' 11. How many radians in an angle whose arc is 12 and radius 10? How many degrees? 12. Show that sin (x + 1 r) + sin (x - 1 7r) = sin x. Supply the two missing quantities in each of the following: P a R A 13 2.5 10 in. - 14.25 - 50 in. - 15 -- 12 ft.ft. 6 in. 16 - - 42 in. 1O3-0 17 - 100 - 37~ 18. If a railroad track has a rise of 1 ft. in 750 ft., what angle does the track make with the horizontal? 19. If a railroad makes an angle of 1~ 30' with the horizontal, what is its rise in one half mile? 20. An irrigating ditch should have a fall of at least - in. per rod. What angle does the bottom of the ditch make with the horizontal? 21. If the moon is at a distance of 240,000 mi. from the earth and the radius of the moon subtends an angle of 16' as seen from the earth, what is the radius of the moon in miles? 22. If the sun is at a distance of 92,800,000 mi. from the earth, and the diameter of the sun subtends an angle of 32.4' as viewed from the earth, what is the radius of the sun in miles? 23. The planet Mars has a diameter of 4200 miles. When Mars is nearest the earth, its diameter subtends an angle of 24.5" as seen from CIRCULAR MEASURE 147 the earth. What is the distance of Mars from the earth at such a time? 24. Find the numerical value of 3 sin r -4 cos 6 tan 7r + cot * 4 6 3 2 25. Make up two practical problems. in each of which a right triangle is solved by the use of radians as in Exs. 17-21. We shall now illustrate the use of radians, or the circular measure of angles, (1) in tracing the graphs of trigonometric functions, (2) in solving trigonometric equations. GRAPHS OF TRIGONOMETRIC FUNCTIONS 104. Graph of sin x. To form what is called the graph of sin x use the equation y = sin x and also a pair of rectangular axes (see Art. 54). In the equation y = sin x, let x have convenient successive values and find the corresponding values of y. Lay off each corresponding pair of values of x and y as the abscissa and ordinate of a point. Draw a continuous curve through the terminal points thus located. It is usually convenient to make the scale of the drawing such that a unit space of the cross-section paper stands 6 Thus, if we desire to make a graph of y = sin x we may take the following corresponding values of x and y: x=O, y=0. x=, y==5 x= = -.5. 6 =6 Y =.86_. V_. x =, / =.86~. x= / =- -.86+. 3 y 3 2 X2 y=1. X=-, y=-1. 2~rr 2~rr _ 11\~~=_.862 x,= 3=.86+. X =_27r y=- =-.86+. =3~r y = 2 r3 A =- 26+. 5w 0 x =, y =.5. x =-5, y -e. 6 6 x = 7r, y = 0, etc. x = - 7r, y - 0, etc. 148 TRIGONOMETRY Using these results, the curve AOBCDE (Fig. 83) is obtained. as the graph of sin x. Such a figure shows at a glance the changes in the values of sin x as x changes in value. - T-T-,-TT-T- -T —T-V-T'-T!!!, I I. I I.._. III FIG. 83. 105. Graphs of Other Trigonometric Functions. By treating the equations y = cos x, y = tan x, y = sec x, etc, similarly, the graphs of the other trigonometric functions may be constructed. Y B D! I I I I I / [ I I [ I I I [I I t- - -+ + ----- I I --- I. ---- II I I — I I — I -- I I-. —+ ---- ----- 4 -- -~ - -—,.-.-+ -.-_I- -+ ---- +- +-++-4.- - t --- - +- - F — -+ ---+ -+-[-+-f - 4-4 — - 4_ — - t --- - - [ --- - -+ —. -- - l- -t-. -- - 4- - — 4 — __r!!'!,! [/ i [.!,w~,/[!,!_ FIG. 84. It is important to observe in constructing the graph of tan x, that, as x- y either + oc or - o. For as we proceed from x = O and make x- y +; but as we7 proceed from x = 0 and make x y + c but ase we proceed from x = rr and make x ' ' y - oc. Hence we 2 CIRCULAR MEASURE 149 obtain as part of the graph of tan x the curve AOB, CO'D of Fig. 84. EXERCISE 43 Graph each of the following: 1. y =sin x. 2. y = cos x. 3. y = tan x. 4. y =cot x. 5. y= sec x. 6. y = csc x. 7. y= sinx.-. 8. y=sin 2x. 9. y= tan I x. 10. y = sil x -- cos x. 11. y = silln - cos x. 12. y=-/sin x. 13. y = sin2 x. 14. y = +sin x. 15. y = --- cos x. 16. y = x+ sin x. 106. Solutions of Trigonometric Equations. greater than 360, i.e. than 2 r radians. Ex. 1. Find the values of x less than 2 rr shall satisfy the equation sin x = -. Since sin 30~= 1 and also sin 150~ - i Answers not radians which 7r 5 r x=- or - radians, Ans. 6 6 Ex. 2. Solve 4 cos x- 3 sec x 0 for values of x less than 2 7r. 4cos x- - = 0. cos x 4 cos2 x- 3 = 0. cos x = ~ 2- v3. Hence, x= 30~, 150~, 210~, 330~, 7r 5 7 - 7r or x= ' 7-' 6 r radians, Ans. 6' 6 6' 6 107. Answers Unlimited. Ex. 1. Solve the equation cos x = 1. One value of x is 60~ and another value is - 60~. But if 360~ be added to or subtracted from the value of an angle, the value of the function is unchanged. 150 TRIGONOMETRY Hence, x = 2 n7r ~ radians, where n is zero or any positive or 3 negative integer. Ex. 2. Solve the equation sin x - cs x + 3 = 0. Solving the equation, we obtain, sin x =- 2, Since the sine of an angle cannot be greater than 1, no angle corresponds to the value - 2. For sinl = 1 = 2 nr -, (2n+ 1)7r-, Ans. EXERCISE 44 Solve each of the following equations, expressing the answers in radians, by use of 7r. 1. cot2 0 - 3. 12. cot+l =cos 2x. I J 2. tan2 0 = 3. 3. cot2 = 1. 4. sin2 0 = 3 5. cot 0= 2 cos 0. 6. cos 0 + sec 0= - 7. 3 sin2 x + cos2 x =. 8. 3 cot2 x + tan2 x = 4. 9. cos x = sin 2 x. 10. cos 2x + sin x =4 sin2 x. 11. sin 2 x = tan2 x. cot x- 1 13. 2 sin2 x - sin x = sin 2 x-cos.x. 14. cos 2 x + cosx = 0. 15. tan (45~ + x) + tan (45~ - x) = 4. 16. 2 csc2 x - V3 cot x = 5. 17. sin 3 x = sin 5 x - sin x. 18. cos 3 x - cos x = cos 2 x. 19. sin 5 x - sin x= cos 3 x. 20. cos 3x-cosx=-sin 2x. 21. sin 5 x+ sin 3 x sinx = 0. 22. cos 5 x cos 3 x + cos x= 0. 108. Simultaneous Trigonometric Equations. Ex. 1. Solve x sin y= a s - j-for x and y. x cos y= b Dividing the first equation by the second, a a tan y = -.. y = whose tan is -, Ans. b b (For a briefer way of expressing this result see Chapter IX.) CIRCULAR MEASURE 151 From this result the value of y may be obtained. When y is known x can be obtained from either of the original equations. a b Thus x= —, or x= - sin y cos y Ex. 2. Solve for x and y the equations, xcosA+y sinA=a.. x sin A-y cos A=b. Multiply equation (1) by cos A, then x cos2 A + y sin A cos A = a cos A. Multiply equation (2) by sin A, then x sin2 A - y sin A cos A = b sin A... (1).... (2)..... (3) I* * (4) Add (3) and (4), using the fact that sin2 A + cos2A = 1. then x= a cos A +b sin A, I and similarly, y= sin A- cos A.ns and similarly, y = a sin A - b cos A. j EXERCISE 45 Solve for x and 0, or for x and y: 1. x cos 0 = 86.65, x sin 0=50. 2 x sin 0 = 118.96, x cos 0 = 160.78. x tan 0 = 816.95, x sin 0 = 426.3. 4. x sin y = 4, x cos y = 8.. x sin 30~ + y cos 45~ = 53.28, * x cos 30~ + y sin 45~ = 71.58. 6. $ sin 48~ + y cos 19~ = 2634.1, L x cos 48~ + y sin 19~ = 1320.3. 7 sin x + sin y = 1.573, [Use Art. 71.] cos x + cos y = 1.207. sin x - sin y =.2154, cos x - cos y —.1231. 9. x sin (0- 21.5~) = 771.1, x cos ( - 32.5~) = 766.:0. IxcosA-ysinA= a, ( x sin A +ycos A = b. CHAPTER IX INVERSE TRIGONOMETRIC FUNCTIONS 109. Anti-sine. If y is an angle and x its sine, the relation between x and y may be expressed in either of two ways: 1/ x (1) =siny, or (2) y= sin-' x, which reads "y is the angle whose sine is x" or " y is the antiFIG. 85. sine of x." One or the other of methods (1) or (2) is used according as the angle, or its sine, has the leading place in the discussion. Thus if the angle, or y, is more prominent, x = sin y is used; but if the sine, x, is more prominent, y= sin- x is used. The pupil should carefully discriminate between sin-lx and the -1 1 power of sin x. The latter is expressed thus, (sin x)-1. Thus, - sin x (sin x)-1, and not sin-1 x. But (sin x)-2 may be written sin-2. 110. Other Anti-trigonometric Functions. Similarly.cos-' x means " the angle whose cosine is x"; tan- x means " the angle whose tangent is x." Let the pupil state the meaning of cot-1 xcsc- x, vers-t x. It is evident that sin (sin- x) = x, since the sine of the angle whose sine is x must be x. Similarly cos (cos-l x) = x, etc. Hence there is a similarity in form between a(ac-)x = x, and sin (sin-l x) =x. It is because of this similarity that the system of symbols described above is used to express the anti-trigonometric functions. 152 INVERSE TRIGONOMETRIC FUNCTIONS 153 A much better symbolism for "y equals the angle whose sine is x" would seem to be "y = Z sin x," and if the pupil has difficulty in grasping the principles of this chapter, it may be well for him to use this latter method of writing inverse functions till he becomes familiar with their nature. 111. Values of Inverse Trigonometric Functions. The direct and inverse trigonometric functions have an important difference with reference to the number of values which satisfy them. Thus, if y=sin 30~, y has a single value, ~; but if x= sin-' 1, x can have an indefinite number of values, viz.: 30~, 150~, 390~, 510~, etc.; or x=2n ~7+6, (2n+ 1)7r- (SeeArt.107,Ex.2.) For many purposes it is customary to limit the values of an inverse circular function to the smallest value that will satisfy a given expression. Thus, if 0= tan- 1, we take =-45~. 112. Given an Anti-trigonometric Function, to find the other Related Functions. Ex. 1. Given = tan- ~, find sin 0; that is, find sin (tan-l -). 0 = tan-l 2 may be converted into the form tan 0 =- for which a diagram may be con- 2 structed (Fig. 86)..*. sill 0= - J3 =3v3 ~... sin 0-_ -2FIG. 86... sin (tan-1 2) = 2AV3 Ans. Ex. 2. Find sin 2(cos-1' ). Let x be the angle whose cosine is ~. Then cos x= 1 sin x = V/ - - = 2... sin 2x = 2 sin x cos x 2(2 2)1 = Hence, sin 2(cos-1 -) = 4-/V2, Ans. 154 TRIGONOMETRY Ex. 3. If 0 = tan-' a, express the direct and inverse functions of 0 in terms of a. tan 0 = a, hence 0 = tan-l a. cot= 0, 0 cot-l1 a a sec 0= V + a2, 0 = sec-l + a2. a cos 0= 6 ___, 0=cos-, _1__ a a /^ A/1 + a2 V/1 + V 2 1 sin 0 =, = sin-' FIG. 87. V/1 + a2 V + a2' es e 0 + a=cs csc=,_ - -c, sc-S 1 +/a. a a Ordinarily only the positive value of each radical is used. 113. Inverse Trigonometric Functions of Two Angles. Ex. 1. Find sin (sin-' + cos-~ -). Let x = sin-l'. /. sin x-=, 3 cos x -- 1 3.0 Let y - cos -. 2 ' 1 cos y = 2, -= i / 2.'. sin y -- ~x/5...sin y -= V5. FIG. 88. FIG. 89. Then sin (sin-' ~ + cos-1 -) = sin (x + y) = sin x cos y + cos x sin y =. 2 + 1A/3 * /r = (2+ /15), Ans. Ex. 2. Prove that sin- a + cos- a = 2 Using the method of Ex. 1, show that sin (sin-l a + cos-' a) = 1 = sin 2. Ex. 3. Show that tan-l a + t tn-l b =an-1 a+b b 1 -ab Let x= tan- a..'. a = tan, y = tan-l b. a ~~/ bXa~ flab.'. b = tan y. But /w_-1 _ -1 tan(x+y)= tanx+ta FIG. 90. FIG. 91. 1 - tan x tan y INVERSE TRIGONOMETRIC FUNCTIONS 155.. tan (tan-' a + tan-' b) = -a + b or tan-l a + tan-' b = tan-' a + b 1 - ab' 1- ab 114. Solution of Trigonometric Equations by Use of Inverse Trigonometric Functions. It is sometimes useful to express the answer obtained by solving a trigonometric equation in terms of an inverse function. Ex. Solve 6 cos2 x-cos x = 2. Factoring, (2 cos x + 1)(3 cos x - 2) = 0..'. cs X = - 2.'. x = cos- (- ), Cos-, Ans. EXERCISE 46 If the pupil has any difficulty in grasping any one of the following problems, it will be well for him to translate the symbols of the problem into general language before attempting the solution. Thus Ex. 2 would read " find the cosine of the angle whose cotangent is 3," and might be written in the form "find cos Z cot " (see Art. 110). Express the following angles first in degrees and then in radians: 1. cos-1 V/2, tan-1 /3, sin- 1 sec- V2, csc — 3, cot- 3 cos-1 -, sec- 2, sin — A/3, cot-I 3, tan-1 /3. Find the value of: 2. cos (cot-l I). 3. tan (sin-l5 15). 4. see (tan-l 8,). 5. sin (cot-1 a). 6. cot(cos-1 a). b 7. tan (2 sin-l 2). Show that: 8. sin (2 tan —l 5). 9. cos(2 sec-l 7). 10. sin (- cos-' ). 11. cot ( — tan-1 5-). 12. sin (3 sin-'1 ) 13. sin (sin-'1 - cos-' 2). 14. tan (tan-' 2 + cot-' 3). + 1 - 1 2 t 15. tan-i +tan-11 3 4. 16. tan-12 +tan-l 1 2 2 a34n 22 17. sin-l' -- + sin-1 3- = sin-l1. 18. cos-1 3 + cos-1 53= Cos-l (- 1 -19. tan-l ' + tan 8 = tan-l 7 20. cot-1 a + cot-l b = cot-l ab - 1 b +a 156 TRIGONOMETRY Prove that: 21. sin (sin-' 4 + cot-1 4)= 1. 22. (cos-1 15 + tan-' ) = s1in-1 7 2x 23. sin (2 tanT' x) = 2. 1+ X2 24. sin-l x cot-l1 — x2 25. cos-1 a - cos-l b = cos-1 (ab + -<1 - a2 - b2 + a2b2). 26. 3 cos-l x = cos-1 (4 x3 - 3 x). 27. 3 sin-l x = sin-' (3 x- 4 xs). 28. tan-l a - tan-l b = 1 + ab 29. sin-' a + sin-l b = cos-l(-l1 - a2- b 2 - ab). Express the value of each of the following in its most general form: 30. sin-' 1 35. cos-1 V/3. 31. tan-' 1 3. 36. tan-' o. 32. cos-1- V2. 37. cot-' V3. 33. cot-l 1/3. 38. sec-1 2. 34. sin-'1 V/3. 39. sin-l( —). 2a 40. Prove that tan (2 tan-l a) =1 - a' 2a 41. Prove sin (2 tan-l a) = 1 a42. If cos- x == 2 cos-l x, find x. 43. Express the following angles in the inverse notation: 30~, 60~, 90~, 45~, 0~; n 180~, n 90~. Can each of these angles be expressed in more than one way in the inverse notation? 44. Who first, and at what time, brought inverse circular functions into use in their present form (see p. 173)? 45. At what time did the circular method of measuring angles come into use (see p. 167)? CHAPTER X COMPUTATION OF TABLES TRIGONOMETRIC SERIES sin x tan x 115. Limiting values of sn and tan It is important sinx tan x to determine the values which -- and x approach when X x x'0, x being the value of an angle expressed in circular measure (radians). p Take any angle A OP (Fig. 92) less than 90~ and denote it by x; construct the angle AOP' equal to o -- M A- T AOP, and draw the tangents PT and P'T. These tangents will meet at i on OA produced. Draw PP'. p Then OT is L to PP' at its middle FIG. 92. point A. By geometry, arc PP'> chord PP'; also are PP'< P T+ P'T. Hence arc PA > P/, and arc PA < PT. arc PA PM are PA PI' > >o and < OP >OP- OP OP.. x > sin x, and x < tan x. x x 1.'. — > 1 and < — sin x sin x cos x sin x.. cesx< < 1. 157 158 TRIGONOMETRY sin x sin x As x 0, cos x 1, hence - =n 1, since x lies between x x cos x and 1. Hence as x ' 0, limit (sin = 1. \ x This result may also be stated thus, as x 0, sin x x. Ao tan x sin x _ sin x\ ( \ Also -- -- ) = x x cos x x cos sinx. 1. But as x= 0, 1, and or. x cos x 1 Hence tn x I x 1, or 1. an x Or, as x 0, limit (t 1 x arc AP Since the number of radians in x= A, it follows OA that as the angle x = 0, the number of radians in x sin x, and also - tan x. In practical work, when x<2~, sinx and tanx may be taken as = p without appreciable error. 116. Computation of the Tables of Trigonometric Functions. Since, as x 0, sinx and x approach equality (Art. 115), the circular measure of a small angle is the same as the sine of that angle to a large number of decimal places. By the use of methods which are beyond the scope of this book it is found that the value of sin 1' and the circular measure of 1' coincide for the first fourteen decimal places. Hence in constructing tables which are to be correct for the first five decimal places, there will be no error in taking sin 1' = 1' (in radians). But, by Art. 101, '= 3141592+ radians =.0002908882+ radians. 180 x 60 Hence sin 1'=.0002908882+. COMPUTATION OF TABLES 159 But cos 1'= '1- sin2 1'= /1- (.0002908882+)2' =.9999999577+. sin 2'= 2 sin 1' cos 1'= 2 x (.0002909-)(.9999999577+) =.000582+. sin 3' = sin (2' + 1') = sin 2' cos 1' + cos 2' sin 1'. From this the value of sin 3' may be computed. In like manner the sines of all angles less than 90~ may be obtained. The cosines of these angles may be obtained similarly, or by use of the formula cos x = sin (90~-x). The tangents of these angles may be computed by the use sin x of the formula tan x si. To obtain the cotangents, the COS x formula cot x tan (90~- x) may be used. The above method of computing sines and cosines may be abbreviated thus: sin (x + y) + sin (x - y) = 2 sin x cos y. (Art. 71) Let x= a + 2 b, and y= b. Then, by substitution, sin (a + 3 b) + sin (a + b) = 2 sin (a + 2 b) cos b. Whence sin (a+ 3 b)= 2 sin (a+2b) cosb-sin(a+b)... (1) In like manner, cos (a + 3 b) = 2 cos (a + 2 b) cos b - cos (a + b).. (2) Let b= 1' in (1) and (2). sin (a + 3') = 2 sin (a + 2') cos 1'- sin (a + 1')... (3) cos (a + 3') =2 cos (a + 2') cos 1' -cos (a + 1')... (4) Letting a=- 1', 0, 1', 2',... in succession, we obtain from (3) sin 2'= 2 sin 1' cos 1'. sin 3' = 2 sin 2' cos 1'- sin 1'. sin 4' = 2 sin 3' cos 1' - sin 2', etc. 160 TRIGONOMETRY Similarly from (4), cos 2'= 2 cos 1' -1. cos 3'= 2 cos 2' cos 1'- cos 1'. cos 4' = 2 cos 3' cos 1'- cos 2', etc. 117. Computation by the Use of Series. The computation of the numerical values of the trigonometric functions is, however, performed much more expeditiously by the use of certain trigonometric series than by the above method. The demonstration of these series lies beyond the scope of this work. The series are as follows: 3 5 7 X" r r sin x -3- 5 7 X2 X4 X6 coS x = 1- - +- - + L2 4 16 x3 2 x5 17 x3 tanx=x+= +- + + 315 3 15 315 The student is aided in recalling these series by the fact that sin (-x) = -sin x (Art. 63); hence sin x must equal a series composed of odd powers of x. The same is true of tan x. But since cos (- x) = cos x, cos x must equal a series composed of even powers of x. 118. Analytical Trigonometry. Theory of Functions. When trigonometry is treated in the way indicated in certain preceding articles, it ceases to be merely an instrument for solving triangles and becomes the theory of quantities varying in certain periodic or rhythmic ways. Also by the use of the so-called imaginary quantities, the subject of trigonometry is still further extended. Thus, for instance, denoting - 1 by the symbol i, it is shown that (cos x + i sin x)n = cos nx + i sin nx (called De Moivre's Theorem). COMPUTATION OF TABLES 161 By the aid of this theorem and similar principles, trigonometry gains much additional power. This branch of the subject is termed analytical trigonometry (though it is sometimes treated as a part of higher algebra). When trigonometry is extended in these various ways, it is also looked upon as a part of the larger subject, the theory of functions. EXERCISE 47 1. By use of De Moivre's Theorem obtain the formulas for sin 3 x and cos 3 x. By use of this theorem we obtain But (cos x +'i sin x)3 = cos 3 x + i sin 3 x. (cos x + i sin x)3 = cos3 x + 3 i sin x cos x + 3 i2 sin2 x cos x+ i' cos3 x...cos 3 x + i sin 3 x = cos3 x - 3 sin2 x cos x i (3 cos2 sin x - sin3 x). By a theorem of algebra, in an identical equation containing both real and imaginary quantities, the sum of the reals in one member is equal to the sum of the reals in the other member, and so with imaginaries. Hence, cos 3 x = os x - 3 si2 x cos x = 4 cos3 x — 3 cos x sin 3 x = 3 cos2 x sin x - sin x = 3 sin x -4 sin3 x. In like manner, by De Moivre's Theorem, prove: f sin 4 = 2 sin 2 x (1 - 2 sin2 x), cos 4 x = 8 cos4 - 8 cos2 x +. f sin 5 x = 16 sin5 x -20 sin3x + 5 sin x, 3 cos 5 =16 cos x — 20 cos3 +5 cos x. 4. sin 7 x = 7 sin x- 56 sin3 x + 112 sin5 x- 64 sin7 x. n(n - 1) 5. cos nx = cosn- 2 cosUn-2 sill2 n(n - 1)(.7 - 2)( - 3) o + ]4 +. - cos)4 an sin4 x +.... 6. sin nx = n cosn-1 x sin x - n(n - 1 cos"-3 sin3 x 13 n (n-1) (n -2) (n -3) (n -4) cOsn-5 si x - 15 2 tanx 7. tan 2x= 2tan 1 - tan2 x 8. Find the value of sin 225~ by use of the formula for sin 5 x in Ex. 3. CHAPTER XI HISTORY OF TRIGONOMETRY 119. Epochs in the History of Trigonometry. The beginnings, or germs, of Trigonometry are found in the Rhind Papyrus, now preserved in the British Museum. This papyrus, the oldest known mathematical document, was written by a scribe named Ahmes about 1400 B.C., and is a copy, so the writer states, of a more ancient work, dating, say, 3000 B.C., or several centuries before the time of Moses. In dealing with pyramids, Ahmes makes use of two of the trigonometrical ratios, viz.: that between a lateral edge of a pyramid and diagonal of the base, corresponding to the cosine of an angle; and another which corresponds to the trigonometrical tangent of the angle made by the lateral face of a pyramid with the plane of the base. This use of ratios is, however, too crude to be regarded as scientific trigonometry. We have the following principal epochs in the scientific development of Trigonometry: 1. Greek (at Island of Rhodes and Alexandria), 150 B.c.200 A.D. 2. Arab (in western Asia and in Spain), 650 A.D.-1200 A.D. 3. Hind o, 450 A.D.-1100 A.D. 4. European, 1200 A.D. - We shall also find the three following principal stages in the development of trigonometry: I. (150 B.c.-1400 A.D.) Spherical Trigonometry studied as a part of Astronomy, with incidental use of Plane Trigonometry. 162 HISTORY OF TRIGONOMETRY 163 II. (1400 A.D.-1700 A. D.) Plane and Spherical Trigonometry studied as a part of Geometry. III. (1700 A.D.- ) Trigonometry as an independent science. PRINCIPAL MAKERS OF TRIGONOMETRY 120. Hipparchus. The founder of trigonometry as a science was Hipparchus, a Greek, born about 180 B.C. ir Bithynia in the northern part of Asia Minor. Hipparchus studied at Alexandria and afterward retired to the Island of Rhodes, where he did his principal work. He was primarily an astronomer and determined, for instance, the length of the year to within six minutes. He created trigonometry as a tool or aid in his astronomical work. Hence the trigonometry used by him was almost exclusively spherical. 121. P; lemy (87 A.D.-165 A.D.). The next great name in the history of trigonometry is that of Ptolemy, also a Greek. He lived and did his work in Egypt at Alexandria. Like Hipparchus, Ptolemy was primarily an astronomer and used trigonometry merely as an aid in his astronomical investigations. He wrote a treatise on mathematical and astronomical topics, now known as the Almagest,* which was the standard authority in astronomy for 1200 years. The Almagest contains thirteen books, the first of which treats mainly of trigonometry. 122. Regiomontanus (or Johann Miller, 1436-1476 A.D.) was a German and studied at the University of Vienna. After doing important work in Germany he was called to Rome by the Pope to reform the calendar and was assassinated while in that city. The ephemerides calculated by X Ptolemy entitled his work jLueyio-rq7 uaOauartK' v7-wrtaTts, or " Greatest Mathematical Collection." The book was translated by the Arabs into their language and used by them as a text-book. The name Almagest comes from a blending of the Arabic article " al" (the) with the Greek word gfeCtir- (greatest). 164 TRIGONOMETRY Regiomontanus were used by Columbus in crossing the Atlantic. Regiomontanus wrote a text-book entitled De Triangulis, in which he freed the subject of trigonometry from its astronomical bondage. Though he made trigonometry a part of geometry, he presented the subject essentially in the form in which it is customary even yet to make a first presentation of it to pupils. Several other Germans, as Pitiscus, Rheticus, and several French and English mathematicians made important contributions to the development of trigonometry, but the thinker who first put the subject on a firm modern basis was 123. Euler (1707-1783), born in Basle, Switzerland. Euler's life as a scientific worker was spent mainly at St. Petersburg and Berlin. Through his writings and influence trigonometry was established as an independent science. Since Euler, a large number of mathematicians have made contributions to trigonometry in the larger sense, that is, considered as a branch of the theory of functions, which has been mentioned merely in an incidental way in this book. HISTORY OF TRIGONOMETRICAL FUNCTIONS AND THEIR NOTATION 124. Sine. During all the early history of trigonometry, the trigonometric functions were regarded as lines, not as ratios. Hipparchus (120 B.c.) used but one trigonometric function. This was the chord subtended o0 Aby double the angle, and it therefore corresponded in a general way to the sine of an angle. Thus, the angle AOP was regarded as determined by FIG. 93. the chord PQ. Ptolemy (150 A.D.) treated angles by the same method as Hipparchus, that is, by use of the chord of the double angle. This method introduced unnecessary labor in two ways: first, it made it necessary to double each angle dealt with, in HISTORY OF TRIGONOMETRY 165 order to get the required chord; second, it made it necessary to divide by two each angle obtained as the result of a process. The Hindoos regarded an angle as determined by the semichord of twice the angle; thus by them in the above figure the angle AOP would be regarded as determined by PR. This is the method which is used at present when the sine is regarded as a line. The Arabs also determined the angle by the semichord of twice the angle, one of their writers remarking that the use of the semichord C saves the continual doubling" mentioned above. Rheticus (Germany, 1514-1.576) was the first to consider the right triangle OPR as independent of any arc or circle. He defined the trigonometric functions as ratios of the sides of the right triangle, but this improvement was not adopted by other mathematicians until the time of Euler. Euler also defined the sine and other trigonometric functions as ratios between the sides of a right triangle. He was thus able to make them functions of the angle only and to treat them as pure numbers. In this way, trigonoinetry became an independent science. 125. Other Functions. The Egyptians used the cosine and cotangent, in effect. Hero, of Alexandria (110 B.c.), in effect, used a table of cotangents by which to determine the areas of regular polygons. The Hindoos used the versed sine and cosine as well as the sine. The Arabs invented the tangent, cotangent, and secant, though these functions were afterward neglected and reinvented in Europe. Regiomontanus rediscovered the tangent and cotangent. Rheticus, using the simple right triangle, had the secant and cosecant suggested to him by it. 166 TRIGONOMETRY 126. Notation of the Trigonometric Functions. The Egyptians used the word segt for both the ratios employed by them (cosine and tangent). The Hindoos called the chord jiva; the semi-chord, or sine, ardhajya, and later, jiva also; the cosine they termed katijya, and the versed sine utkramajya. The Hindoo word for sine, jiva, the Arabs transliterated as jiba, which resembled an Arabic word, jaib, meaning an indentation or gulf. The Arabs in time substituted the latter familiar word for the former artificial one. Hence, when the Arabic mathematical works were translated into Latin, the term jaib was designated by the Latin word sinus (which means " gulf "). Later, Rheticus, in his use of the right triangle, termed the sine the perpendicular, and the cosine the basis. By others the cosine was sometimes termed the sinus rectus secundus, and sometimes the complementi sinus. Gunter (England, 1580-1626) was the first to use the word cosine, which he obtained by contracting the words " complementi sinus." The Arabs called the tangent umbra, and the secant, diameter umbrae, as a result of their use of these functions in connection with the shadows of tall objects. Later in Europe the tangent was sometimes spoken of as the umbra recta, and the cotangent as the umbra versa. The words tangent and secant for the corresponding trigonometric functions were first used by Thomas Finck (Denmark, 1583). Gunter, who invented the word cosine, also invented the word cotangent. Girard (Holland, 1590-1633) was the first to use the abbreviations sin, tan, sec, etc. These abbreviations, however, were not generally accepted till they were taken up (1748) by Simpson in England and Euler in Germany. HISTORY OF TRIGONOMETRY 167 HISTORY OF TRIGONOMETRICAL TABLES 127. History of Methods of Measuring Angles. The division of the circumference of a circle into 360 degrees, each degree into 60 minutes, and each minute into 60 seconds, is due to the Babylonians. This system of angular measurement was transmitted from the Babylonians to the Greeks, Hindoos, and Arabs. The terms minutes and seconds are derived from their Latin names which were in full "partes minutse primne" and "partes minutae secundve." This so-called sexagesimal notation also came to be applied to other lines and quantities than the circumference of a circle as we shall see later. The Hindoos developed the Babylonian sexagesimal method into a rude form of the circular method of measuring angles (see Art. 128). The circular method in its present form (use of radians, etc.) came into use in the early part of the eighteenth century. The inventors of the metric system of weights and measures at the time of the French Revolution proposed to divide the right angle into 100 equal parts called "grades," and to subdivide the grade decimally, but this system never came into practical use. At present the custom of dividing a right angle into 90 degrees, and then dividing each degree decimally (instead of into minutes and seconds), is growing in favor. 128. Notation used in Trigonometric Tables. As decimal fractions in their present form are a comparatively modern invention, in the early history of Trigonometry the values of the trigonometrical functions were necessarily expressed in some other way. Thus the Greeks used sexagesimal fractions in expressing the lengths of the lines which were their trigonometrical functions. Ptolemy divided the diameter of the circle into 120 equal parts, each of these parts into 60 minutes, and each minute into 60 seconds. For instance, where we would write sine 18=.3090, Ptolemy wrote chord 36~ = 37~ 4' 55". The Hindoos divided the radius of the circle into 3148 168 TRIGONOMETRY equal parts, 3148 being the number of minutes in an are equal to the radius. Hence the Hindoos made an approach to the circular measure for angles, the number denoting the radius, however, in their use of the relations being determined by the angle rather than the unit angle by the radius. Regiomontanus in forming his tables first used a radius of 600,000, but later he used a purely decimal scale, 10,000,000 being the radius. Hence his work may be regarded as a transition from the sexagesimal to the decimal scale. 129. Computation of Trigonometrical Tables. Hipparchus (120 B.c.) computed a table of chords for different angles. This table, however, has been lost. Ptolemy in his Almagest gives a table of chords (computed in sexagesimal fractions carried out to a point equivalent to 5 decimal places) for every 2~ of the quadrant, the table being remarkably accurate. Hero of Alexandria (110 B.C.) gives a table of cotangents calculated for cot(2-) when n 3, 4,... 12. The Hindoos (530 A.D.) computed a table of sines for every 33~ of the quadrant. The Arabs (Bagdad, 980 A.D.) formed a table of sines for every 1o2 and also a table of tangents and cotangents. The printing press was invented about the year 1450. Shortly afterward the Germans took up the problem of computing very full and exact trigonometric tables, and to their industry we owe our tables essentially in their present form. Peuerbach (1423-1461), teacher of Regiomrontanus, computed a table of sines for every 10' with 600,000 as a radius (i.e. six-place tables). Regiomontanus constructed a table of sines with 6,000,000 and another with 10,000,000 as the radius. Regiomontanus also constructed a table of tangents for every 1' with 100,000 as a radius. HISTORY OF TRIGONOMETRY 169 Apian (1495-1552) made a table of sines for every 1' with a radius equal to 100,000. Rheticus computed tables of sines, tangents, and secants for every 10" with radius equal to 10,000,000,000; and later a table of sines with radius equal to 10'. He began tables of tangents and secants on the same scale, but died before completing them. In this work he employed several computers for twelve years and spent large sums of money. When completed by his pupil, Otho, and published, these tables made a volume of 1468 pages. Pitiscus (1561-1613) computed tables of sines, tangents, secants, cosines, cotangents, cosecants, with radius equal to 1025. By annexing tables of proportional parts, he facilitated interpolations. It is to be remembered that each time we use trigonometric tables we use again the labor of these indefatigable workers; or, to put it another way, by a species of kindly foresight on the part of these men we find a large part of our work already done for us by them. Lord Napier of Scotland published his invention of logarithms in 1614. Immediately upon this invention, logarithmic tables of sines, cosines, tangents, and cotangents were formed. These tables were printed in 1633. 130. Methods of Computing Trigonometric Tables. Hipparchus and Ptolemy in constructing their tables of chords used the theorem of geometry which reads; If a quadrilateral be inscribed in a circle, A the product of the diagonals equals the sum of the products of the oppo- site sides;" i.e. (Fig. 94) AC x BD = BC x AD- +CD x AB. By means of this theorem, if the chords of two FiG. 94. arcs are known (as of 45~, 30~), the chords of the sum and of the difference of those arcs (i.e. of 75~ and 15~) can be corn 170 TRIGONOMETRY puted. Hence the theorem in a rough way is equivalent to the trigonometrical formulas for sin (A ~ B) and cos (A ~ B) (Art. 71). The theorem was also applied by Ptolemy to the problem of finding the chord of half an arc when the chord of the whole arc was known. Both the Hindoos and Germans in computing their tables of trigonometric functions used methods which were essentially the same as those given in Art. 116. As has been said, much more expeditious methods are now at the service of the computer, and these methods have been used in verifying and correcting the tables as at first computed. SOLUTION OF TRIANGLES 131. Greeks (see Ptolemy's Almagest, Book 1) made spherical trigonometry primary and fundamental. Plane trigonometry was developed only as a part or detail of spherical trigonometry. The methods of solving spherical triangles used by the Greeks were mainly geometrical and comparatively awkward. These methods are derived from the principles of projection, and when applied to right spherical triangles become equivalent to four of the ten formulas which are included in Napier's Rule for Circular Parts. In plane trigonometry, as treated by the Greeks, a right triangle was solved by inscribing the triangle in a circle. An oblique triangle was solved by resolving it into right triangles. The fundamental principle in the solution of plane oblique triangles, viz. that the sides are to each other as the double chords of double the angles opposite (i.e. as sines of angles opposite) was used implicitly by Ptolemy, but was not stated by him in so many words. In one of the examples solved in the Almagest, three sides of an oblique triangle are given, and the triangle is solved by finding the segments of one of the sides made by a perpendicular on it from the opposite vertex. HISTORY OF TRIGONOMETRY 171 To show how spherical trigonometry led the Greeks to plane trigonometry, we may mention one of the problems occurring in their treatment of the former subject, viz: To divide a given arc into two parts so that the chords of the doubles of those arcs shall have a given ratio. Stated in terms of modern notation this problem is, Given x +-y sin x c a given angle (j), to find x and y so that -in. Stated with refersin y b ence to the triangle ABC, this problem becomes one in Case II of oblique plane triangles; for / C =180~ - (x + y) = 180~ -j, A = x, ZB=y; BC=a; AC=b.. The Hindoos, like the Greeks, made use of trigonometry only as an aid in the study of astronomy. They solved both plane and spherical triangles, but treated plane trigonometry as a mere detail of spherical trigonometry. 132. The Arabs also gave spherical trigonometry the leading place in the study of the subject. They simplified Ptolemy's method of solving spherical triangles, discovered cos a -cos b cos c that in spherical triangles cosA= A b - s, and to sin b sin c the four of the ten formulas included in Napier's Rule for Circular Parts, which Ptolemy had implicitly known, added two others, viz.: cos B = cos b sin A, cos c = cot A cot B. The Arabs, however, developed no general theory for the solution of plane or spherical triangles. 133. Regiomontanus separated plane from spherical trigonometry and made plane trigonometry primary. In his treatise he begins with the right triangle, solves it by using the sine function only, and then solves equilateral and isosceles triangles by resolving them into right triangles. He also solves oblique triangles much as is done at present. His treatment of spherical trigonometry, however, is far less general and satisfactory. Romanus (Belgium, 1561-1625) condensed the twenty-six cases of spherical trigonometry then in use into six cases. 172 TRIGONOMETRY 134. Lord Napier (Scotland, 1550-1617) reduced the solution of right spherical triangles to ideal simplicity by his Rule for Circular Parts. This has been commended as perhaps "the happiest example of artificial memory that is known." He also simplified the solution of oblique spherical triangles by his discovery of the formulas known as Napier's Analogies. 135. Notation of Triangles. To Euler is due the method of denoting the angles of a triangle by the capital letters A, B, C, and the sides opposite by the small letters a, b, c. 136. The theory of the complete spherical triangle, that is, of the triangle in which the length of the sides is not necessarily less than 180~, was developed by Gauss (Germany, 1777-1855) and Moebius (Germany, 1790-1868), but such triangles are not much used in practice. 137. Spheroidal trigonometry, that is, the theory of triangles on the surface of a spheroid has great practical importance because of its use in surveying large portions of the earth's surface, as in the coast and geodetic surveys in different countries. DEVELOPMENT OF GONIOMETRY 138. Greeks. As has been stated (Art. 130), the geometrical methods used by the Greeks in constructing tables of chords were in a rough way equivalent to a use of the formulas for sin (A ~ B), cos (A ~ B), and sin i A. 139. The Hindoos knew the identical equation sin A + cosA 1. They also used the formula sin ~ A= V1719(3438-cos A), where 3438 is the radius of the circle. This is equivalent to the formula sin - A = \- A 22 HISTORY OF TRIGONOMETRY 173 In computing trigonometric tables they appear to have used the formula sin (n + 1) a - sin na = sin na - sin (n - 1) a - sin na cosec a. This formula is not quite accurate and was probably arrived at inductively. 140. The Arabs knew the relations tan sin cot cos cos sin and were also able to solve an equation like tan = a, obtaining sin = a Vl+a2 141. Rheticus obtained the formulas sin 2 A = 2 sin A cos A, sin 3 A = 3 sin A - 4 sinlA. Romnanus discovered the formula for sin (A + B). The formulas for sin (A - B) and cos (A ~ B) were published by Pitiscus (1599). 142. Vieta (France, 1540-1603) gave the general formulas for sin nA and cos nA in terms of sin A and cos A. OTHER PROCESSES 143. Trigonometrical Series. The series for sin x and cos x in terms of powers of x and for sin-' x in terms of sin x were known to Sir Isaac Newton before the year 1669. Those for tan x and sec x in teruns of powers of x and for tan- x in terms of powers of tan x were discovered by Gregory (England, 1638-1675) in 1670. 144. Inverse Circular Functions in their general form were introduced by John Bernouilli (1667-1748). 174 TRIGONOMETRY 145. Use of V-1 or i. John Bernouilli first treated trigonometry as a branch of analysis. Among other algebraic methods he introduced the use of V-1, or i, into trigonometry and obtained real results by its use. For instance, by employing V- 1 he obtained a series for tan no in term of powers of tan f. This use of i was followed up by Euler, who among other results obtained the formula (sin x + i cos x) -= sin nx + i cos nx known as De Moivre's Theorem. EXERCISE 48. GENERAL REVIEW 1. Simplify log2 4 + 5 log1 9 + 1 logI0.1 - logIoV001. 2. Compute the value of x from the equation 5 x3-= i/.2784. 3. Also from cos x=(.9387)2. (7.605)3/14.82 4. Also from tan x =(7.605) —14.82 (27.32)i 5. If x is an angle in the first quadrant and cos x=- 8, find the sin x + tan x value of cos x - cot x 6. If x is an angle in the first quadrant and 2 cos x = 2 - sin x, find the value of tan x. 7. If tan x = - find sin 2 x. b 8. If sin y = a and tan y -= b, prove that (1 - a2) (1 + b2)=. 9. ABCD is a square. D is joined to E, the midpoint of AB. Find the trigonometric ratios of Z ECD. 10. Determine the numerical value of sin 18~ by use of the geometric method of inscribing a regular decagon in a circle. 11. If A is an angle in the first quadrant and tan A=, find the value of p cos A — q sin A 2 cos A + q sin A 12. Which of the following statements are possible and which impossible: (1) 16 sin x 1. (2) 4see 0 1. (3) 7tany= 30. GENERAL REVIEW EXERCISE 175 13. Prove that sec x + tan x -= se2 sec x tan x. tan x + sec x vers2 2 sin x 14. Prove that v — 2 sin x. sin x 1 + cos x 15. Find the numerical value of 3 tan 30~ sec3 60~ sin2 90~ tan245~ + 5 cos 90~. 16. If tan2 45~ - cos2 60~ = y sin 45~ cos 45~ tan 60~, find y. cos2 7r see 7 tan r 17. If xsinmrcos2= 6 3 4 find x. 6 4 7r csc2 - cos - 4 6 Solve each of the following right triangles, given: 18. A = 36~ 18' 6" [36.3~], b = 217.9 ft. 19. b = 315.92 ft., c = 814.23 ft. 21. B = 12~ 15' [12.25], c = 1001.4. 20. c = 900, b = 887. 22. A = 1~ 20' [1.33~], c =872.56. 23. In a right triangle b = 426, A = 38~ 45' [38.75~]. Find a + c and the area. 24. The hypotenuse of a right triangle is 5 ft. and one angle of the triangle is 30~. Solve the triangle and find the area without the use of tables. 25. The area of a regular polygon of 11 sides is 80. Find the side, radius, and apotheni of the polygon. 26. In an isosceles triangle the leg is 21.7 and the area 32.51. Solve the triangle. 27. The legs of a right triangle are to each other as 5: 9. Find the angles of the triangle. 28. On the steepest part of the Mt. Washington railway (Jacob's Ladder), there is a rise of 13- inches for every 3 ft. of track. What angle does the track make with the horizontal? At this rate what would be the rise in one mile of track? Show that in a right triangle: b2 — _a 3 ac2 - a3 29. cos2A= a 30. sin3 A=32 a 31. (sin A - sin B)2 + (cos A + cos B)2 = 2. 176 TRIGONOMETRY 32. Find the other trigonometric functions of A, when cos A = - and A lies between 540~ and 630~. 33. Given sec x =- - and x in the third quadrant, find the value of sin x + tan x l-os x - cot x 34. Find the trigonometric functions of 180~ +x and of 270~- x when tan x =. 35. For what values of x between 0~ and 360~ is sin x- cos x positive, and for what values is it negative? 36. Find the nunerical value of 3 sin2 225~ + 4 sin (- 120~) tan 150~ - ~ cos2 330~ cot 750~ + 5 sin2 180~. 37. For each of the following angles state which of the three principal trigonometric ratios are positive: (1) 460~. (2) -220~. (3) -1200~. (4) 1 6 38. Trace the changes in sign and magnitude of sin A between 0~ and 360~. csc A between 0 and r. cos x between Or and 2 7r. tan A between - 90~ and - 270~. 39. If A is in the third quadrant and tan A= -5, find the value of sin 2 A. 40. Express the cosine of an angle in the second quadrant in terms of (a) each of the other trigonometric functions of the given angle, (b) the cosine of the complement of the angle. 41. If sin A 1 and sin B and A and B are both acute, find the numerical value of tan (A + B); also of tan (A - B). 42. If x is an angle in the second quadrant and sin x =, find the value of sin 2 x + cos 2 x. 20 ( 50 43. Express 2 cos - cos - as a sum or difference. 3 3 44. If sin 1 x =, find the numerical value of cos x. Also of tan x. Prove that: 45. sin (A + B) - sin2 (A- B) = sin 2 A sin 2 B. sin 4 x + sin 3 x. 46. s2 47. sin 50 + sin 10= sin 70~. cos3 x- cos 4 x GENERAL REVIEW EXERCISE 48. sin215~ + cos215~ =1. 49. cos 55~ + sin 25~ = sin 850. 50. sin A + sin 2 A + sin 3 A n 2 A. 50. = tan 2 A. cos A + cos 2 A + cos 3 A 51. - tan2 (450 - 1 + tan2 (45~ - x) 52 in. 27 4+)- sil sin 0 +Y -Sin2 0 Solve each of the following oblique triangles, given: 53. A = 30~ 18' 12" [39.3~], b = 3294, c = 2846. 54. A = 76~ 24' 36" [76.41~], B = 48~ 42' [48.7~], c = 1012. 55. a = 850, b - 760, c = 590. 56. B = 46~ 18' [46.3~], b = 213.76, a = 192.72..57. b = 927, A = 79~, B = 21~ 17' 12" [21.29~]. 58. a = /V3, b = V2, c = V5. 59: A = 51~ 30' [51.5~], a = 294.6, b = 301.7. 60. a = 926.8, b = 842.5, C= 46~ 27' [46.45~]. 61. Solve the triangle in which K=-20.602, a = 214.2, and b = 315.8. 62. The diagonals of a parallelogram are 347 and 264 ft., and the area of the parallelogram is 40.437 sq. ft. Find the sides and angles of the parallelogram. 63. The diagonals of a quadrilateral are 34 and 56, and they intersect at an angle of 67~. Find the area of the quadrilateral. Solve the following equations for answers not greater than 360~ or less than 0~: 64. sec x + tan x = ~ 3. 67. 2 sin x sin 3 x - sin 2 x = 0. 65. sec2 x + cot2 x= -. 3 68. sin 2 0 + sin 0= cos 2 cos. 66. sin 2 x = V-3 cos x. 69. sin 2 y + -/3 cos 2 y = 1. 70. sin (60~ - x) - sin (60~ + x) = ~V3. 71. Give the answers to Exs. 64-70, in the unlimited form. 178 TRIGONOMETRY 72. If 2 cos2 x - 7 cos x + 3 = 0, show that there is only one value for cos X. 73. Find the least possible positive value of 0 which will satisfy the equation 2/3 coss 0 = sin 0. 74. Solve sin x + sin 2 x + sin 3 x -1 + cos x + cos 2 x. 75. If sin 3 x + sin 2 x = sin x, find tan x. 76. Find the length of an arc intercepted by an angle of 2.2 radians at the center of a circle whose radius is 5 ft. How many degrees in this angle? 77. Two angles of a triangle are.5 and.4 radians. Find the third angle in radians and in degrees. 78. The sum of two angles is 2 radians, their difference is 10~. How many radians are there in each of these angles? 79. Prove cos (3s + i- cos (- r - X 2 sin x. 2 2 80. Find the numerical value of 3sin2 r + 4 cos2 57r - tan' -7 2 6 4 3 4 81. If sin ( + sin x — find x. 82. Simplify tan (7 x) - tan 3- + 83. An angle of 30~ at the center of a circle subtends an arc AB of length 7 ft. Find the length of the perpendicular dropped from A on BC. 84. Express each of the following angles in degrees: sin-'1; cos-1 V2; tan-(- 1); sin- (- 1); cos-1 (- -/3). 85. Find tan (cot-1'). 86. Prove that tan-12 + tan-' 1 = 7 -~1 2 87. Find the value of x, if tan-l x + 2 cot-l x = 3 ' 88. How many degrees in sin-' (- V2)? How many radians? 1 89. Prove sin-la = sec-1 I V1- a2 GENERAL REVIEW EXERCISE 179 90. Solve the following for x and y: sin-1 x + sin-' y = 120~. cos- x - cos-1 y = 60~. 91. At a point 50 ft. from the base of a tower the angle of elevation of the top of the tower was found by the use of a transit instrument to be 68~ 18' [68.3~]. If the height of the instrument above the ground was 4.75 ft., what was the height of the tower? 92. If the railway up Pike's Peak rises 7552 ft. in 8: mi., what angle does the railway make with the horizon on the average? 93. Two towers are 240 and 80 ft. high, respectively. From the foot of the second the angle of elevation of the top of the first is 60~. Find, without the use of tables, the angle of elevation of the second from the foot of the first. 94. An unknown force combined with one of 128 lb. produces a resultant force of 200 lb. The resultant makes an angle of 18~ 24' [18.4~] with the known force. Find the magnitude of the unknown force and the angle which it makes with the known force. 95. A tree 82 ft. high stands at one corner of a garden which is in the form of an equilateral triangle. The distance from the top of the tree to the midpoint of the opposite side of the garden is 112 ft. Find a side of the garden. 96. If the earth's radius (3956 mi.) as viewed from the sun subtends an angle of 8.8", find the distance of the earth from the sun. 97. In a circle whose radius is 13.7, find the area of a segment whose angle is radians. 98. In order to determine the breadth of a river, a base line of 500 yd. was measured on one shore, and at each end of the base line the angle included between the base line and a line to a rod on the other bank was measured. These angles were found to be 53~ and 79~ 12' [79.2~], respectively. What was the breadth of the river? 99. If a barn is 40 X 80 ft., and the pitch of the roof is 45~, find the length of the rafters and the area of the entire roof, the horizontal projection of the cornice being 1 ft. 10o. If the sun's angle of elevation is 60~, what angle must a stick make with the horizontal in order that its shadow on a horizontal plane may be the largest possible. 180 TRIGONOMETRY 101. If a railroad rises 1 ft. for every 1000 ft. of its length, what angle does it make with the horizontal? 102. In surveying a circular railroad curve successive chords of 100 ft. each are laid off. Find the radius of the curve, if the angle between two successive chords is 177~. 103. If the diagonal of a regular pentagon is 32.835, what is the radius of the circumscribed circle? 104. The angle x is in the third quadrant and cos x = -; find the value of csc x, tan x, sin ~ x, tan (180~ - x), and sin - x. 105. Find all the values of x between 0~ and 360~ which satisfy the equation sin (30~ - x)= cos (30~ + x). 106. If x is an angle in the second quadrant, prove geometrically that tan (270~ + x) = - cot x. 107. One angle of a rhombus is 60~ and the opposite diagonali is 5 inches. Without the use of tables find the sides of the rhombus and its area. 108. Give a general formula for all angles whose sine is ~. Is - Is -1. 109. Express cos 2 x in terms of each of the functions of x. 110. Express cos A cos B as a sum. 111. If cos A = h, and tan A = k, find the equation connecting h and k. 112. How many radians in each interior angle of a regular hexagon? In each exterior angle? How many degrees in each of these angles? 113. Prove that cos-l -3 +.2 tan- = sin-. 2 fn tan2 + cos2x 114. If sin x = find ta2 2 3 tan" x - cos x 115. In the isosceles right triangle ABC, D is the midpoint of AC. Prove without the use of tables that cot / ABD: cot L DBC =2: 3. 116. If 0 lies between 180~ and 270~, and 3 tan 0 =4, find the value of 2 cot 0 = - 5 cos 0 + sin 0. 117. Is it possible to have an angle whose tan is 503? Whose cos is 43? Whose secant is? Whose sine is 23? 118. Show that cos 80~ + cos 40~ - cos 20~ = 0. 119. That 2 sin (+ sin (- 7 = sin' x - cos2 x. \ y \ ~4/ GENERAL REVIEW EXERCISE 181 120. If sin (60 ~- x) - sin (60~ + x)= -V/3, find tan 2 x. 121. Express 2 sin 9 A sin A in the form of a sum or difference. 122. Find the value of sin-l- + 3tan-'13 - 2 cot- 1 + sec- 1, using values between 0~ and 90~. 123. If tan 2 x -24, find tan x and sin x, it being given that x is an angle in the third quadrant. 124. Find by inspection one value of x when cos.(100 + A) cos (10~ - A) + sin (10~ + A) sin (10~ - A) = cos x. 125. A surveyor standing on a bank of a river observes the angle subtended by a flagpole on the opposite bank to be 33 10' [33.17~] and when he retires 120 ft. from the bank he finds the angle to be 18~ 16' [18.27~]. Find the width of the river. 126. Develop cos (270~ - x - y) in the shortest way. 127. What is the angle of elevation of the sun when the length of the shadow of a pole is V3 times the height of the pole? 128. If tanA = - and sinB= 12, and A is in the third quadrant and B in the second, find sin (A + B), cos (A + B), tan (A + B). 129. At the Panama Canal the Gatun dam has three different slopes: the ratio of the horizontal to the vertical near the base is 16 to 1; in the middle of the dam this ratio is 8 to 1; and at the top the ratio is 4:1. What three different angles does the surface of the dam make with the horizontal? 130. If A is an angle in the first quadrant, and sec2A csc2 A- 4 0, find the numerical value of cot A. 131. If 0 is an angle in the third quadrant, and sec2 0 = 2 + 2 tan 0, find sin 0. 132. Find all the values of x between 0~ and 500~ which satisfy the equation tan (45~ - x) + cot (45 ~- x) = 4. 133. Graph y = sin-' x. 134. Also, y = tan-1 x. 135. From the top of a mountain 3 mi. high, the angle of depression of the horizon is 2~ 13' 50" [2.23~]. Hence determine the diameter of the earth. 136. Can an angle exist such that 9 sin 2 x + 3 sin x = 20? Why? 137. Find the numerical value of tan2 2r + cos2 7 + sin2 3 4 6 138. Find the sines of all angles less than 2r whose tangents are equal to cos 135~. 182 TRIGONOMETRY 139. Given cos 2 + x=a, find cot (- + 140. What is the most general value of x which satisfies both of the equations cot x = - 3 and csc x= - 2. 141. Show that 2 sin ll( + A)cos( + B)=cos (A + B) sin (A-B). 142. Find the length of a circular arc whose radius is 5 ft. and whose subtending angle is 3 units of circular measure. 143. In the triangle ABC, B is 45~, and C is 120~, and a is 40. Without the use of tables find the length of the perpendicular drawn from A to BC produced. 144. Prove that sin x + sin 2x tan x. 1 + cos x + cos 2 x 145. When y =, find the numerical value of sin2 y - cos2 y + 2 tan y - sec2 y. 146. Prove the identity sin-1 y + tan-' y = sin-l (1 + ) - v/1 + Y2 147. Is sin x- 2 cos x + 3 sin x - 6 = 0 a possible equation? 148. A vertical pole stands at the center of a circular millpond and rises 100 ft. above the surface of the water. From a point on the shore the angle of elevation of the top of the pole is 20~. Find the area of the pond. 149. When the planet Venus is most brilliant, its diameter subtends an angle of 40" as seen from the earth. If the diameter of the planet is 7600 mi., what is the distance of the planet from the earth at such a time? 150. Verify the statement 4 cot2 7r + 3 sin 2 csc22 7r 3 tan2 I = 10 3 6 3 3 4 6 3 151. Find the value of sin x, if tan ( + tan - ) + 2 =0. 152. What sign has sin x cos x for the following values of x: 1400, 278~, -356~, -1125? 153. If 1 + sin2 x = 3 sin x cos x, find tan x. 154. If i denotes the angle of incidence of a ray of light falling on water, and r the angle of refraction, and sln = 1.423, find r when i = 34.37~. sin r GENERAL REVIEW EXERCISE 183 a2 + b2 155. When is sin x = a possible, and when impossible? 2ab 156. Show that sin (2 a - f) cos (a - 2/3) - cos (2 a - /)) sin (a - 2 p) = sin (a +- ). '157. Solve sin 2 x - cos 2 x - sin x - cos x = 0. 158. Solve x = sin- -- + tan- 1. 159. Trace the changes in sign and magnitude of sin 30 as x incos 2 0 creases from 0 to. 2 160. Two trains leave a railroad crossing at the same time on straight tracks, including an angle of 21~ 12' (21.2~). If they travel at the rate of 40 and 50 mi. per hour respectively, how far apart will.they be in 45 min.? 161. Show that os 2 B +cos 2A = cot (A + B) cot (A -B). cos 2 B - cos 2 A 162. In a right triangle show that \/- +a — b 2sin ta-b a + b Vcos 2 B tan + A + tan -f A 163. Prove cse A. tan + A)-tan 7-| A 164. In any triangle prove that c = a cos B + b cos A, and hence show that sin (A + B) = sin A cos B + cos A sin B. 165. Determine the angles in a right triangle in which a>b, and c- = a-a -b. 166. Prove cos2 ( -) - 2 cos (x - y) cos x cos y - sin2 - cos2 y. 167. If sin x - cos x + 4 cos2 x = 2, find the ratio of tan x to sec x. 168. If A + B = 225~, prove that ( c ot A 1 cot B~c 2 l + cotA + cotB 2 169. The shadow of a tower is found to be 60 ft. larger when the sun's altitude is 30~ than when it is 45~. Find the height of the tower without the use of tables. 170. A workman is told to make a triangular enclosure having 50, 41, and 21 yd. as its sides. If he makes the first side one yard too long, of what length must he make the other two sides in order to inclose the required area, and keep the perimeter of the triangle unchanged? 171. If sin A is a geometric mean between sin B and cos B, prove cos 2 A = 2 sin (45~ - B) cos (45~ +B). 184 TRIGONOMETRY 172. If the diameter of the earth's orbit about the sun is 186,000,000 miles, and this diameter when viewed from the nearest fixed star subtends an angle of 1.52", find the distance of the star from the earth. 173. In a circle whose radius is 111.3 find the area inclosed between two parallel chords, on the same side of the center whose lengths are 129.3 and 97.4. a - a2 -11 -- b2 174. If 2 tan-1 x = c - cos - b2 find x. + a2 + b' 175. If tan2 (180 - x) - sec (180~ + x) = 5, find cos x. 176. In order to fix the distance between two islands C and D, a base line, AB, 900 ft. long, is measured on the shore. Also, Z BAC was found to be 110~ 50' [110.83~], Z DAB, 67~ 51' [67.85~], Z CBA, 49~ 51 [49.85~], Z ABD, 85~ 19' [85.32~]. What was the distance between the islands? LOGARITHMIC AND TRIGONOMETRIC TABLES EDITED BY FLETCHER DURELL, PH.D. HEAD OF THE MATHEMATICAL DEPARTMENT THE LAWRENCEVILLE SCHOOL NEW YORK CHARLES E. MERRILL CO. 44-60 EAST TWENTY-THIRD STREET 1911 Durell's Mathematical Series Plane Geometry 341 pages, 12mo, half leather..75 cents Solid Geometry 213 pages, 12mo, half leather... 75 cents Plane and Solid Geometry 514 pages, 12mo, half leather... $1.25 Plane Trigonometry 184 pages, 8vo, cloth.... 1.00 Plane Trigonometry and Tables 298 pages, 8vo, cloth... $1.25 Plane Trigonometry, with Surveying and Tables In preparation... Plane and Spherical Trigonometry, with Tables 351 pages, 8vo, cloth...... $1.40 Plane and Spherical Trigonometry, with Surveying and Tables In preparation... Logarithmic and Trigonometric Tables 114 pages, 8vo, cloth... 75 cents Copyright, 1910, by Charles E. Merrill Co. [3] CONTENTS PAGE INTRODUCTION TO TABLES..... 5 TABLES: I. FIVE-PLACE LOGARITHMS OF NUMBERS 1-10,000... 21 II. LOGARITHMS AND COLOGARITHMS OF MUCH-USED NUMBERS 40 [II. FIVE-PLACE LOGARITHMS OF THE SINE, COSINE, TANGENT, AND COTANGENT FOR EACH MINUTE OF THE QUADRANT 41 IV. AUXILIARY FIVE-PLACE TABLE FOR SMALL ANGLES.. 87 V. FOUR-PLACE TABLE OF THE NATURAL SINE, COSINE, TANGENT, AND COTANGENT FOR EVERY TEN MINUTES OF THE QUADRANT..... 91 VI. FOUR-PLACE LOGARIThMS OF NUMBERS 1-2000... 97 VII. FOUR-PLACE LOGARITHMS OF THE TRIGONOMETRIC FUNCTIONS FOR ANGLES OF THE QUADRANT EXPRESSED BY THE DECIMALLY DIVIDED DEGREE..... 103 VIII. CONVERSION OF MINUTES AND SECONDS INTO DECIMAL PARTS OF A DEGREE.... 114 IX. CONVERSION OF DECIMAL PARTS OF A DEGREE INTO MINUTES AND SECONDS....... 114 3 INTRODUCTION TO TABLES 1. Number of DecimalPlaces in Tables. All trigonometric work is based on the results of measurements. But no measurement is accurate beyond the sixth or seventh figure; this is owing to the limitations of our eyesight and sense of touch-perception, and to the ultimate imperfections in all our instruments of measurement. Thus a mile (63,360 inches) can be measured to within -- inch of its true length; an inch can be measured only to within a millionth part of itself, etc. So great a degree of accuracy, however, can be obtained only by applying every possible refinement of accuracy. Ordinary measuring, such, for instance, as that done by a carpenter, is accurate only to the second or third figure, that is, to within T- or 1-,T1 part. Hence it would be absurd for a carpenter or surveyor to use a number like 7.382654 ft.; 7.38+ ft. is sufficient. In 6,543,786, if the figure 6 to the right is 1 inch long, how long would the figure 6 on the left be if its length were made proportional to its value? Hence four-place tables are sufficiently accurate for all ordinary work (such as is done by a land surveyor, or in a physical laboratory under ordinary circumstances). Five-place tables give all the accuracy required except in very rare cases, when six- or seven-place tables may be used. But the latter cases are beyond the scope of this book. TABLE I. FIVE-PLACE LOGARITHMS OF NUMBERS 1-10,000 (pp. 21-39) 2. General Description of Table I. Table I consists of two parts. Part I occupies p. 21 and gives the logarithms (both characteristic and mantissa) of numbers 1-100. Part II occupies pp. 22-39, contains mantissas only, and gives the-e for all numbers from 1 to 10,000. 5 6 TRIGONOMETRY In using Part II the characteristic of each logarithm must be determined and supplied in accordance with the methods stated in Arts. 4 and 5 of Durell's Plane Trigonometry. DIRECT USE OF TABLE I 3. To find the mantissa for a number containing four figures. In the given table the left-hand column (headed NV) is a column of ordinary numbers. The first three figures of the given number whose mantissa is sought are found in this column. In the top row of each page are the figures 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. The fourth figure of the given number is found here. Hence, to obtain the mantissa of 3647, for instance, we take 364 in the first column on page 27 and look along the row beginning with 364 till we come to the column headed 7. The mantissa thus obtained is.56194. The first two figures of the row of mantissas, viz. 56, are supposed to be repeated in connection with each mantissa that follows till another complete mantissa is given. The use of a * indicates that the first two figures of the mantissa are to be taken from the beginning of the line of mantissas which follows. Thus, the mantissa of 1125 is.05115, not.04115. If the number whose mantissa is sought contains less than four figures, in using the tables we regard enough zeros as annexed to the given figures to make up four figures. In Chapter I of Durell's Plane Trigonometry it is shown that doing this does not affect the mantissa. Thus, to find the mantissa of 271, we find the mantissa of 2710, viz..43297. Similarly the mantissa of 7 is the same as that of 7000, viz..84510. 4. To find the mantissa of a number containing five or six figures. Interpolation. The method consists in finding the mantissa for the first four figures and adding a correction for INTRODUCTION TO TABLES 7 the fifth, or for the fifth and sixth figures. This correction is computed on the assumption that the differences in logarithms are proportional to the differences in the numbers to which they. belong. Though this proportion is not strictly accurate, it is sufficiently accurate for practical purposes. Ex. Find the mantissa of 1581.47. m. for 1582 =.19921 Mantissa of 1581 -.19893 m. for 1581 =.19893.00028 x.47 =.00013 Diff. for 1 =.00028 Mantissa of 1581.47 =.19906, Ans. For since an increase of 1 in the number makes an increase of.00028 in the mantissa, an increase of.47 in the number will make an increase of.47 of.00028, that is, of.00013 in the logarithm. As in the mantissa, so in the correction only five places of figures may be used. If the figure in the sixth place of the correction is 5 or a larger number, the figure in the fifth place of the correction is to be increased by 1; if less than 5, the figures after the fifth place are to be rejected. Thus if the above correction had been.000135 it would have been treated as.00014. If it had been.0001346 it would have been treated as 0.00013. The difference between the mantissas of two successive numbers is called the tabular difference. IHence, in general, to find a mantissa for a number containing five or six figures: Obtain from the table the mantissa for the first four figures, and also that for the next higher number, and subtract; Multiply the difference between the two mantissas by the fifth figure (or fifth and sixth figures) expressed as a decimal, and add the result to the mantissa for the first four figures. 5. Hence, to find the log of a given number: Determine the characteristic by Art. 4 or 5, Chapter I; Neglect the decimal point (in the given number) and obtain from the table tke mantissa for the given figures. 8 TRIGONOMETRY Ex. 1. Find log 3.62057. m. of 3.621 =.55883 log 3.620 = 0.55871 m. of 3.620 =.55871.00012 x.57 =.00007.00012 log 3.62057 = 0.55878, Ans. Ex. 2. Find log.078546. m. of 7855 =.89515 log.07854 = 8.89509 - 10 n. of 7854 =.89509.00006 x.6 =.00004.00006 log.078546 = 8.89513 - 10, Ans. For examples to be worked by the pupil, see the first part of Exercise 3 of Durell's Plane Trigonometry. INVERSE USE OF TABLE I 6. To find an antilogarithm, that is, to find the number corresponding to a given logarithm. Since the characteristic depends only on the position of the decimal point and not on the figures forming the given number, the characteristic is neglected at the outset of the process of finding the antilogarithm. (a) If the given mantissa can be found in the table: Take from the table the figures corresponding to the mantissa of the given logarithm; Use the characteristic of the given logarithm to fix the decimal point in the number obtained from the table. Ex. 1. Find the antilogarithm of 1.44138. The figures corresponding to the mantissa.44138 are 2763. Since the characteristic is 1, there are two figures at the left of the decimal point. Hence the antilog 1.44138 = 27.63. Or, if log x = 1.44138, x = 27.63. (b) In case the given mantissa does not occur in the table: Obtain from the table the next lower mantissa with the corresponding four figures of the antilogarithm; INTRODUCTION ITO TABLES 9 Subtract the tabular mantissa from the given mantissa; Divide this difference by the difference between the tabular mantissa and the next higher mantissa in the table; Annex the quotient to the four figures of the antilogarithm obtained from the table; Use the characteristic to place the decimal point in the result. Ex. 1. Find the antilog of 2.42376. The mantissa.42376 does not occur in the table, and the next lower mantissa is.42374. The difference between.42376 and.42374 is.00002. If a difference of 16 in the last two figures of the mantissa makes a difference of 1 in the fourth figure of the antilog, a difference of 2 in the last figure of the mantissa will make a difference of -2- of 1 or.125 (or.13) with respect to the fourth figure of the antilog. Hence we have antilog 2.42376 = 265.313- Ans. 374 16)2.00(.13 -16 40 Ex. 2. If log x= 7.26323 -10, find x. Nearest less mantissa=.26316, whose number is 1833. Tab. diff. = 24. 7 - 24 =.29+. Hence x=.00183329, Ans. The first part of Exercise 4 of Durell's Plane Trigonometry should be worked at this point. TABLE II. LOGARITHMS AND COLOGARITHMS OF MUCH-USED NUMBERS (p. 40) This table explains itself. TABLE III. FIVE-PLACE LOGARITHMS OF TRIGONOMETRIC FUNCTIONS FOR EVERY MINUTE OF THE QUADRANT (pp. 41-86) 7. Description of Table III. This table gives the logarithms of the sine, cosine, tangent, and cotangent of each minute of angle from 0~ up to 90~. 10 TRIGONOMETRY Where - 10 is a part of the characteristic of the log function it is omitted for the sake of economy of space. This omission occurs at the end of the log function of each angle except for log tangents from 45~ to 90~, and log cotangents from 0~ to 45~. For angles between 0 and 45~, the required functions are printed at the top of the columns, the number of degrees at the top of the page, and the number of minutes in the lefthand column. For angles between 45~ and 90~, the required function is printed at the bottom of the columns, the number of degrees at the bottom of the page, and the number of minutes in the right-hand column. Thus, log sin 26~ 37' = 9.65130 -10 (p. 68). log tan 67~ 48' = 0.38924 (p. 64). log sin 58~ 16' = 9.92968 -10 (p. 73). log cot 12~ 23' = 0.65845 (p. 54). Let the pupil determine why each column of the table has the name of a trigonometric function at the top and the name of the corresponding co-function at the bottom of the column. Let him also determine why -10 is to be annexed at the end of some log trigonometric functions as taken from the tables, and not at the end of others. DIRECT USE OF TABLE III 8. Given the degrees, minutes, and seconds of an angle, to find a logarithmic trigonometric function of the angle. After finding the log function for the given number of degrees and minutes, the log function for the given number of degrees, minutes, and seconds is found by interpolation. Ex. 1. Find the log sin 37~ 42' 53". The log sin 37~ 42' is 9.78642, and the difference between this and log sin 37~ 43' is 16. Since an increase of 1' in the angle makes an increase of 16 in the INTRODUCTION TO TABLES 11 last two places of the log sin, an increase of 53" or -53 of 1' will make an increase of 58 of 16 in the log of the function. Hence we have log sin 37~ 42'= 9.78642 -10 Diff. for 53"=- 5 of 16 = 14 log sin 37~ 42' 53" = 9.78656 - 10 Ex. 2. Find the log sin 53~ 27' 18". log sin 53~ 27' = 9.90490 - 10 Diff. for 18" = of 9 = 3 log sin 53~ 27' 18" = 9.90493 -10 Ex. 3. Find log cos 23~ 48' 12". Since the cosine of an angle decreases as the angle increases, the log of 23~ 49' is less than the log cos 23~ 48'. Hence the correction for 12" must be subtracted from the log cos 23~ 48'. Thus log cos 23~ 48' = 9.96140 -10 Diff. for 12" = 12 of 5 - 1 log cos 23~ 48' 12" = 9.96139-10 Ex. 4. Find log cot 57~ 18' 43". log cot 57~ 18' 9.80753 -10 Diff. for 43" = 28 x 4 - 20 log cot 57~ 18' 43" = 9.80733 -10 Hence, in general, Obtain from the table the log function for the given number of degrees and minutes; Also obtain from the table the log function for the angle, 1 minute greater; find the difference between these two log functions; multiply this difference by no. secon; this will give the correction for seconds; Add the correction for seconds in case of sine and tangent (direct functions); Subtract the correction in case of cosine and cotangent (complementary functions). 12 TRIGONOMETRY 9. Log Secants. To find the log secant of an angle, use 1 the formula sec x =... log sec x = 0 + colog cos x. COS X Thus log sec 39~ 28' 23" = colog cos 39~ 28' 23". But log cos 39~ 28' 23" = 9.88757 - 10. colog cos 39~ 28' 23" or log sec 39~ 28' 23" = 0.11243. 10. Log Functions of Angles greater than 90~. By the methods of Chapter IV, a trigonometric function of any angle greater than 90~ can be reduced to a trigonometric function of an angle less than 90~. Thus, since sin A = sin (180~ - A), sin 113~ 27'= sin 66~ 33'..'. log sin 113~ 27' = log sin 66~ 33' = 9.96256 -10. Also cos A = - cos (180~ - A). Hence, log cos A = log cos (180~ - A)(n), the small n being annexed to show that the function whose log is being used is a negative quantity. Thus log cos 142~ 18' = log cos 37~ 42' (n) = 9.78642 - 10 (n). At this point work the first part of Exercise 14 of Durell's Plane Trigonometry. INVERSE USE OF TABLE III 11. Given the logarithm of a function to find the corresponding acute angle (or find antilog sin, antilog cos, etc. or Zlog sin, Zlog cos, etc.) Obtain from the table, if possible, the number of degrees and minutes corresponding to the given logarithmic function. Ex. If log tan A = 9.92535 - 10, find the angle A. By consulting the table, tangent column, we find that A =40~ 6'. Or antilog tan 9.92535 - 10 = 40~ 6'. If the given logarithmic function does not occur in the table; INTRODUCTION TO TABLES 13 Obtain from the table the next less logarithm of the same function, noting the corresponding number of degrees and minutes; subtract this logarithm from the given logarithm; Divide the diference so obtained by the tabular difference for 1' and multiply by 60"; the result will be the correction, in seconds, to be added in case of sine and tangent, and subtracted in case of cosine and cotangent, to the angle already noted. Ex. 1. Find antilog sin 9.78538 - 10. Z log sin 9.78538 -10 = 37~ 35'+ 9.78527 -10 11 Since a difference of 16 in the log makes a difference of 1' (or of 60") in the angle, a difference of 11 in the log makes a difference of 1 of 60", or 41", in the angle..'o antilog sin 9.78538-10 =37 35' 41", Ans. Ex. 2. Find antilog cos 9.96623 - 10. antilog cos 9.96623 - 10 = 22 19' - 9.96619-10 4 of 60" = 48" 5 antilog cos 9.96623 - 10 = 22 18' 12", Ans. Ex. 3. Find antilog cot 0.57603. antilog cot 0.57603 = 14~ 52' - 0.57601 2 of 60"=2" 51 antilog cot 0.57603 = 14~ 51' 58", Ans. Ex. 4. Find antilog cos 9.60172 - 10. antilog cos 9.60172 - 10= 66~ 27t9.60157 -10 of 60" = 31", 29 antilog cos 9.60172-10 = 66~ 26' 29", Ans. 14 TRIGONOMETRY At this point work the first part of Exercise 15 of Durell's Trigonometry. TABLE IV. AUXILIARY FIVE-PLACE TABLE FOR SMALL ANGLES (pp. 87-89) 12. The Auxiliary Table of Logarithms of Sine and Tangent for Small Angles is needed because when an angle is smaller than 2~, the logarithms of the sine and tangent vary so rapidly that ordinary methods of interpolation are not sufficiently accurate. (The same is true for the cosine, cotangent, and tangent when the angle is between 88~ and 90~, but there are other indirect methods of meeting such cases.) Table IV is based on Art. 115 of Plane Trigonometry, where it is shown that the sine (or tangent) of a small angle is approximately the same in value as the number of radians in the angle. Hence, for example, to find sine 1~ 21' 37", we divide the number of seconds in 1~ 21' 37" by the number of seconds in a radian, viz. 206,265. This process is facilitated by Table IV. The column headed " in this table gives the number of seconds in each angle containing an exact number of minutes, and hence is an aid in converting any given angle into seconds. In the column headed S' is given the log of 206,265 (viz. 5.31443), modified by a slight correction owing to the change in the slight differences between the sine of a small angle and the radian measure of that angle. Similarly the column headed T' gives log of 206,265 in use of the tangent. (The columns headed S and T give the cologs corresponding to the S' and T' columns.) The column headed log sin gives the log sin or final answer for each even minute, these numbers being needed also in guiding the work in the inverse use of the table. Hence INTRODUCTION TO TABLES 15 13. To find the log sin or tangent of an angle less than 2~. Find the number of seconds in the given angle and find the log of this number in Table I; Add to this log the corresponding log in column S or T according as the log sin or log tan is desired. Ex. Find log sin 1~ 26' 13". 1~ 26' 13" 5173" log 5173 = 3.71374 S (or colog 206265) = 4.68553- 10.. log 1~ 26' 13" = 8.39927 - 10, Ans. 14. To find the angle corresponding to a given log sine or log tangent (less than 8.54282 - 10). Look up in the L. Sin column the number nearest in size to the given log; and set down the number on the same row with this in column S' or T', according as the given function is a sine or tangent; Add the given log function to the number set down from the table; Find the antilog of the result; this will be the number of seconds in the required angle. Ex. Find antilog tan 8.39307. In L. Sin column, the nearest number is 8.39310. Corresponding to this is T'= 5.31434 Given tan = 8.39307 antilog 13.70741 = 5098" = 1~ 24' 58", Ans. The reason for the above process is seen from the fact that 5098" sin of required Z = 2065" 206265". 206265 x (sin of required Z) = 5098"... log 206265 + 8.39307 = log 5098". 16 TRIGONOMETRY 15. Other Uses of the Auxiliary Table IV. The log cosine of an angle between 88~ and 90~ changes so rapidly as to make direct interpolation inaccurate. In such cases use the formula cos A = sin (90- A). Thus, for example, log cos 88~ 47' = log sin 1~ 13', and the value of log sin 1~ 13' can be obtained by Art. 14. The log cot A, when A is between 88~ and 90~, may be obtained similarly. Also, if A is an angle between 88~ and 90~, the log tan A changes so rapidly that interpolation is inaccurate. In this case use tan A -= cot A log tan A = colog cot A. = colog tan (90~ - A). Thus, for example, log tan 88~ 47' = colog tan 1~ 13', etc. At this point work the first part of Exercise 16 of Durell's Trigonometry. TABLE V. FOUR-PLACE TABLE OF THE NATURAL SINE, COSINE, TANGENT, AND COTANGENT FOR EVERY TEN MINUTES OF THE QUADRANT (pp. 91-96) 16. Method of using Table V. By natural trigonometric functions are meant the actual numerical (not logarithmic) values of these functions. Thus 1 is the natural sine of 30~. Interpolation for this table is made in the same general way as for Table V. Ex. Find natural sine 27~ 48'. N. Sine 27~ 40' 0.4643 80 of 26-= 21 N. Sine 27~ 48' = 0.4664, Alns. TABLE VI. FOUR-PLACE TABLE OF LOGARITHMS OF NUMBERS 1-2000 (pp. 97-101) 17. Method of using Table VI. In using the four-place log of a number, when the first significant figure of the number is 1, use pp. 100-101; otherwise use pp. 98-99. INTRODUCTION TO TABLES 17 In finding the antilog of a four-place log, if the given log is less than.3010, use pp. 100-101; otherwise use pp. 98-99. At this point work the latter part of Exercises 3 and 4 of Durell's Plane Trigonometry. TABLE VII. FOUR-PLACE LOGARITHMIC TABLE OF THE TRIGONOMETRIC FUNCTIONS FOR ANGLES OF THE QUADRANT EXPRESSED IN DECIMALLY DIVIDED DEGREES (pp. 103-113) 18. Method of using Table VII. The explanation of the methods of using Table III given in Arts. 8-11 of this Introduction apply in general to the use of Table VII. Hence we need only illustrate by examples the application of these methods to the table in hand. Ex. 1. Find log sin 48.34~. log sin 48.4~ = 9.8738 - 10 log sin 48.3~ = 9.8731 - 10 log sin 48.3~ = 9.8731 - -10 - of 7 = 3 7 log sin 48.34 = 9.8734 - 10, Ans. Ex. 2. Find the antilog tan 0.2165. log tan 0.2165 = 58.7~+ 2161 - of 10 = 2+ 17 / log tan 0.2165 = 58.72~, Ans. At this point work the latter part of Exercises 14 and 15 of Durell's Trigonometry. 19. Four-place Log Functions of Angles near 0~ or 90~. As is explained in Art. 12 of this Introduction, when an angle is less than 2~, the logarithms of the sine and tangent vary so rapidly that ordinary methods of interpolation are not sufficiently accurate. To get an accurate log function in this case we use the result obtained in Art. 106 of Plane Trigonometry, viz: sine or tangent of a very small Z x -n.. Z x in degrees = no. radians in Z x, or = 57.2960 18 TRIGONOMETRY.. log sin (or tan) of small Z x=log x +colog 57.296 = log x+ 8.2419 - 10. 1 57.296~ Also when x is small cot x= - = tan cx x in degrees. log cot small Z x= 1.7581 + colog x. Interpolation also is not accurate for log cos, log tan, log cot, of angles between 88~ and 90~. When A is an angle between 88~ and 90~ proceed as follows: cos A = sin (90~ - A)..'. log cos A = log sin (90~ - A) = 8.2419 - 10 + log (90 - A). cot A = tan (90~ -A)..-. log cot A = log tan (90~ -A)=8.2419- 10 + log (90~ -A). tan A = -.. log tan A = 1.7581 -log (90~- A). cot A Ex. 1. Find sin 0.876~. log 0.876~ = 9.9425 - 10 colog 57.296~ = 8.2419 - 10.'. log sin 0.876~ = 8.1844 - 10, Ans. Ex. 2. Find z log sin 7.9592 - 10. 17.9592 - 20 8.2419 - 10 antilog 9.7173 - 10 = 0.522~-.'. Z log sin 7.9592 - 10 = 0.522~-, Ans. At this point work the latter part of Exercise 16 of Durell's Trigonometry. TABLE VIII. TABLE FOR CONVERTING MINUTES AND SECONDS INTO THE DECIMAL PART OF A DEGREE (p. 114) 20. The method of using Table VIII is evident from the form of the table, but it should be remembered that in each INTRODUCTION TO TABLES 19 decimal equivalent ending in a significant figure the last figure is supposed to repeat indefinitely. Hence, for example, we have 36~ 46' = 36.766~+ = 36.77~ Also 35~ 43' = 35.716~ 20" =.006~.'. 35~ 43' 20" = 35.722~ = 35.72~, Ans. TABLE IX. TABLE FOR CONVERTING THE DECIMAL PARTS OF A DEGREE INTO MINUTES AND SECONDS (p. 114) 21. The method of using Table IX is also evident from the table itself. I II 1 TABLE I COMMON LOGARITHMS OF NUMBERS PART I LOGARITHMS (WITH CHARACTERISTICS) OF NUMBERS 1-100 I N. Log. 0 - Infinity 1 0.00 000 2 0.30 103 3 0.47 712 4 0.60 206 5 0.69 897 6 0.77 815 7 0.84 510 8 0.90 309 9 0.95 424 10 1.00 000 '11 1.04 139 12 1.07 918 13 1.11 394 14 1.14 613 15 1.17 609 16 1.20 412 17 1.23 045 18 1.25 527 19 1.27 875 20 1.30 103 21 1.32 222 22 1.34 242 23.36 173 24 1.38 021 25 1.39 794 26 1.41 497 27 1.43 136 28 1.44 716 29 1.46 240 30 1.47 7 N. Log. N. Log. N. Log. Jt I I 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 59 it^ 1.47 712 1.49 136 1.50 515 1.51 851 1.53 148 1.54 407 1.55 630 J.56 820 1.57 978 1.59 106 1.60 206 1.61 278 1.62 325 1.63 347 1.64 345 1.65 321 1.66 276 1.67 210 1.68 124 1.69 020 1.69 897 1.7^0 757 I,.72 428 1..74.036./4 819 1.75 587 1.76 343 1.77 085 1.77 815 I 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 1.77 815 1.78 533 1.79 239 1.79 934 1.80 618 1.81 291 1.81 954 1.82 607 1.83 251 1.83 885 1.84 510 1.85 126 1.85 733 1.86 332 1.86 923 1.87 506 1.88 081 1.88 649 1.89 209 1.89 763 1.90 309 1.90 849 1.91 381 1.91 908 1.92 428 1.92 942 1.93 450 1.93 952 1.94 448 1.94 939 1.95 424 90 91 92 93 94 95 96 97 98 99 100 1.95 424 1.95 904 1.96 379 1.96 848 1.97 313 1.97 772 1.98 227 1.98 677 1.99 123 1.99 564 2.00 000 I 0 5i _- A - [21] 0 PART II MANTISSAS OF NUMBERS 1-10,000 N. 0 1 2 3 4 5 6 7 8 9 i- -- I - l m -1-I 100 01 02 03 04 05 06 07 08 09 110 11 12 13 14 15 16 17 18 19 120 '21 22 23 24 25 26 27 28 29 130 31 32 33 34 35 36 37 38 39 140 41 42 43 44 45 46 47 48 49 150 00 000 432 860 01 284 703 02 119 531 938 03 342 743 04 139 - '532 922 05 308 690 06 070 446 819 07 188 555 043 475 903 326 745 160 572 979 383 782 179, 087 518 945 368 787 202 612 *019 423 822 218 130 561 988 410 828 243 653 *060 463 862 258 173 604 *0300, 452 870 284 694 *100 503 902 297 217 647 *072 494 912 325 735 *141 543 941 336 260 689 *115 536 953 366 776 *181 583 981 376 303 732 *157 578 995 407 816 *222 623 *021 415 346 775 *199 620 *036 449 857 *262 663 *060 454 389 817 *242 662 *078 490 898 *302 703 *100 493 571 961 346 729 108 483 856 225 591 610 999 385 767 145 521 893 262 628 650 *038 423 805 183 558 930 298 664 689 *077 461 843 221 595 967 335 700 727 *115 500 881 258 633 *004 372 737 766 *154 538 918 296 670 *041 408 773 805 *192 576 956 333 707 *078 445 809 844 *231 614 994 371 744 *115 482 846 883 *269 652 *032 408 781 *151 518 882 918 954 990 *027 *063 1*099 *135 08 279 636 991 09 342 691 10 037 380 721 11 059 314 672 *026 377 726 072 415 755 093 350 707 *061 412 760 106 449 789 126 386 743 -*096 447 795 140 483 823 160 422 778 *132 482 830 175 517 857 193 458 814 *167 517 864 209 551 890 227 493 849 *202 552 899 243 585 924 261 *171 529 884 *237 587 934 278 619 958 294 *207 565 920 *272 621 968 312 653 992 327 *243 600 955 *307 656 *003 346 687 *025 361 3~4 428 461 494 528 561 594 628 661 - 694 28.F~~~:]F~JF 727 12 057 385 710 13 033 354 672 988 14 301 760 090 418 743 066 386 704 *019 333 793 123 450 775 098 418 735 *051 364 826 156 483 808 130 450 767 *082 395 860 189 516 840 162 481 799 *114 426 893 222 548 872 194 513 830 *145 457 - 926 254 581 905 226 545 862 *176. 489 959 287 613 937 258 577 893 *208 520 992 320 646 969 290 609 925 *239 551 *024 352 678 *001 322 640 956 *270 582 613 644 675 706 737 768 799 829 860 891 613/4(75 1067-78- 799Q~Q66 Q 9 922 15 229 534 836 16 137 435.732 17 02'~319 609 953 259 564 866 167 465 761 056 348 638 983 290 594 897 19' 791 085 377 667 *014 320.52 I 27 j 27' i 52 820 114 406 696 *045 351 655 256 850 143 435 725 *076 *106 381 412 685 715 * 017: 16.... 6': 513 871 i 909 73. ' 202 464 493 754 782 *137 442 746 *047 346 643 938 231 522 811. *168 473 776 *077 376 673 967 260 551 840 *198 503 806 *107 406 702 997 289 580 869 I_ II _I __ _ _ -- I -I I N. 0 1 2 3 4 5 6 7T 8 9!11_I i."! [22] m N. 0 1 2 3 4 5 6 7 8 9 im i m m m m -i mI m ---- 150 51 52 53 54 55 56 57 -58 59 160 61 62 63 64 65 66 67 68 69 170 71 72 73 74 75 76 77 78 79 180 81 82 83 84 85 86 87 88 89 190 j.L 92 93 94 95 96 97 98 99 200 17 609 898 18 184 469 752 19 033 312 590 866 20 140 412 683 952 21 219 484 748 22 011 272 531 789, I I,,. I i l I~~~~ 638 667 696 725 -1 926 213 498 780 061 340 618 893 167 439 710 978 245 511 775 037 298 557 814 955 241 526 808 089 368 645 921 194 466 737 *005 272 537 801 063 324 583 840 984 270 554 837 117 396 673 948 222 493 763 *032 299 564 827 089 350 608 866 *013 298 583 865 145 424 700 976 249 520 790 *059 325 590 854 115 376 634 891 I 754 782 1811 840 869 *041 327 611 893 173 451 728 *003 276 548 817 *085 352 617 880 141 401 660 917 *070 355 639 921 201 479 756 *030 303 575 844 *112 378 643 906 167 427 686 943 *099 384 667 949 229 507 783 *058 330 602 871 *139 405 669 932 194 453 712 968 *127 412 696 977 257 535 811 *085 358 629 898 *165 431 696 958 220 479 737 994 *156 441 724 *005 285 562 838 *112 385 656 925 *192 458 722 985 246 505 763 *019 23 045 070 096 121 147 300 553 805 24 055 304 551 797 25 042 285 325 578 830 080 329 576 822 066 '310 350 603 855 105 353 601 846 091 334 376 629 880 130 378 625 871 115 358 401 654 905 155 403 650 895 139 382 527 551 575 600 624 ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~... 768 26 007 245 482 717 951 27 184.6 346 875 28 103 330 556 78Q 29 003 226 447 667 885 30 103 792 031 269 505 741 975 207 439 669 816 055 293 529 764 998 231 462 692 840 079 316..t553 788 *021 254 485 715 864 102 340 576 811 *045 277 508 738 426 679 930 180 428 674 920 164 406 452 704 955 204 452 699 944 188 431 172 648 672 696 720 744......-. _ 198 I 223 249_ 274 477 729 980 229 477 724 969 212 455 502 754 *005 254 502 748 993 237 479 528 779 *030 279 527 773 *018 261 503 888 126 364 600 834 *068 300 531 761 912 150 387 623 858 *091 323 554 784 935 17.4. 411 647 881 *114 346 577 807 959 198 435 670 905 *138 370 600 830 983 221 458 694 928 *161 393 623 852 898 921 944 967 989 *012 *035 *058 *081 126 353 578 803 026 248 469 688 907 125 149 375 601 825 048 270 491 710 929 146 171 398 623 847 070 292 513 732 951 168 194 421 646 870 092 314 535 754 973 190 217 443 668 892 115 336 557 776 994 211 240 466 691 914 137 358 579 798 *016 233 262 488 713 937 159 380 601 820 *038 255 285 511 735 959 181 403 623 842 *060 276 307 533 758 981 203 425 645 863 *081 298 *N. | 0O l _6 17 _ _ i _..i i L W L_~ [23]. N. 200 01 02 03 04 05 06 07 08 09 210 11 12 13 14 15 16 17 18 19 220 21 22 23 24 25 26 27 28 29 230 31 32 33 34 35 36 37 38 39 240 41 42 43 44 45 46 47 48 49 250 o 1 2 3 4 15 61 7 8 9 - m - - mI ~m -m~ 30 103 125 146 1168 1190 211 233 255 1 276 298 320 535 750 963 31 175 387 597 806 32 015 341 557 771 984 197 408 618 827 035 363 578 792 *006 218 429 639 848 056 384 600 814 *027 239 450 660 869 077 406 621 835 *048 260 471 681 890 098 428 643 856 *069 281 492 702 911 118 449 664 878 *091 302 513 723 931 139 471 685 899 *112 323 534 744 952 160 492 707 920 *133 345 555 765 973 181 514 728 942 *154 366 576 785 994 201 222 243 263 284 305 1 325 346 366 387 408 _a I-4 6124 3515 4 36 8 0 428 634 838 33 041 244 445 646 846 34 044 449 654 858 062 264 465 666 866. 064 469 675 879 082 284 486 686 885 084 490 695 899 102 304 506 706 905 104 510 715 919 122 325 526 726 925 124 531 736 940 143 345 546 746 945 143 552 756 960 163 365 566 766 965 163 572 777 980 183 385 586 786 985 183 593 797 *001 203 405 606 806 *005 203 613 818 *021 224 425 626 826 *025 223 242 262 282 301 321 341 361 380 400 420 _1.6 I28 _ 0 2)3116)30)401 439 635 830 35 025 218 411 603 793 984 459 655 850 044 238 430 622 813 *003 479 674 869 064 257 449 641 832 *021 498 694 889 083 276 468 660 851 *0'40 518 713 908 102 295 488 679 870 *059 537 733 928 122 315 507 698 889 *078 557 753 947 141 334 526 717 908 *097 577 772 967 160 353 545 736 927 *116 - 596 792 986 180 372 564 755 946 *135 616 811 *005 199 392 583 774 965 *154 36 173 192 211 229 248 1 267 286 305 324 342 361 549 736 922 37 107 291.475 658 840 38 021 202 382 561 739 917 39 094 270 445 620 794 380 568 754 940 125 310 493 676 399 586 773 959 144 328 511 694 858 1876 039 220 399 578 757 934 111 287 463 637 811 057 238 417 596 775 952 129 305 480 655 829 418 605 791 977 162 346 530 712 894 075 256 435 614 792 970 146 322 498 672 846 436 624 810 996 181 365 548 731 912 093 274 453 632 810 987 164 340 515 690 863 455 642 829 *014 199 383 566 749 931 112 292 471 650 828 *005 182 358 533 707 881 474 661 847 *033 218 401.585 767 949 130 31 489 668 846 *023 199 375 550 724 38983 493 680 866 *051 236 420 603 967 14,8 3250 50,i 686 863 *041 217 393 568 742 915 511 698 884 *070 254 438 621 803,346 525 703 881 *058 235 410 585 759 933 530 717 903 *088 273 457 639 822 643 54 3 721 899 *076 252 428 602 777 950 N. 0 -1 2 3 4 5 7 r1 [a24 I I N. I 0 1 2 3 1 4 5 6 7 8 9 i n i n - I -H 250 51 52 53 54 55 56 57 58 59' 260 61 62 63 64 65 66 67 68 69 270 71 72 73 74 75 76 77 78 79 280 81 82 83 84 85 86 87 88 89 290 91 92 93 94 95 96 97 98 99 300 F -w 39 794 967 40 140 312 483 654 824/ 993 41 162 330 811 829 846 863 881 898 915 933 950 985 157 329 500 671 841 *010 179 347 *002 175 346 518 688 858 *027 196 363 *019 192 364 535 705 875 *044 212 380 *037 209 381 552 722 892 *061 229 397 1 *054 226 398 569 739 909 *078 246 414 *071 243 415 586 756 926 *095 263 430 *088 261 432 603 773 943 *111 280 447 *106 278 449 620 790 960 *128 296 464 *123 295 466 637 807 976 *145 313 481 I 42 497 514 531 547 564 581 597 614 631 647 664 681 697 714 731 747 764 780 797 814 830 847 863 880 896 913 929 946 963 979 996 *012 *029 *045 *062 *078 *095 *111 *127 *144 2 160 177 193 210 226 243 259 275 292 308 325 341 357 374\ 390 406 423 439 455 472 488 504 521 537 553 570 586 602 619 635 651 667 684 700 716 732 749 765 781 797 813 830 846 862 878 894 911 927 943 959 975 991 *008 *024 *040 *056 *072 *088 *104 *120 3 136 152 169 185 201 217 233 249 265 281 '297 313 329 345 361 377 393 409 425 441 457 473 489 505 521 537 553 569 584 600 616 632 648 664 680 696 712 727 743 759 775 791 807 823 838 854 870 886 902 917 933 949 965 981 996 *012 *028 *044 *059 *075 4-091 107 122 138 154 170 185 201 -217 232 248 264 279 295 311 326 342 358 373 389 404 420 436 451 467 483 498 514 529 545 560 576 592 607 623 638 654 669 685 700 716 731 747 762 778 793 809 824 840 855 -1 871 45 025 179 332 484 637 -788 939 46 090 240 389 538 687 835 982 47 129 - -276 422 567 712 886 040 194 347 500 652 803 954 105 255 404 553 702 850 997 144 -290 436 582 902 056 209 362 515 667 818 969 120 270 419 568 716 864 *012 159 305 451 596 917 071 225 378 530 682 834 984 135 285 434 583 731 879 *026 173 319 465 611 932 086 240 393 545 697 849 *000 150 300 449 598 746 894 *041 188 334 480 625 948 102 255 408 561 712 864 *015 165 315 464 613 761 909 *056 202 349 494 640 963 117 271 423 576 728 879 *030 180 330 479 627 776 923 *070 217 363 509 654 979 133 286 439 591 743 894 *045 195 345 494 642 790 938 *085 232 378 524 669 994 148 301 454 606 758 909 *060 210 359 509 657 805 953 *100 246 392 538 683 *010 163 317 469 621 773 924 *075 225 374 523 672 820 967 *114 261 407 553 698 727 741 756 770 784 799 813 828 842 E i -I.. I * RMVM.~ 0 1 i 2 3 4 5 6 7 8 9 U mmU - I - U- U- U * * [25] 0N. 0 2 1 5 6 7 58 9 m - m mI 1 mm -- 300 01 02 03 04 05 06 07 08 09 310 11 12 13 14 15 16 17 18 19 320 21 22 23 24 25 26 27 28 29 330 31 32 33 34 35 36 37 38 39 340 41 42 43 44 45 46 47 48 49 350 47 712 727 741 756 770 1 784 799 813 828 842 857 48 001 144 287 430 572 714 855 996 49 136 871 015 159 302 444 586 728 869 *010 885 029 173 316 458 601 742 883 *024 900 044 187 330 473 615 756 897 *038 914 058 202 344 487 629 770 911 *052 929 073 216 359 501 643 785 926 *066 943 087 230 373 515 657 799 940 *080 958 101 244 387 530 671 813 954 *094 972 116 259 401 544 686 827 968 *108 986 130 273 416 558 700 841 982 *122 150 164 178 192 1 206 220 234 248 262 276 415 554 693 831 969 50 106 243 379 290 429 568 707. 845 982 120 256 393 304 443 582 721 859 996 133 270 406 318 457 596 734 872 '010 147 284 420 332 471 610 748 886 *024 161 297 433 346 485 624 762 900 *037 174 311 447 360 499 638 776 914 *051 188 325 461 374 513 651 790 927 *065 202 338 474 388 527 665 803 941 *079 215 352 488 402 541 679 817 955 *092 229 365 501, 515 529 542 556 569 583 596 610 623 637 651 786 920 51 055 188 322 455 587 720 664 799 934 068 202 335 468 601 733 678 813 947 081 215 348 481 614 746 691 826 961 095 228 362 495 627 759 705 840 974 108 242 375 508 640 772 718 853 987 121 255 388 521 654 786 732 866 *001 135 268 402 534 667 799 745 880 *014 148 282 415 548 680 812 759 893 *028 162 295 428 561 693 825 772 907 *041 175 308 441 574 706 838 851 865 878 891 904 917 930 943 957 970 851 865187. 89 1 4 9 9 / 43 jQ57 7 983 52 114 244 375 504 634 763 892 53 020 996 127 257 388 517 647 776 905 033 *009 140 270 401 530 660 789 917 046 *022 153 284 414 543 673 802 930 058 *035 166 297 427 556 686 815 943 071 *048 179 310 440 569 699 827 956 084 *061 192 323 453 582 711 840 969 097 *075 205 336 466 595 724 853 982 110 *088 218 349 479 608 737 866 994 122 *101 231 362 492 621 750 879 *007 135 148 161 173 1 86 199 212 224 237 250 263 275 403 529 656 782 908 54 033 158 283 288 415 542 668 794 920 045 170 295 301 428 555 681 807 933 058 183 307 314 441 567 694 820 945 070 195 320 326' 453 580 706 832 958 083 208 332 339 466 593 719 845 970 095 220 345 352 479 605 732 857 983 108 233 357 364 491 ' 618 744 870 995 120 245 370 377 504 631 757 882 *008 133 258 382 390 517 643 769 895 *020 145 270 394 407 419 432 444 456 469 481 494 506 518 N. 0 1 2 3 4 5 6 7 8 9.... m m mi I m m I m [26] N. 1 2 3 4 6 7 58 9 _-] I I I _ - I I - _ _ 350 51 52 53 54 55 56 57 58 59 360 61 62 63 64 ' 65 66 67 68 69 370 71 72 73 74 75 76 77 78 79 380 81 82 83 84 85 86 87 88 89 390 91 92 93 94 95 96 97 98 99 400 54 407 419 432 444 456 1 469 481 494 506 518 531 654 777 900 55 023 145 267 388 509 543 667 790 913 035 157 279 400 522 555 679 802 925 047 169 291 413 534 568 691 814 937 060 182 303 425 546 580 704 827 949 072 194 315 437 558 593 716 839 962 084 206 328 449 570 605 728 851 974 096 218 340 461 582 617 741 864 986 108 230 352 473 594 630 753 876 998 121 242 364 485 606 642 765 888 *011 133 255 376 497 618 630 642 654 666 678 1 691 703 715 727 739 -~~~~ _ _- I _ _ — _ _.. 751 871 991 56 110 229 348 467 585 703 763 883 *003 122 241 360 478 597 714 775 895 *015 134 253 372 490 608 726 787 907 *027 146 265 384 502 620 738 799 919 *038 158 277 396 514 632 750 811 931 *050 170 289 407 526 644 761 823 943 *062 182 301 419 538 656 773 835 955 *074 194 312 431 549 667 785 847 967 *086 205 324 443 561 679 797 859 979 *098 217 336 455 573 691 808 820 832 844 855 867 879 891 902 914 926 _2.3.. 5 67 89 81 0 2 937 57 054 171 287 403 519 634 749 864 949 066 183 299 415 530 646 761 875 961 078 194 310 426 542 657 772 887 972 089 206 322 438 553 669 784 898 984 101 217 334 449 565 680 795 910 996 113 229 345 461 576 692 807 921 *008 124 241 357 473 588 703 818 933 *019 136 252 368 484 600 715 830 944 *031 148 264 380 496 611 726 841 955 *043 159 276 392 507 623 738 852 967 978 990 *001 *013 *024 *035 *047 *058 *070 *081 58 092 206 320 433 546 659 771 883 995 59 106 104 218 331 444 557 670 782 894 *006.118 115 229 343 456 569 681 794 906 *017 129 127 240 354 467 580 692 805 917 *028 140 138 252 365 478 591 704 816 928 *040 151 149 263 377 490 602 715 827 939 *051 162 161 274 388 501 614 726 838 950 *062 173 172 286 399 512 625 737 850 961 *073 184 184 297 410 524 636 749 861 973 *084 195 195 309 422 535 647 760 872 984 *095 207 218 329 439 550 660 770 879 988 60 097 229 340 450 561 671 780 890 999 108 240 351 461 572 682 791 901 *010 119 251 362 472 583 693 802 912 *021 130 262 373 483 594 704 813 923 *032 141 273 384 494 605 715 824 934 *043 152 284 395 506 616 726 835 945 *054 163 295 406 517 627 737 846 956 *065 173 306 417 528 638 748 857 966 *076 184 318 428 539 649 759 868 977 *086 195 206 217 2281 239 249 260 1 271 282 293 304 N. I 0 1 2 3 4 1 6 7 8 9 m m~ ~ ~ ~~~~~~ ~~~~~~~~~~~~~~~~~~~~~~~~~~ rom mIlm [27] N. ~0 1 2 3 4 15 - 6 7 8 9 I 400 01 02 03 04 05 06 07 08 09 410 11 12 13 14 15 16 17 18 19 420 21 22 23 24 25 26 27 28 29 430 31 32 33 34 35 36 37 38 39 440 41 42 43 44 45 46 47 48 49 450 60 206 314 423 531 638 746 853 * 959 61 066 172 278 384 490 595 700 805 909 -62 014 118 221 325 428 531 634 737 839 941 63 043 144 246 217 1 228 239 249 260 271 282 293 304 325 433 541 649 756 863 970 077 183 336 444 552 660 767 874 981 087 194 347 455 563 670 778 885 991 098 204 358 466 574 681 788 895 *002 109 215 369 477 584 692 799 906 *013 119 225 379 487 595 703 810 917 *023 130 236 390 498 606 713 821 927 *034 140 247 401 509 617 724 831 938 *045 151 257 412 520 627 735 842 949 *055 162 268 289 395 500 606 711 815 920 024 128 232 300 405 511 616 721 826 930 034 138 242 310 416 521 627 731 836 941 045 149 252 321 426 532 637 742 847 951 055 159 263 331 437 542 648 752 857 962 066 170 273 342 448 553 658 763 868 972 076 180 284 352 458 563 669 773 878 982 086 190 294 363 469 574 679 784 888 993 097 201 304 374 479 584 690 794 899 *003 107 211 315 335 439 542 644 747 849 951 053 155 256 346 449 552 655 757 859 961 063 165 266 356 459 562 665 767 870 972 073 175 276 366 469 572 675 778 880 982 083 185 286 377 480 583 685 788 890 992 094 195 296 387 490 593 696 798 900 *002 104 205 306 397 I500 603 706 808 910 *012 114 215 317 408 511 613 716 818 921 *022 124 225 327 418.~21 624 726 829 931 *033 134 236 337 347 357 367 377 387 397 407 417 428 438 448 548 649 749 849 949 64 048 147 246 345 444 542 640 738 836 933 65 031 128 225 - 321 458 558 659 759 859 959 058 157 256 355 468 568 669 769 869 969 068 167 266 365 478 579 679 779 879 979 078 177 276 375 488 589 689 789 889 988 088 187 286 385 498 599 699 799 899 998 098 197 296 395 508 609 709 809 909 *008 108 207 306 404 518 619 719 819 919 *018 118 217 316 414 528 629 729 829 929 *028 128 227 326 424 538 639 739 839 939 *038 137 237 335 434 454 552 650 748 846 943 040 137 234 464 562 660 758 856 953 050 147 244 473 572 670 768 865 963 060 157 254 483 582 680 777 875 972 070 167 263 493 591 689 787 885 982 079 176 273 503 601 699 797 895 992 089 186 283 513 611 709 807 904 *002 099 196 292 523 621 719 816 914 *011 108 205 302 532 631 729 826 924 *021 118 215 312 331 341 350 360 369 379 - 389 398 408 N. - o I1 2 3 4 6: 7 7 8 -9 I - I [28] N. 450 51 52 53 54 55 56 57 58 59 460 61 62 63 64 65 66 67 68 69 470 71 72 73 74 75 76 77 78 79 480 81 82 83 84 85 86 87 88 89 490 91 92 93, 94 95 96 97 98 99 500 s0 1 2. 3 4 5 6 7 8 9 65 321 -418 514 610 706 801O 896 992 66 087 181 276 -331- 341 350- 360 I 369 379 389 398 408 427 523 619 715 811.906 *001 096 191 437 533 629 725 820 916 *011 106 200 447 543 639 734 830 925 *020 115 210 456 552 648 744 839 935 *030 124 219 466 562 658 753 849 944 *039 134 229 475 571 667 763 858 954 *049 143 238 485 581 677 772 868 963 *058 153 247 495 591 686 782 877 973 *068 162 257 504 600 696 792 887 982 *077 172 266 285 295 304 314 323. 332 342 351 361 6, 6E 370 380 389 398 408 417 427 436 445 455 464 474 483 492 502 511 521 530 539 549 558 567 577 586 596 605 614 624 633 642 652 661 671 680 689 699 708 717 727 736 745 755 764 773 783 792 801 811 820 829 839 848 857 867 876 885 894 904 913 922 932 941 950 960 969 978 987 997 *006 *015 7 025 034 043 052 062 071 080 089 099 108 117 127 136 145 154 164 173 182 191 201 210 219 228 237 247 256 265 274 284 293 302 311 32,1 330 339 348 357 367 376 385 394 403 413 422 431- 440 449 459 468 477 486 495 504 514 523 532 541 550 560 569 578:587 596 605 614 624 633 642 651 660 669 679 688 697 706 715 724 733 742 752 761 770 779 788 797 806 815 825 834 843 852 861 870 879 888 897 906 916 925 934 -943 952 961 970 979 988 997 *006 *015 *024 3 034 043 052 061 070 079 088 097 106 115 124 215 305 395 485 -574 664 753 842 931 69 020 108 197 285 373 461 548 636 723 810 897 133 224 314 404 494 583 673 762 851 940 142 233 323 413 502 592 681 771 860 949 151 242 332 422 511 601 690 780 869 958 160 251 341 431 520 610 699 789 878 966 169 260 350 440 529 619 708 797 886 975 178 269 359 449 538 628 717 806 895 984 187 278 368 458 547 637 726 815 904 993 196 287 377 467 556 646 735 824 913 *002 205 296 386 476 565 655 744 833 922 *011 028 037 046 055 064 073 082 090 099 117 205 -294 -381 469 557 644 — 732 819 906 126 214 302 390 478 566 653 740 827 914 135 223 311 399 487 574 662 749 836 923 144 232 320 408 496 583 671 758 845 932 152 241 329 417 504 592 679 767 854 940 161 249 338 425 513 601 688 775 862 949 170 258 346 434 522 609 697 784 871 958 179 267 355 443 531 618 705 793 880 966 188 276 364 452 539 627 714 801 888 975 N.; 0 1 2 3 4 5 6 7 8 9 -. - ~ m ~~~~~~~~~~~~ m u m m m~~ ~ ~ ~ ~ i i [291 . N. O 1 2 3 |4 61 7 8 9 i i -ii- I m ~ m m m -IMF 500 01 02 03 04 05 06 07 08 09 510 11 12 13 14 15 16 17 13 19 520 21 22 23 24 25 26 27 28 29 530 31 32 33 34 35 36 37 38 39 540 41 42 43 44 45 46 47 48 49 550 69 897 984 70 070 157 243 329 415 501 586 672 906 992 079 165 252 338 424 509 595 680 914 923 932 1 940 949 *001 088 174 260 346 432 518 603 689 *010 096 183 269 355 441 526 612 697 *018 105 191 278 364 449 535 621 706 I *027 114 200 286 372 458 544 629 714 *036 122 209 295 381 467 552 638 723 *044 131 217 303 389 475 561 646 731 *053 140 226 312 398 484 569 655 740 958 966 975 *062 148 234 321 406 492 578 663 749 757 766 774 783 791 800 808 817 825 834. - - I _ I - - -- I - -- -- I -- I --- I -- I 842 927 71 012 096 181 265 349 433 517 851 935 020 105 189 273 357 441 525 859 944 029 113 198 282 366 450 533 868 952 037 122 206 290 374 458 542 876 961 046 130 214 299 383 466 550 885 969 054. 139 223 307 391 475 559 893 978 063 147 231 315 399 483 567 902 986 071 155 240 324 408 492 575 910 995 079 164 248 332 416 500 584 919 *003 088 172 257 341 425 508 592 I 600 609 617 625 634 642 650 659 667 675........ _. _ - _. _ 684 767 850 933 72 016 099 181 263 346 692 775 858 941 024 107 189 272 354 700 784 867 950 032 115 198 280 362 709 792 875 958 041 123 206 288 370 717 800 883 966 049 132 214 296 378 725 809 892 975 057 140 222 304 387 734 817 900 983 066 148 230 313 395 742 825 908 991 074 156 239 321 403 750 834 917 999 082 165 247 329 411 759 842 925 *008 090 173 255 337 419 428 436 444 452 460 469 477 485 493 501 -— ~~~~~~2 8-1 ~~~~~~~~4~~~ ~~~ ~~~F ~~~~ ~~~F 1 69 477~~~~~~~~~. 509 591 673 754 835 916 997 73 078 159 518 599 681 762 843 925 *006 086 167 526 607 689 770 852 933 *014 094 175 534 616 697 779 860 941 *022 102 183 542 624 705 787 868 949 *030 111 191 550 632 713 795 876 957 *038 119 199 558 640 722 803 884 965 *046 127 207 567 648 730 811 892 973 *054 135 215 575 656 738 819 900 981 *062 143 223 583 665 746 827 908 989 *070 151 231 239 247 255 263 272 280 288 296 304 312 320 328 336 344 352 360 368 376 384 392 400 408 416 424 432 440 448 456 464 472 480 488 496 504 512 520 528 536 544 552 560 568 576 484 592 600 608 616 624 632 640 648 656 664 672 679 687 695 703 711 719 727 735 743 751 759 767 775 783 791 799 807 815 823 830 838 846 854 862 870 878 886 894 902 910 918 926 933 941 949 957 965 973 981 989 997 *005 *013 *020 *028 74 036 044 052 060 068 -I 076 084 1 092 1 099 107 _. |1I I _ I _ I I N. O 1 2 3 4 6 7 8 9 - m m m u m m m m m m~~~~~~~~~~~~~~ [30] N. 0 2 3 4 6 7 8 9 ___ I mm - mm ii m.mm - m mm mmmm! 550 51 52 53 54 55 56 57 58 59 560 61 62 63 64 65 66 67 68 69 570 71 72 73 74 75 76 77 78 79 580 81 82 83 84 85 86 87 88 89 590 91 92 93 94 95 96 97 98 99 600 N. 74 036 044 052 060 j 115 194 273 351 429 507 586 663 741 819 896 974 75 051 128 205 282 358 435 511 123 202 280 359 437 515 593 671 749 827 904 981 059 136 213 289 366 442 519 131 210 288 367 445 523 601 679 757 834 912 989 066 143 220 297 374 450 526 139 218 296 374 453 531 609 687 764 842 920 997 074 151 228 305 381 458 534 068 147 225 304 382 461 539 617 695 772 850 927 *005 082 159 236 312 389 465 542 076 084 I 155 233 312 390 468 547 624 702 780 858 935 *012 089 166 243 320 397 473 549 162 241 320 398 476 554 632 710 788 865 943 *020 097 174 251 328 404 481 557 092 170 249 327 406 484 562 640 718 796 873 950 *028 105 182 259 335 412 488 565 099 178 257 335 414 492 570 648 726 803 881 958 *035 113 189, 266 343 420 496 572 107 186 265 343 421 500 578 656 733 811 889 966 *043 120 197 274 351 427 504 580 656 732 808 884 959 *035 110 185 260 335 I - 587 664 740 815 891 967 76 042 118 193 268 595 671 747 823 899 974 050 125 200 275 603 679 755 831 906 982 057 133 208 283 610 686 762 838 914 989 065 140 215 290 I I -1 618 694 770 846 921 997 072 148 223 298 626 702 778 853 929 *005 080 155 230 305 633 709 785 861 937 *012 087 163 238 313 641 717 793 868 944 *020 095 170 245 320 648 724 800 876 952 *027 103 178 253 328 343 418 492 567 641 716 790 864 938 77 012 085 350 425 500 574 649 723 797 871 945 019 093 358 433 507 582 656 730 805 879 953 026 100 365 440 515 589 664 738 812 886 960 034 107 1 I I 373 448 522 597 671 745 819 893 967 041 115 I 380 455 530 604 678 753 827 901 975 048 122 388 462 537 612 686 760 834 908 982 056 129 395 470 545 619 693 768 842 916 989 063 137 -1 403 477 552 626 701 775 849 923 997 070 144 410 485 559 634 708 782 856 930 *004 078 151 159 232 305 379 452 525 597 670 743 815 166 240 313 386 459 532 605 677 750 822 173 247 320 393 466 539 612 685 757 830 181 254 327 401 474 546 619 692 764 837 -1 188 262 335 408 481 554 627 699 772 844 I 195 269 342 415 488 561 634 706 779 851 203 276 349 422 495 568 641 714 786 859 210 283 357 430 503 576 648 721 793 866 217 291 364 437 510 583 656 728 801 873 225 298 371 444 517 590 663 735 808 880 0 1 2 3 4 5 6 7 8 9.1~~ ~ ~~~ m m m m - m — mtm m m. m [31] I N. I 0 1 2 3 4 5 O 7 8 9 - - - in... -. -. - I 600 01 02 03 04 05 06 07 08 09 610 11 12 13 14 15 16 17 18 19 620 21 22 23 24 25 26 27 28 29 630 31 32 33 34 35 36 37 38 39 640 41 42 43 44 45 46 47 48 49 650 77 815 822 830- 837 844 851 859 1866 873 880 887 960 78 032 104 176 247 319 390 462 895 967 039 111 183 254 326 398 469 902 974 046 118 190 262 333 405 476 909 981 053 125 197 269 340 412 483 916 988 061 132 204 276 347 419 490 924 996 068 140 211 283 355 426 497 931 *003 075 147 219 290 362 433 504 938 *010 082 154 226 297 369 440 512 945 *017 089 161 233 305 376 447 519 952 *025 097 168 240 312 383 455 526 mm 533 540 547 554 561 569 576 583 590 597........ _..AI. -. 1. 604 675 746 817 888 958 79 029 099 169 611 682 753 824 895 965 036 106 176 618 689 760 831 902 972 043 113 183 625 696 767 838 909 979 050 120 190 633 704 774 845 916 986 057 127 197 640 711 781 852 923 993 064 134 204 647 718 789 859 930 *000 071 141 211 654 725 796 866 937 *007 078 148 218 661 732 803 873 944 *014 085 155 225 668 739 810 880 951 *021 092 162 232 I 239 246 253 260 267 274 281 288 295 302 309 316 323 330 337 344 351 358 365 372 379 386 393 400 -407 414 421 428 435 442 449 456 463 470 477 484 491 498 505 511 518 525 532 539 546 553 560 567 574 581 588 595 602 609 616 623 630 637 644 650 657 664 671 678 685 692 699 706 713 720 727 734 741 748 754 761 768 775 782 789 796 803 810 817 824 831 837 844 851 858 865 872 879 886 893 900 906 913 920 927 934 941 948 955 962 969 975 982 989 996 ____,,__ ___ ___ ___ 996 80 003 072 140 209 277 346 414 482 550 010 079 147 216 284 353 421 489 557 017 085 154 223 291 359 428 496 564 024 092 161 229 298 366 434 502 570 030 099 168 236 305 373 441 509 577 037 106 175 243 312 380 448 516 584 044 113 182 250 318 387 455 523 591 051 120 188 257 325 393 462 530 598 058 127 195 264 332 400 468 536 604 065 134 202 271 339 407 475 543 611 618 625 632 638 645 652 659 665 672 679 ~ I I, —:61 72 6 -9 686 754 821 889 956 81 023 090 158 224 693 760 828 895 963 030 097 164 231 699 767 835 902 969 037 104 171 238 706 774 841 909 976 043 111 178 245 713 781 848 916 983 050 117 184 251 720 787 855 922 - 990 057 124 191 258 726 794 862 929 996 064 131 198 265 733 801 868 936 *003 070 137 204 271 740 808 875 943 *010 077 144 211 278 747 814 882 949 *017 084 151 218 285 351! 291 298 305 311 318 325 331 338 345 - - *_ - 1 - _ - - N. 0 I 2 -3 4 5 6 7 8 9 -i - L - - - [32] N. I 0 1 2 3 4 5 6 7 8 9 ia -ii 650 51 52 53 54 55 56 57 58 59 660 61 62 63 64 65 66 67 68 69 670 71 72 73 74 75 76 77 78 79 680 81 82 83' 84 85 86 87 88 89:90 91 92 93 94 95 96 97 98 99 700 N. 81 291 298 305 311 318 325 331 338 345 351 358 425 491 558 624 690 757 823 889 365 431 498 564 631 697 763 829 8.95 371 438 505 571 637 704 770 836 902 378 445 511 578 644 710 776 842 908 385 451 518 584 651 717 783 849 915 981 391 458 525 591 657 723 790 856 921 987 398 465 531 598 664 730 796 862 928 994 405 471 538 604 671 737 803 869 935 *000 411 478 544 611 677 743 809 875 941 *007 418 485 551 617 684 750 816 882 948 *014 954 961 968 974 -1 82 020 086 151 217 282 347 413 478 543 027 092 158 223 289 354 419 484 549 033 099 164 230 295 360 426 491 556 040 105 171 236 302 367 432 497 562 046 112 178 243 308 373 439 504 569 053 119 184 249 315 380 445 510 575 060 125 191 256 321 387 452 517 582 066 132 197 263 328 393 458 523 588 073 138 204 269 334 400 465 530 595 079 145 210 276 341 406 471 536 601 607 614 620 627 633 640 646 653 659 666 672 737 802 866 930 995 83 059 123 187 251 679 743 808 872 937 *001 065 129 193 257 685 750 814 879 943 *008 072 136 200 264 692 756 821 885 950 *014 078 142 206 270 698 763 827 892 956 *020 085 149 213 276 705 769 834 898 963 *027 091 155 219 283 711 776 840 905 969 *033 097. 161 225 289 718 782 847 911 975 *040 104 168 232 296 724 789 853 918 982 *046 110 174 238 302 730 795 860 924 988 *052 117 181 245 308 315 321 327 334 340 347 353 359 366 372 378 385 391 398 404 410 417 423 429 436 442 448 455 461 467 474 480 487 493 499 506 512 518 525 531 537 544 550 556 563 569 575 582 588 594 601 607 613 620 626 632 639 645 651 658 664 670 677 683 689 696 702 708 715 721 727 734 740 746 753 759 765 771 778 784 790 797 803 809 816 822 828 835 841 847 853 860 866 872 879 885 891 897 904 910 916 923 929 935 942 i -- ~ ~ ~ ~ - - - - - - 948 84 011 073 136 198 261 323 386 448 954 017 080 142 205 267 330 392 454 960 023 086 148 211 273 336 398 460 967 029 092 155 217 280 342 404 466 973 036 098 161 223 286 348 410 473 979 042 105 167 230 292 354 417 479 985 048 111 173 236 298 361 423 485 992 055 117 180 242 305 367 429 491 998 061 123 186 248 311 373 435 497 *004 067 130 192 255 317 379 442 504 I 510 0 516 522 1 528 535 541 1 547 553 559 566 1 213 415 6 7 8 9 i I ~[33] I ~N. ' 0 1 2 3 4 5 6 7 8 9 -i miimm - - 700 01 02 03 04 05 06 07 08 09 710 11 12 13 14 15 16 17 18 19 720 21 22 23 24 25 26 27 28 29 730 31 32 33 34 35 36 37 38 39 740 41 42 43 44 45 46 47 48 49 750 84 510 572 634 696 757 819 880 942 85 003 065 516 578 640 702 763 825 887 948 009 071 522 584 646 708 770 831 893 954 016 077 528 590 652 714 776 837 899 960 022 083 535 597 658 720 782 844 905 967 028 089 541 603 665 726 788 850 911 973 034 095 547 609 671 733 794 856 917 979 040 101 553 615 677 739 800 862 924 985 046 107 559 621 683 745 566 628 689 751 I 807 868 930 991 052 114 813 874 936 997 058 120 126 132 138 144 150 156 163 169 175 181 187 193 199 205 211 217 224 230 236 242 248 254 260 266 272 278 285 291 297 303 309 315 321 327 333 339 345 352 358 364 370 376 382 388 394 400 406 412 418 425 431 437 443 449 455 461 467 473 479 485 491 497 503 509 516 522 528 534 540 546 552 558 564 570 576 582 588 594 600 606 612 618 625 631 637 643 649 655 661 667 673 679 685 691 697 703 709 715 721 727 733 739 745 751 757 763 769 775 781 788 794 854 914 974 86 034 094 153 213 273 332 800 860 920 980 040 100 159 219 279 338 806 866 926 986 046 106 165 225 285 344 812 872 932 992 052 112 171 231 291 818 878 938 998 058 118 177 237 297 I 824 884 944 *004 064 124 183 243 303 830 890 950 *010 070 130 189 249 308 836 896 956 *016 076 136 195 255 314 842 902 962 *022 082 141 201 261 320 848 908 968 *028 088 147 207 267 326 -I I 350 356 362 368 374 380 386 392 390 404 410 415 421 427 433 439 445 451 457 463 469 475 481 487 493 499 504 510 516 522 528 534 540 546 552 558 564 570 576 581 587 593 599 605 6 617 623 629 635 641 646 652 658 664 670 676 682 688 694 700 705 711 717 723 729 735 741 747 753 759 764 770 776 782 788 794 800 806 812 817 823 829 835 841 847 853 859 864 870 876 882 888 894 900 906 911 917 923 929 935 941 947 953 958 964 970 976 -,__.. _., i X. X...........0 982 87 040 099 157 216 274 332 390 448 988 046 105 163 221 280 994 052 111 169 227 286 999 058 116 175 233 291 *005 064 122 181 239 297 I. *011 070 128 186 245 303 *017 075 134 192 251 309 *023 081 140 198 256 315 373 431 489 *029 087 146 204 262 320 379 437 495 *035 093 151 210 268 326 384 442 500 I m m 338 396 454 344 349 355 361 367 402 408 413 419 425 460 466 471 i477 483 A V_ 506 512 518 523 529 535 541 547 552 558 m I N. 0 1 2 3 4 5 6 7 8 9 [34] N. 0 1 2 3 415 6 7 8 9 - - - _~ _mm[ 750 51 52 53 54 55 56 57 58 59 760 61 62 63 64 65 66 67 68 69 770 71 72 73 74 75 76 77 18 79 780 81 82 83 84 85 86 87 88 89 790 91 92 93 94 95 96 97 98 99 800 87 506 512 518 523 529 564 622 679 737 795 852 910 967 88 024 570 628 685 743 800 858 915 973 030 576 633 691 749 806 864 921 978 036 581 639 697 754 812 869 927 '984 041 587 645 703 760 818 875 933 990 047 081 138 195 252 309 366 423 480 536 593 087 144 201 258 315 372 429 485 542 598 093 150 207 264 321 377 434 491 547 604 098 156 213 270 326 383 440 497 553 610 104 161 218 275 332 389 446 502 559 615 593 651 708 766 823 881 938 996 053 599 656 714 772 829 887 944 *001 058 604 662 720 777 835 892 950 *007 064 I - 535 541 547 552 558 610 668 726 783 841 898 955 *013 070 616 674 731 789 846 904 961 *018 076 110 167 224 281 338 395 451 508 564 621 116 173 230 287 343 400 457 513 570 627 121 178 235 292 349 406 463 519 576 632 127 184 241 298 355 412 468 525 581 638 133 190 247 304 360 417 474 530 587 643 649 655 660 666 672 I: 677 683 689 694 700 705 762 818 874 930 986 89 042 098 154 711 767 824 880 936 992 048 104 159 717 773 829 885 941 997 053 109 165 722 779 835 891 947 *003 059 115 170 728 784 840 897 953 *009 064' 120 176 734 790 846 902 958 *014 070 126 182 739 795 852 908 964 *020 076 131 187 745 801 857 913 969 *025 081 137 193 750 807 863 919 975 *031 087 143 198 756 812 868 926 981 *037 092 148 204 L 209 215 221 226 232 237 243 248 254 260 265 271 276 282 287 293 298 304 310 315 321 326 332 337 343 348 354 360 365 371 376 382 387 393 398 404 409 415 421 426 432 437 443 448 454 459 465 470 476 481 487 492 498 504 509 515 520 526 531 537 542 548 553 559 564 570 575 581 586 592 597 603 609 614 620 625 631 636 642 647 653 658 664 669 675 680 686 691 697 702 708 713 719 724 730 735 741 746 752 757 I 763 818 873 927 982 90 037 091 146 200 255 768 823 878 933 988 042 097 151 206 260 774 829 883 938 993 048 102 157 211 266 779 834 889 944 998 053 108 162 217 271 785 840 894 949 *004 059 113 168 222 276 I 790 845 900 955 *009 064 119 173 227 282 796 851 905 960 *015 069 124 179 233 287 801 807 856 911 966 *020 075 129 184 238 293 862 916 971 *026 080 135 189 244 298 812 867 922 977 *031 086 140 195 249 304 -1 V 309 314 320 325 331 336 342 347 352 358 I E a 9 -. 0 I I I - 6 7 N. I 0 1 2 3 4 5 6 7 8 9 [35] m N. 800 01 02 03 04 05 06 07 08 09 810 11 12 13 14 15 16 17 18 19 820 21 22 23 24 25 26 27 28 29 830 31 32 33 34 35 36 37 38 39 840 41 42 43 44 45 46 47 48 49 850 N. 0 1 2 3 14 15 6 7 8 9 90 309 314 320 325 331 336 342 347 363 417 472 526 580 634 687 741 795 369 423 477 531 585 639 693 747 800 374 428 482 536 590 644 698 752 806 380 434 488 542 596 650 703 757 811 385 439 493 547 601 655 709 763 816 390 445 499 553 607 660 714 768 822 396 450 504 558 612 666 720 773 827 401 455 509 563 617 671 725 779 832 352 407 461 515 569 623 677 730 784 838 358 412 466 520 574 628 682 736 789 843 849 902 956 91 009 062 116 169 222 275 328 381 434 487 540 593 645 698 751 803 855 908 960 92 012 065 117 169 221 273 324 376 854 907 961 014 068 121 174 228 281 334 387 440 492 545 598 651 703 756 808 861 859 913 966 020 073 126 180 233 286 339 392 445 498 551 603 656 709 761 814 866 865 918 972 025 078 132 185 238 291 344 870 924 977 030 084 137 190 243 297 350 875 929 982 036 089 142 196 249 302 355 881 934 988 041 094 148 201 254 3.07 360 886 940 993 046 100 153 206 259 312 365 891 945 998 052 105 158 212 265 318 371 897 950 *004 057 110 164 217 270 323 376 397 403 450 455 503 556 609 661 714 766 819 871 508 561 614 666 719 772 824 876 408 461 514 566 619 672 724 777 829 882 413 466 519 572 624 677 730 782 834 887 418 471 524 577 630 682 735 787 840 892 424 477 529 582 635 687 740 793 845 897 429 482 535 587 640 693 745 798 850 903 -1 913 965 018 070 122 174 226 278 330 381 918 971 023 075 127 179 231 283 335 387 924 976 028 080 132 184 236 288 340 392 929 981 033 085 137 189 241 293 345 397 934 986 038 091 143 195 247 298 350 402 939 991 044 096 148 200 252 304 355 407 944 997 049 101 153 205 257 309 361 412 950 *002 054 106 158 210 262 314 366 418 955 *007 059 111 163 215 267 319 371 423 428 433 438 443 449 454 459 464 469 474 480 485 490 495 500 505 511 516 521 526 531 536 542 547 552 557 562 567 572 578 583 588 593 598 603 609 614 619 624 629 634 639 645 650 655 660 665 670 675 681 686 691 696 701 706 711 716 722 727 732 737 742 747 752 758 763 768 773 778 783 788 793 799 804 809 814 819 824 829 834 840 845 850 855 860 865 870 875 881 886 891 896 901 906 911 916 921 927 932 937 ~~~~~~~~~I.... 942 947 952 957 962 967 973 978 983 988 - _ I - - I - - I - I - I I - _ - 0 1 2 3 4 5 6 7 8 9 m m L [36] N. 0 1 2 3 4 6 7 8 9 i n11 i n1 I i, in -- I - 850 51 52 53 54 55 56 57 58 59 860 61 62 63 64 65 66 67 68 69 870 71 72 73 74 75 76 77 78 79 880 81 82 83 84 85 86 87 88 89 890 91 92 93 94 95 96 97 98 99 900 92 942 947 952 957 962 993 93 044 095 146 197 247 298 349 399 998 049 100 151 202 252 303 354 404 *003 054 105 156 207 258 308 359 409 *008 059 110 161 212 263 313 364 414 *013 064 115 166 217 268 318 369 420 450 455 460 465 470 500 551 601 651 702 752 802 852 902 952 94 002 052 101 151 201 250 300 349 399 448 498 547 596 645 694 743 792 841 890 505 556 606 656 707 757 807 857 907 957 007 057 106 156 206 255 305 354 404 453 503 552 601 650 699 748 797 846 895 510 561 611 661 712 762 812 862 912 962 012 062 111 161 211 260 310 359 409 458 507 557 606 655 704 753 802 851 900 515 566 616 666 717 767 817 867 917 967 017 067 116 166 216 265 315 364 414 463 512 562 611 660 709 758 807 856 905 520 571 621 671 722 772 822 872 922 972 022 072 121 171 221 270 320 369 419 468 517 567 616 665 714 763 812 861 910 *018 069 120 171 222 273 323 374 425 *024 075 125 176 227 278 328 379 430 967 973 978 475 480 485 490 495 983 988 *029 080 131 181 232 283 334 384 435 *034 085 136 186 237 288 339 389 440 *039 090 141 192 242 293 344 394 445 526 576 626 676 727 777 827 877 927 977 027 077 126 176 226 275 325 374 424 473 522 571 621 670 719 768 817 866 915 531 581 631 682 732 782 832 882 932 982 032 082 131 181 231 280 330 379 429 478 527 576 626 675 724 773 822 871 919 536 586 636 687 737 787 837 887 937 987 037 086 -136 186 236 285 335 384 433 483 532 581 630 680 729 778 827 876 924 541 591 641 692 742 792 842 892 942 992 042 091 141 191 240 290 340 389 438 488 537 586 635 685 734 783 832 880 929 546 596 646 697 747 797 847 897 947 997 047 096 146 196 245 295 345 394 443 493 542 591 640 689. 738 787 836 885 934 I 930 988 95 036 085 134 182 231 279 328 376 944 993 041 090 139 187 236 284 332 381 949 998 046 095 143 192 240 289 337 386 954 *002 051 100 148 197 245 294 342 390 959 *007 056 105 153 202 250 299 347 395 963 *012 061 109 158 207 255 303 352 400 968 *017 066 114 163 211 260 308 357 405 973 *022 071 119 168 216 265 313 361 410 978 *027 075 124 173 221 270 318 366 415 983 *032 080 129 177 226 274 323 371 419 I 424 429 434 439 444 448 453 1458 463 468 - m I - N. 0 1I 2 1 3 4 5 6 7 8 9 [37] I. I0 1 4 i 68 9 N. I 0 1i 2 3 4 5 6 7 8 9 900 01 02 03 04 05 06 07 08 09 910 11 12 13 14 15 16 17 18 19 920 21 22 23 24 25 26 27 28 29 930 31 32 33 34 35 36 37 38 39 940 41 42 43 44 45 46 47 48 49 950 95 424 429 1 434 439 444 1 448 1 453 458 463 468 472 521 569 617 665 713 761 809 856 904 477 525 574 622 670 718 766 813 861 909 482 530 578 626 674 722 770 818 866 914 487 535 583 631 679 727 775 823 871 918 492 540 588 636 684 732 780 828 875 923 497 545 593 641 689 737 785 832 880 928 501 550 598 646 694 742 789 837 885 933 506 554 602 650 698 746 794 842 890 938 511 559 607 655 703 751 799 847 895 942 516 564 612 660 708 756 804 852 899 947 952 999 96 047 095 142 190 237 284 332 957 *004 052 099 147 194 242 289 336 961 *009 057 104 152 199 246 294 341 966 *014 061 109 156 204 251 298 346 971 *019 066 114 161 209 256 303 350 976 *023 071 118 166 2-13 261 308 355 980 *028 076 123 171 218 265 313 360 985 *033 080 128 175 223 270 317 365 990 *038 085 133 180 227 275 322 369 995 *042 090 137 185 232 280 327 374 379 426 473 520 567 614 661 708 755 802 384 431 478 525 572 619 666 713 759 806 388 435 483 530 577 624 670 717 764 811 393 440 487 534 581 628 675 722 769 816 398 445 492 539 586 633 680 727 774 820 402 450 497 544 591 638 685 731 778 825 407 454 501 548 595 642 689 736 783 830 412 459 506 553 600 647 694 741 788 834 417 464 511 558 605 652 699 745 792 839 421 468 515 562 609 656 703 750 797 844 848 895 942 988 97 035 081 128 174 220 267 853 900 946 993 039 086 132 179 225 271 858 904 951 997 044 090 137 183 230 276 862 909 956 *002 049 095 142 188 234 280 867 914 960 *007 053 100 146 192 239 285 872 918 965 *011 058 104 151 197 243 290 876 923 970 *016 063 109 155 202 248 294 881 928 974 *021 067 114 160 206 253 299 886 932 979 *025 072 118 165 211 257 304 890 937 984 *030 077 123 169 216 262 308 313 317 322 327 331 336 340 345 350 354 313/111 I _321 31133 1O5 350 35 359 405 451 497 543 589 635 681 727 772 364 410 456 502 548 594 640 685 731 777 368 414 460 506 552 598 644 690 736 782 373 419 465 511 557 603 649 695 740 786 377 424 470 516 562 607 653 699 745 791 382 428 474 520 566 612 658 704 749 795 387 433 479 525 571 617 663 708 754 800 391 437 483 529 575 621 667 713 759 804 396 442 488 534 580 626 672 717 763 809 400 447 493 539 585 630 676 722 768 813 N. 0 1 2 3 4 5 6 7 8 9! m i i mi mi mi [38] 1 N. 0 1 2 3 4 5 6 7 8 9 - - I - - - -mmm~ I 950 51 52 53 54 55 56 57 58 59 960 61 62 63 64 65 66 67 68 69 970 71 72 73 74 75 76 77 78 79 980 81 82 83 84 85 86 87 88 89 990 91 92 93 94 95 96 97 98 99 1000 97 772 818 864 909 955 98 000 046 091 137 182 777 823 868 914 959 005 050 096 141 186 782 827 873 918 964 009 055 100 146 191 -I 786 832 877 923 968 014 059 105 150 195 791 836 882 928 973 019 064 109 155 200 795 841 886 932 978 023 068 114 159 204 227 272 318 363 408 453 498 543 588 632 677 722 767 811 856 900 945 989 99 034 078 I -1 232 277 322 367 412 457 502 547 592 637 682 726 771 816 860 905 949 994 038 083 236 281 327 372 417 462 507 552 597 641 686 731 776 820 865 909 954 998 043 087.. 241 286 331 376 421 466 511 556 601 646 691 735 780 825 869 914 958 *003 047 092 I - 245 290 336 381 426 471 516 561 605 650 695 740 784 829 874 918 963 *007 052 096 250 295 340 385 430 475 520 565 610 655 I 700 744 789 834 878 923 967 *012 056 100 -I 800 845 891 937 982 028 073 118 164 209 254 299 345 390 435 480 525 570 614 659 I 804 850 896 941 987 032 078 123 168 214 259 304 349 394 439 484 529 574 619 664 I - - I 809 855 900 946 991 037 082 127 173 218 263 308 354 399 444 489 534 579 623 668 813 859 905 950 996 041 087 132 177 223 268 313 358 403 448 493 538 583 628 673 - 704 749 793 838 883 927 972 *016 061 105 -1 709 753 798 843 887 932 976 *021 065 109 713 758 802 847 892 936 981 *025 069 114 I 717 762 807 851 896 941 985 *029 074 118 -1 - -1 123 167 211 255 300 344 388 432 476 520 127 171 216 260 304 348 392 436 480 524 131 176 220 264 308 352 396 441 484 528 -1 -1 136 180 224 269 313 357 401 445 489 533 140 185 229 273 317 361 405 449 493 537 I 145 189 233 277 322 366 410 454 498 542 149 - I I I 564 568 572 577 581 607 651 695 739 782 826 870 913 957 00 000 612 656 699 743 787 830 874 917 961 004 -I 616 660 704 747 791 835 878 922 965 009 621 664 708 752 795 839 883 926 970 013 -1 --- I 625 669 712 756 800 843 887 930 974 017 585 193 238 282 326 370 414 458 502 546 590 634 677 721 765 808 852 896 939 983 026 I 594 154 198 242 286 330 374 419 463 506 550 I 1 202 247 291 335 379 423 467 511 555 I 158 I 207 251 295 339 383 427 471 515 559 603 162 I I 599 629 673 717 760 804 848 891 935 978 022 638 682 726 769 813 856 900 944 987 030 642 686 730 774 817 '861 904 948 991 035 647 691 734 778 822 865 909 952 996 039 I N.~ 0~! 2 3 41 U 7 8 9 N. [ L ' 1 2 3 4 5 6 7 8 9 [39] TABLE II LOGS AND COLOGS OF CERTAIN MUCH-USED NUMBERS NUMBER LOGARITHM COLOGARITIHM 2 0.3010300 9.6989700-10 3 0.4771213 9.5228787-10 '/2 0.1505150 9.8494850-10 V/3 0.2385607 9.7614393-10 r 0.4971499 9.5028501-10 7r2 0.9942997 9.0057003-10 2 7r 0.7981799 9.2018201-10 /7r T0.2485749 9.7514251-10 57.2957795 1.7581226 8.2418774-10 206264.806 5.3144251 4.6855749-10 FIVE PLACE 2 0.30103 9.69897-10 3 0.47712 9.52288-10 XV2 0.15052 9.84948-10 V/3 0.23856 9.76144-10 7 0.49715 9.50285-10 7r 0.99430 9.00570-10 2 r 0.79818 9.20182-10 /w- 0.24857 9.75143-10 57.2957795 1.75812 8.24188-10 206264.806 5.31443 4.68557-10 FoUR PLACE 2 0.3010 9.6990-10 3 0.4771 9.5229-10 V2 0.1505 9.8495-10 V3 0.2386 9.7614-10 r 0.4971 9.5029-10 7r2 0.9943 9.0057-10 2 0.7982 9.2018-10 V\/-r 0.2486 9.7514-10 57.2956695 1.7581 8.2419-10 206264.806 5.3144 4.6858-10 [40] TABLE III FIVE-PLACE LOGARITHMS OF THE SINE, COSINE, TANGENT, AND COTANGENT FOR EACH MINUTE OF THE QUADRANT [41] - T I I I V ' I I I I L. Sin. L. Tang. L. Cotg. L. Cos. T i- -m 0o 0 oo co 0.00 000 60 1 6.46 373 6.46 373 3.53 627 0.00 000 59 2 6.76 476 6.76 476 3.23 524 0.00 000 58 3 6.94 085 6.94 085 3.05 915 0.00 000 57 4 7.06 579 7.06 579 2.93 421 0.00 000 56 5 7.16 270 7.16 270 2.83 730 0.00 000 55 6 7.24 188 7.24 188 2.75 812 0.00 000 54 7 7.30 882 7.30 882 2.69 118 0.00 000 53 8 7.36 682 7.36 682 2.6a 318 0.00 000 52 9 7.41 797 7.41 797 2.58 203 0.00 000 51 10 7.46 373 7.46 373 2.53 627 0.00 000 50 11 7.50 512 7.50 512 2.49 488 0.00 000 49 12 7.54 291 7.54 291 2.45 709 0.00 000 48 13 7.57 767 7.57 767 2.42 233 0.00 000 47 14 7.60 985 7.60 986 2.39 014 0.00 000 46 15 7.63 982 7.63 982 2.36 018 0.00 000 45 16 7.66 784 7.66 785 2.33 215 0.00 000 44 17 7.69 417 7.69 418 2.30 582 9.99 999 43 18 7.71 900 7.71 900 2.28 100 9.99 999 42 19 7.74 248 7.74 248 2.25 752 9.99 999 41 20 7.76 475 7.76 476 2.23 524 9.99 999 40 21 7.78 594 7.78 595 2.21 405 9.99 999 39 22 7.80 615 7.80 615 2.19 385 9.99 999 38 23 7.82 545 7.82 546 2.17 454 9.99 999 37 24 7.84 393 7.84 394 2.15 606 9.99 999 36 25 7.86 166 7.86 167 2.13 833 9.99 999 35 26 7.87 870 7.87 871 2.12 129 9.99 999 34 27 7.89 509 7.89 510 2.10 490 9.99 999 33 28 7.91 088 7.91 089 2.08 911 9.99 999 32 29 7.92 612 7.92 613 2.07 387 9.99 998 31 30 7.94 084 7.94 086 2.05 914 9.99 998 30 31 7.95 508 7.95 510 2.04 490 9.99 998 29 32 7.96 887 7.96 889 2.03 111 9.99 998 28 33 7.98 223 7.98 225 2.01 775 9.99 998 27 34 7.99 520 7.99 522 2.00 478 9.99 998 26 35 8.00 779 8.00 781 1.99 219 9.99 998 25 36 8.02 002 8.02 004 1.97 996 9.99 998 24 37 8.03 192 8.03 194 1.96 806 9.99 997 23 38 8.04 350 8.04 353 1.95 647 9.99 997 22 39 8.05 478 8.05 481 1.94 519 9.99 997 21 40 8.06 578 8.06 581 1.93 419 9.99 997 20 41 8.07 650 8.07 653 1.92 347 9.99 997 19 42 8.08 696 8.08 700 1.91 300 9.99 997 18 43 8.09 718 8.09 722 1.90 278 9.99 997 17 44 8.10 717 8.10 720 1.89 280 9.99 996 16 45 8.11 693 8.11 696 1.88 304 9.99 996 15 46 8.12 647 8.12 651 1.87 349 9.99 996 14 47 8.13 581 8.13 585 1.86 415 9.99 996 13 48 8.14 495 8.14 500 1.85 500 9.99 996 12 49 8.15 391 8.15 395 1.84 605 9.99 996 11 50 8.16 268 8.16 273 1.83 727 9.99 995 10 51 8.17 128 8.17 133 1.82 867 9.99 995 9 52 8.17 971 8.17 976 1.82 024 9.99 995 8 53 8.18 798 8.18 804 1.81 196 9.99 995 7 54 8.19 610 8.19 616 1.80 384 9.99 995 6 55 8.20 407 8.20 413 1.79 587 9.99 994 5 56 8.21 189 8.21 195 1.78 805 9.99 994 4 57 8.21 958 8.21 964 1.78 036 9.99 994 3 58 8.22 713 8.22 720 1.77 280 9.99 994 2 59 8.23 456 8.23 462 1.76 538 9.99 994 1 60 8.24 186 8.24 192 1.75 808 9.99 993 0 89~ 1 L. Cos. L. Cotg. I L. Tang. L. Sin. I f L [42] 1 i Mr I L.Sin. | L.Tang. Ml 1 L. Cotg. 1 L, Cos. m m 0 8.24 186 8.24 192 1.75 808 9.99 993 60 1 8.24 903 8.24 910 1.75 090 9.99 993 59 2 8.25 609 8.25 616 1.74 384 9.99 993 58 3 8.26 304 8.26 312 1.73 688 9.99 993 57 4 8.26 988 8.26 996 1.73 004 9.99 992 56 5 8.27 661 8.27 669 1.72 331 9.99 992 55 6 8.28 324 8.28 332 1.71 668 9.99 992 54 7 8.28 977 8.28 986 1.71 014 9.99 992 53 8 8.29 621 8.29 629 1.70 371 ' 9.99 992 52 9 8.30 255 8.30 263 1.69 737 9.99 991 51 10 8.30 879 8.30 888 1.69 112 9.99 991 50 11 8.31 495 8.31 505 1.68 495 9.99 991 49 12 8.32 103 8.32 112 1.67 888 9.99 990 48 13 8.32 702 8.32 711 1.67 289 9.99 990 47 14 8.33 292 8.33 302 1.66 698 9.99 990 46 15 8.33 875 8.33 886 1.66 114 9.99 990 45 16 8.34 450 8.34 461 1.65 539 9.99 989 44 17 8.35 018 8.35 029 1.64 971 9.99 989 43 18 8.35 578 8.35 590 1.64 410 9.99 989 42 19 8.36 131 8.36 143 1.63 857 9.99 989 41 20 8.36 678 8.36 689 1.63 311 9.99 988. 40 21 8.37 217 8.37 229 1.62 771 9.99 988 39 22 8.37 750 8.37 762 1.62 238 9.99 988 38 23 8.38 276 8.38 289 1.61 711 9.99 987 37 24 8.38 796 8.38 809 1.61 191 9.99 987 36 25 8.39 310 8.39 323 1.60 677 9.99 987 35 26 8.3.9 818 8.39 832 1.60 168 9.99 986 34 27 8.40 320 8.40 334 1.59 666 9.99 986 33 28 8.40 816 8.40 830 1.59 170 9.99 986 32 29 8.41 307 8.41 321 1.58 679 9.99 985 31 88c 30 8.41 792 8.41 807 1.58 193 9.99 985 30 31 8.42 272 8.42 287 1.57 713 9.99 985 29 32 8.42 746 8.42 762 1.57 238 9.99 984 28 33 8.43 216 8.43 232 1.56 768 9.99 984 27 34 8.43 680 8.43 696 1.56 304 2.99 984 26 35 8.44 139 8.44 156 1.55 844 9.99 983 25 36 8.44 594 8.44 611 1.55 389 9.99 983 24 37 8.45 044 8.45 061 1.54 939 9.99 983 23 38 8.45 589 8.45 507 1.54 493 9.99 982 22 39 8.45 930 8.45 948 1.54 052 9.99 982 21 40 8.46 366 8.46 385 1.53 615 9.99 982 20 41 8.46 799 8.46 817 1.53 183 9.99 981 19 42 8.47 226 8.47 245 1.52 755 9.99 981 18 43 8.47 650 8.47 669 1.52 331 9.99 981 17 44 8.48 069 8.48 089 1.51 911 9.99 980 16 45 8.48 485 8.48 505 1.51 495 9.99 980 15 46 8.48 896 8.48 917 1.51 083 9.99 979 14 47 8.49 304 8.49 325 1.50 675 9.99 979 13 48 8.49 708 8.49 729 1.50 271 9.99 979 12 49 8.50 108 8.50 130 1.49 870 9.99 978 11 50 8.50 504 8.50 527 1.49 473' 9.99 978 10 51 8.50 897 8.50 920 1.49 080 9.99 977 9 52 8.51 287 8.51 310 1.48 690 9.99 977 8 53 8.51 673 8.51 696 1.48 304 9.99 977 7 54 8.52 055 8.52 079 1.47 921 9.99 976 6 55 8.52 434 8.52 459 1.47 541 9.99 976 5 56 8.52 810 8.52 835 1.47 165 9.99 975 4 57 8.53 183 8.53 208 1.46 792 9.99 975 3 58 8.53 552 8.53 578 1.46 422 9.99 974 2 59 8.53 919 8.53 945 1.46 055 9.99 974 1 60 8.54 282 8.54 308 1.45 692 9.99 974 0 I I L. Cos. I -m -IF -I L. Cotg. L. Tang. L. Sin. ' &M Lri la [43] - Pi -I I L. Sin. T a a F F _ —I L. Tang. L. Cotg. L. Cos. 0 8.54 282 8.54 308 1.45 692 9.99 974 60 1 8.54 642 8.54 669 1.45 331 9.99 973 59 2 8.54 999 8.55 027 1.44 973 9.99 973 58 3 8.55 354 8.55 382 1.44 618 9.99 972 57 4 8.55 705 8.55 734 1.44 266 9.99 972 56 5 8.56 054 8.56 083 1.43 917 9.99 971 55 6 8.56 400 8.56 429 1.43 571 9.99 971 54 7 8.56 743 8.56 773 1.43 227 9.99 970 53 8 8.57 084 8.57 114 1.42 886 9.99 970 52 9 8.57 421 8.57 452 1.42 548 9.99 969 51 10 8.57 757 8.57 788 1.42 212 9.99 969 50 11 8.58 089 8.58 121 1.41 879 9.99 968 49 12 8.58 419 8.58 451 1.41 549 9.99 968 48 13 8.58 747 8.58 779 1.41 221 9.99 967 47 14 8.59 072 8.59 105 1.40 895 9.99 967 46 15 8.59 395 8.59 428 1.40 572 9.99 967 45 16 8.59 715 8.59 749 1.40 251 9.99 966 44 17 8.60 033 8.60 068 1.39 932 9.99 966 43 18 8.60 349 8.60 384 1.39 616 9.99 965 42 19 8.60 662 8.60 698 1.39 302 9.99 964 41 20 8.60 973 8.61 009 1.38 991 9.99 964 40 21 8.61 282 8.61 319 1.38 681 9.99 963 39 22 8.61 589 8.61 626 1.38 374 9.99 963 38 23 8.61 894 8.61 931 1.38 069 9.99 962 37 24 8.62 196 8.62 234 1.37 766 9.99 962 36...~~~~~~~~ 25 26 27 28 29 8.62 497 8.62 795 8.63 091 8.63 385 8.63 678 I 8.62 535 8.62 834 8.63 131 8.63 426 8.63 718 1.37 465 1.37 166 1.36 869 1.36 574 1.36 282 9.99 961 9.99 96. 9.99 960 9.99 960 9.99 959 35 34 33 32 31 30 31 32 33 34 8.63 968 8.64 256 8.64 543 8.64 827 8.65 110 8.64 009 8.64 298 8.64 585 8.64 870 8.65 154 1.35 991 1.35 702 1.35 415 1.35 130 1.34 846 9.99 959 9.99 958 9.99 958 9.99 957 9.99 956 30 29 28 27 26 2 35 8.65 391 8 65 435 1.34 565 9.99 956 25 36 8.65 670 8.65 715 1.34 285 9.99 955 24 37 8.65 947 8.65 993 1.34 007 9.99 955 23 38 8.66 223 8.66 269 1.33 731 9.99 954 22 39 8.66 497 8.66 543 1.33 457 9.99 954 21 40 8.66 769 8.66 816 1.33 184 9.99 953 20 41 8.67 039 8.67 087 1.32 913 9.99 952 19 42 8.67 308 8.67 356 1.32 644 9.99 952 18 43 8.67 575 8.67 624 1.32 376 9.99 951 17 44 8.67 841 8.67 890 1.32 110 9.99 951 16 45 8.68 104 8.68 154 1.31 846 9.99 950 15 46 8.68 367 8.68 417 1.31 583 9.99 949 14 47 8.68 627 8.68 678 1.31 322 9.99 949 13 48 8.68 886 8.68 938 1.31 062 9.99 948 12 49 8.69 144 8.69 196 1.30 804 9.99 948 11 50 8.69 400 8.69 453 1.30 547 9.99 947 10 51 8.69 654 8.69 708 1.30 292 9.99 946 9 52 8.69 907 8.69 962 1.30 038 9.99 946 8 53 8.70 159 8.70 214 1.29 786 9.99 945 7 54 8.70 409 8.70 465 1.29 535 9.99 944 6 55 8.70 658 8.70 714 1.29 286 9.99 944 5 56 8.70 905 8.70 962 1.29 038 9.99 943 4 57 8.71 151 8.71 208 1.28 792 9.99 942 3 58 8.71 395 8.71 453 1.28 547 9.99 942 2 59 8.71 638 8.71 697 1.28 303 9.99 941 1 60 8.71 880 8.71 940 1.28 060 9.99 940 0 _ I _~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ L. Cos. L. Cotg. L. Tang. L. Sin. I. [44] or~ or ' L. Sin. L. Tang. | L. Cotg. I L. Cos. m -JIL 30 0 8.71 880 8.71 940 1.28 060 9.99 940 60 1 8.72 120 8.72 181 1.27 819 9.99 940 59 2 8.72 359 8.72 420 1.27 580 9.99 939 58 3 8.72 597 8.72 659 1.27 341 9.99 938 57 4 8.72 834 8.72 896 1.27 104 9.99 938 56 5 8.73 069 8.73 132 1.26 868 9.99 937 55 6 8.73 303 8.73 366 1.26 634 9.99 936 54 7 8.73 535 8.73 600 1.26 400 9.99 936 53 8 8.73 767 8.73 832 1.26 168 9.99 935 52 9 8.73 997 8.74 063 1.25 937 9.99 934 51 10 8.74 226 8.74 292 1.25 708 9.99 934 50 11 8.74 454 8.74 521 1.25 479 9.99 933 49 12 8.74 680 8.74 748 1.25 252 9.99 932 48 13 8.74 906 8.74 974 1.25 026 9.99 932 47 14 8.75 130 8.75 199 1.24 801 9.99 931 46 15 8.75 353 8.75 423 1.24 577 9.99 930 45 16 8.75 575 8.75 645 1.24 355 9.99 929 44 17 8.75 795 8.75 867 1.24 133 9.99 929 43 18 8.76 015 8.76 087 1.23 913 9.99 928 42 19 8.76 234 8.76 306 1.23 694 9.99 927 41 20 8.76 451 8.76 525 1.23 475 9.99 926 40 21 8.76 667 8.76 742 1.23 258 9.99 926 39 22 8.76 883 8.76 958 1.23 042 9.99 925 38 23 8.77 097 8.77 173 1.22 827 9.99 924 37 24 8.77 310 8.77 387 1.22 613 9.99 923 36 25 8.77 522 8.77 600 1.22 400 9.99 923 35 26 8.77 733 8.77 811 1.22 189 9.99 922 34 27 8.77 943 8.78 022 1.21 978 9.99 921 33 28 8.78 152 8.78 232 1.21 768 9.99 920 32 29 8.78 360 8.78 441 1.21 559 9.99 920 31 8 c 30. 8.78 568 8.78 649 1.21 351 9.99 919 30 31 8.78 774 8.78 855 1.21 145 9.99 918 29 32 8.78 979 8.79 061 1.20 939 9.99 917 28 33 8.79 183 8.79 266 1.20 734 9.99 917 27 34 8.79 386 8.79 470 1.20 530 9.99 916 26 35 8.79 588 8.79 673 1.20 327 9.99 915 25 36 8.79 789 8.79 875 1.20 125 9.99 914 24 37 8.79 990 8.80 076 1.19 924 9.99 913 23 38 8.80 189 8.80 277 1.19 723 9.99 913 22 39 8.80 388 8.80 476 1.19 524 9.99 912 21 40 8.80 585 8.80 674 1.19 326 9.99 911 20 41 8.80 782 8.80 872 1.19 128 9.99 910 19 42 8.80 978 8.81 068 1.18 932 9.99 909 18 43 8.81 173 8.81 264 1.18 736 9.99 909 17 44 8.81 367 8.81 459 1.18 541 9.99 908 16 45 8.81 560 8.81 653,1.18 347 9.99 907 15 46 8.81 752 8.81 846 1.18 154 9.99 906 14 47 8.81 944 8.82 038 1.17 962 9.99 905 13 48 8.82 134 8.82 230 1.17 770 9.99 904 12 49 8.82 324 8.82 420 1.17 580 9.99 904 11 50 8.82 513 8.82 610 1.17 390 9.99 903 10 51 8.82 701 8.82 799 1.17 201 9.99 902 9 52 8.82 888 8.82 987 1.17 013 9.99 901 8 53 8.83 075 8.83 175 1.16 825 9.99 900 7 54 8.83 261 8.83 361 1.16 639 9.99 899 6 55 8.83 446 8.83 547 1.16 453 9.99 898 5 56 8.83 630 8.83 732 1.16 268 9.99 898 4 57 8.83 813 8.83 916 1.16 084 9.99 897 3 58 8.83 996 8.84 100 1.15 900 9.99 896 2 59 8.84 177 8.84 282 1.15 718 9.99 895 1 60 8.84 358 8.84 464 1.15 536 9.99 894 0 D m -m L. Cos. L. Cotg. L. Tang. L. Sin. ' 0 m 5 - U - - r n a - m [45] - 40 -MI ~O ii IB -r L. Sin. L. Tang. L. Cotg. L. Cos. --— I I I III I I M 0 8.84 358 8.84 464 1.15 536 9.99 894 60 1 8.84 539 8.84 646 1.15 354 9.99 893 59 2 8.84 718 8.84 826 1.15 174 9.99 892 58 3 8.84 897 8.85 006 1.14 994 9.99 891 57 4 8.85 075 8.85 185 1.14 815 9.99 891 56 5 8.85 252 8.85 363 1.14 637 9.99 890 55 6 8.85 429 8.85 540 1.14 460 9.99 889 54 7 8.85 605 8.85 717 1.14 283 9.99 888 53 8 8.85 780 8.85 893 1.14 107 9.99 887 52 9 8.85 955 8.86 069 1.13 931 9.99 886 51 10 8.86 128 8.86 243 1.13 757 9.99 885 50 11 8.86 301 8.86 417 1.13 583 9.99 884 49 12 8.86 474 8.86 591 1.13 409 9.99 883 48 13 8.86 645 8.86 763 1.13 237 9.99 882 47 14 8.86 816 8.86 935 1.13 065 9.99 881 46 15 8.86 987 8.87 106 1.12 894 9.99 880 45 16 8.87 156 8.87 277 1.12 723 9.99 879 44 17 8.87 325 8.87 447 1.12 553 9.99 879 43 18 8.87 494 8.87 616 1.12 384 9.99 878 42 19 8.87 661 8.87 785 1.12 215 9.99 877 41 20 8.87 829 8.87 953 1.12 047 9.99 876 40 21 8.87 995 8.88 120 1.11 880 9.99 875 39 22 8.88 161 8.88 287 1.11 713 9.99 874 38 23 8.88 326 8.88 453 1.11 547 9.99 873 37 24 8.88 490 8.88 618 1.11 382 9.99 872 36 25 8.88 654 8.88 783 1.11 217 9.99 871 35 26 8.88 817 8.88 948 1.11 052 9.99 870 34 27 8.88 980 8.89 111 1.10 889 9.99 869 33 28 8.89 142 8.89 274 1.10 726 9.99 868 32 29 8.89 304 8.89 437 1.10 563 9.99 867 31 30 8.89 464 8.89 598 1.10 402 9.99 866 30 31 8.895625 8.89 760 1.10 240 9.99 865 29 32 8.89 784 8.89 920 1.10 080 9.99 864 28 33 8.89 943 8.90 080 1.09 920 9.99 863 27 34 8.90 102 8.90 240 1.09 760 9.99 862 26 35 8.90 260 8.90 399 1.09 601 9.99 861 25 36 8.90 417 8.90 557 1.09 443 9.99 860 24 37 8.90 574 8.90 715 1.09 285 9.99 859 23 38 8.90 730 8.90 872 1.09 128 9.99 858 22 39 8.90 885 8.91 029 1.08 971 9.99 857 21 40 8.91 040 8.91 185 1.08 815 9.99 856 20 41 8.91 195 8.91 340 1.08 660 9.99 855 19 42 8.91 349 8.91 495 1.08 505 9.99 854 18 43 8.91 502 8.91 650 1.08 350 9.99 853 17 44 8.91 655 8.91 803 1.08 197 9.99 852 16 45 8.91 807 8.91 957 1.08 043 9.99 851 15 46 8.91 959 8.92 110 1.07 890 9.99 850 14 47 8.92 110 8.92 262 1.07 738 9.99 848 13 48 8.92 261 8.92 414 1.07 586 9.99 847 12 49 8.92 411 8.92 565 1.07 435 9.99 846 11 50 8.92 561 8.92 716 1.07 284 9.99 845 10 51 8.92 710 8.92 866 1.07 134 9.99 844 9 52 8.92 859 8.93 016 1.06 984 9.99 843 8 53 8.93 007 8.93 165 1.06 835 9.99 842 7 54 8.93 154 8.93 313 1.06 687 9.99 841 6 55 8.93 301 8.93 462 1.06 538 9.99 840 5 56 8.93 448 8.93 609 1.06 391 9.99 839 4 57 8.93 594.8.93 756 1.06 244 9.99 838 3 58 8.93 740 8.93 903 1.06 097 9.99 837 2 59 8.93 885 8.94 049 1.05 951 9.99 836 1 60 8.94 030 8.94 195 1.05 805 9.99 834 0 ]l.,l ii iii- - u I L. Cos. L. Cotg. I L. Tang. L. Sin. I / PI I i i i ii - [46] a i I L.Sin. T Yr a, I L. Tang. L. Cotg. L. Cos. 0 8.94 030 8.94 195 1.05 805 9.99 834 60 1 8.94 174 8.94 340 1.05 660 9.99 833 59 2 8.94 317 8.94 485 1.05 515 9.99 832 58 3 8.94 461 8.94 630 1.05 370 9.99 831 57 4 8.94 603 8.94 773 1.05 227 9.99 830 56 5 8.94 746 8.94 917 1.05 083 9.99 829 55 6 8.94 887 ~8.95 060 1.04 940 9.99 828 54 7 8.95 029 8.95 202 1.04 798 9.99 827 53 8 8.95 170 8.95 344 1.04 656 9.99 825 52 9 8.95 310 8.95 486 1.04 514 9.99 824 51 10 8.95 450 8.95 627 1.04 373 9.99 823 50 11 8.95 589 8.95 767 1.04 233 9.99 822 49 12 8.95 728 8.95 908 1.04 092 9.99 821 48 13 8.95 867 8.96 047 1.03 953 9.99 820 47 14 8.96 005 8.96 187 1.03 813 9.99 819 46 15 8.96 143 8.96 325 1.03 675 9.99 817 45 16 8.96 280 8.96 464 1.03 536 9.99 816 44 17 8.96 417 8.96 602 1.03 398 9.99 815 43 18 8.96 553 8.96 739 1.03 261 9.99 814 42 19 8.96 689 8.96 877 1.03 123 9.99 813 41 20 8.96 825 8.97 013 1.02 987 9.99 812 40 21 8.96 960 8.97 150 1.02 850 9.99 810 39 22 8.97 095 8.97 285 1.02 715 9.99 809 38 23 8.97 229 8.97 421 1.02 579 9.99 808 37 24 8.97 363 8.97 556 1.02 444 9.99 807 36 25 8.97 496 8.97 691 1.02 309 9.99 806 35 26 8.97 629 8.97 825 1.02 175 9.99 804 34 27 8.97 762 8.97 959 1.02 041 9.99 803 33 28 8.97 894 8.98 092 1.01 908 9.99 802 32 29 8.98 026 8.98 225 1.01 775 9.99 801 31 SXc 30 8.98 157 8.98 358 1.01 642 9.99 800 30 31 8'~8 288 8.98 490 1.01 510 9.99 798 29 32 8.98 419 8.98 622 1.01 378 9.99 797 28 33 8.98 549 8.98 753 1.01 247 9.99 796 27 34 8.98 679 8.98 884 1.01 116 9.99 795 26 35 8.98 808 8.99 015 1.00 985 9.99 793 25 36 8.98 937 8.99 145 1.00 855 9.99 792 24 37 8.99 066 8.99 275 1.00 725 9.99 791 23 38 8.99 194 8.99 405 1.00 595 9.99 790 22 39 8.99 322 8.99 534 1.00 466 9.99 788, 21 40 8.99 450 8.99 662 1.00 338 9.99 787 20 41 8.99 577 8.99 791 1.00 209 9.99 786 19 42 8.99 704 8.99 919 1.00 081 9.99 785 18 43 8.99 830 9.00 046 0.99 954 9.99 783 17 44 8.99 956 9.00 174 0.99 826 9.99 782 16 45 9.00 082 9.00 301 0.99 699 9.99 781 15 46 9.00 207 9.00 427 0.99 573 9.99 780 14 47 9.00 332 9.00 553 0.99 447 9.99 778 13 48 9.00 456 9.00 679 0.99 321 9.99 777 12 49 9.00 581 9.00 805 0.99 195 9.99 776 11 50 9.00 704 9.00 930 0.99 070 9.99 775 10 51 9.00 828 9.01 055 0.98 945 9.99 773 9 52 9.00 951 9.01 179 0.98 821 9.99 772 8 53 9.01 074 9.01 303 0.98 697 9.99 771 7 54 9.01 196 9.01 427 0.98 573 9.99 769 6 55 9.01 318 9.01 550 0.98 450 9.99 768 5 56 9.01 440 9.01 673 0.98 327 9.99 767 4 57 9.01 561 9.01 796 0.98 204 9.99 765 3 58 9.01 682 9.01 918 0.98 082 9.99 764 2 59 9.01 803 9.02 040 0.97 960 9.99 763 1 60 9.01 923 9.02 162 0.97 838 9.99 761 0 L. Cos. I L. Cotg. L. Tang. L. Sin. I 0 h [47 ~I C umm" r '_ I L. Sin. Im~ r L. Tang. III I r I L. Cotg. L. Cos., I r I 0 9.01 923 9.02 162 0.97 838 9.99 761 60 1 9.02 043 9.02 283 0.97 717 9.99 760 59 2 9.02 163 9.02 404 0.97 596 9.99 759 58 3 9.02 283 9.02 525 0.97 475 9.99 757 57 4 9.02 402 9.02 645 0.97 355 9.99 756 56 5 9.02 520 9.02 766 0.97 234 9.99 755 55 6 9.02 639 9.02 885 0.97 115, 9.9!9753 54 7 9.02 757 9.03 005 0.96 995 9.99 752 53 8 9.02 874 9.03 124 0.96 876 9.99 751 52 9 9.02 992 9.03 242 0.96 758 9.99 749 51 10 9.03 109 9.03 361 0.96 639 9.99 748 50 11 9.03 226 9.03 479 0.96 521 9.99 747 49 12 9.03 342 9.03 597 0.96 403 9.99 745 48 13 9.03 458 9.03 714 0.96 286 9.99 744 47 14 9.03 574 9.03 832 0.96 168 9.99 742 46 15 9.03 690 9.03 948 0.96 052 9.99 741 45 16 9.03 805 9.04 065 0.95 935 9.99 740 44 17 9.03 920 9.04 181 0.95 819 9.99 738 43 18 9.04 034 9.04 297 0.95 703 9.99 737 42 19 9.04 149 9.04 413 0.95 587 9.99 736 41 20 9.04 262 9.04 528 0.95 472 9.99 734 40 21 9.04 376 9.04 643 0.95 357 9.99 733 39 22 9.04 490 9.04 758 0.95 242 9.99 731 38 23 9.04 603 9.04 873 0.95 127 9.99 730 37 24 9.04 715 9.04 987 0.95 013 9.99 728 36,,~~~~~~~~~~~~~~~~~~~~~~~~ 1! I 25 26 27 ~28 29 9.04 828 9.04 940 9.05 052 9.05 164 9.05 275 I 9.05 101 9.05 214 9.05 328 9.05 441 9.05 553 0.94 899 0.94 786 0.94 672 0.94 559 0.94 447 I 9.99 727 9.99 726 9.99 724 9.99 723 9.99 721 I 35 34 33 32 31 III S 0' I 30 9.05 386 9.05 666 0.94 334 9.99 720 30 31 9.05 497 9.05 778 0.94 222 9.99 718 29 32 9.05 607 9.05 890 0.94 110 9.99 717 28 33 9.05 717 9.06 002 0.93 998 9.99 716 27 34 9.05 827 9.06 113 0.93 887 9.99 714 26 35 9.05 937 9.06 224 0.93 776 9.99 713 25 36 9.06 046 9.06 335 0.93 665 9.99 711 24 37 9.06 155 9.06 445 0.93 555 9.99 710 23 38 9.06 264 9.06 556 0.93 444 9.99 708 22 39 9.06 372 9.06 666 0.93 334 9.99 707 21 40 9.06 481 9.06 775 0.93 225 9.99 705 20 41 9.06 589 9.06 885 0.93 115 9.99 704 19 42 9.06 696 9.06 994 0.93 006 9.99 702 18 43 9.06 804 9.07 103 0.92 897 9.99 701 17 44 9.06 911 9.07 211 0.92 789 9.99 699 16 45 9.07 018 9.07 320 0.92 680 9.99 698 15 46 9.07 124 9.07 428 0.92 572 9.99 696 14 47 9.07 231 9.07 536 0.92 464 9.99 695 13 48 9.07 337 9.07 643 0.92 357 9.99 993 12 49 9.07 442 9.07 751 0.92 249 9.99 692 11 -50 9.07 548 9.07 858 0.92 142 9.99 690 10 51 9.07 653 9.07 964 0.92 036 9.99 689 9 52 9.07 758 9.08 071 0.91 929 9.99 687 8 53 9.07 863 9.08 177 0.91 823 9.99 686 7 54 9.07 968 9.08 283 0.91 717 9.99 684 6 55 9.08 072 9.08 389 0.91 611 9.99 683 5 56 9.08 176 9.08 495 0.91 505 9.99 681 4 57 9.08 280 9.08 600 0.91 400 9.99 680 3 58 9.08 383 9.08 705 0.91 295 9.99 678 2 59 9.08 486 9.08 810 0.91 190 9.99 677 1 60 9.08 589 9.08 914 0.91 086 9.99 675 0 _ ~ ~~~ ~~ ~ ~~~~~~~~~~~~~~~~~~~ iI.. _ 0 i L. Cos. ElI -m — m 0 L. Cotg. L. Tang. L. Sin. 1 Ol MA M& [ 48] I iL Mr NVVA ' L. Sin. L. Tang. L. Cotg. r L. Cos. r r 1 0 - it — m 0 9.08 589 9.08 914 0.91 086 9.99 675 60 1 9.08 692 ' 9.09 019 0.90 981 9.99 674 59 2 9.08 79~ 9.09 123 0.90 877 9.99 672 58 3 9.08 897 9.09 227 0.90 773 9.99 670 57 4 9.08 999 -9.09 330 0.90 670 9.99 669 56 5 9.09 101 9.09 434 0.90 566 9.99 667 55 6 9.09 202 9.09 537 0.90 463 9.99 666 54 7 9.09 304 9.09 640 0.90 360 9.99 664 53 8 9.09 405 9.09 742 0.90 258 9.99 663 52 9 9.09 506 9.09 845 0.90 155 9.99 661 51 10 9.09 606 9.09 947 0.90 053 9.99 659 50 11 9.09 707 9.10 049 0.89 951 9.99 658 49 12 9.09 807 9.10 150 0.89 850 9.99 656 48 13 9.09 907 9.10 252 0.89 748 9.99 655 47 14 9.10 006 9.10 353 0.89 647 9.99 653 46 15 9.10 106 9.10 454 0.89 546 9.99 651 45 16 9.10 205 9.10 555 0.89 445 9.99 650 44 17 9.10 304 9.10 656 0.89 344 9.99 648 43 18 9.10 402 9.10 756 0.89 244 9.99 647 42 19 9.10 501 9.10 856 0.89 144 9.99 645 41 20 9.10 599 9.10 956 0.89 044 9.99 643 40 21 9.10 697 9.11 056 0.88 944 9.99 642 39 22 9.10 795 9.11 155 0.88 845 9.99 640 38 23 9.10 893 9.11 254 0.88 746 9.99 638 37 24 9.10 990 9.11 353 0.88 647 9.99 637 36 25 9.11 087 9.11 452 0.88 548 9.99 635 35 26 9.11 184 9.11 551 0.88 449 9.99 633 34 27 9.11 281 9.11 649 0.88 351 9.99 632 33 28 9.11 377 9.11 747 0.88 253 9.99 630 32 29 9.11 474 9.11 845 0.88 155 9.99 629 31 82 30 9.11 570 9.11 943 0.88 057 9.99 627 30 31 9.11 666 9.12 040 - 0.87 960 9.99 625 29 32 9.11 761 9.12 138 0.87 862 9.99 624 28 33 9.11 857 9.12 235 0.87 765 9.99 622 27 34 9.11 952 9.12 332 0.87 668 9.99 620 26 35 9.12 047 9.12 428 0.87 572 9.99 618 25 36 9.12 142 9.12 525 0.87 4.75 9.99 617 24 37 9.12 236 9.12 621 0.87 379 9.99 615 23 38 9.12 331 9.12 717 0.87 283 9.99 613 22 39 9.12 425 9.12 813 0.87 187 9.99 612 21 40 9.12 519 9.12 909 0.87 091 9.99 610 20 41 9.12 612 ' 9.13 004 0.86 996 9.99 608 19 42 9.12 706 9.13 099 0.86 901 9.99 607 18 43 9.12 799 9.13 194 0.86 806 9.99 605 17 44 9.12 892 9.13 289 0.86 711 9.99 603 16 45 9.12 985 9.13 384 0.86 616 9.99 601 15 46 9.13 078 9.13 478 0.86 522 9.99 600 14 47 9.13 171 9.13 573 0.86 427 9.99 598 13 48 9.13 263 9.13 667 0.86 333 9.99 596 12 49 9.13 355 9.13 761 0.86 239 9.99 595 11 50 9.13 447 9.13 854 0.86 146 9.99 593 10 51 9.13 539 9.13 948 0.86 052 9.99 591 9 52 9.13 630 9.14 041 0.85 959 9.99 589 8 53 9.13 722 9.14 134 0.85 866 9.99 588 7 54 9.13 813 9.14 227 0.85 773 9.99 586 6 55 9.13 904 9.14 320 0.85 680 9.99 584 5 56 9.13 994 9.14 412 0.85 588 9.99 582 4 57 9.14 085 9.14 504 0.85 496 9.99 581 3 58 9.14 175 9.14 597 0.85 403 9.99 579 2 59 9.14 266 9.14 688 0.85 312 9.99 577 1 60 9.14 356 9.14 780s 0.85 220 9.99 575 O 0 I I I I L. Cos. ] L. Cotg. L. Tang. -1 I L. Sin. I I I m A w M 1I [49] -Y I * Y ___ __ ___ L. Sin. L. Tang. L. Cotg. T L. Cos.:_ I I l S 0 9.14 3o6 9.14 780 0.85 220 9.99 575 60 1 9.14 445 9.14 872 0.85 128 9.99 574 59 2 9.14 535 9.14 963 0.85 037 9.99 572 58 3 9.14 624 9.15 054 0.84 946 9.99 570 57 4 9.14 714 9.15 145 0.84 855 9.99 568 56 5 9.14 803 9.15 236 0.84 764 9.99 566 55 6 9.14 891 9.15 327 0.84 673 9.99 565 54 7 9'14 980 9.15 417 0.84 583 9.99 563 53 8 9.15 069 9.15 508 0.84 492 9.99 561 52 9 9.15 157 9.15 598 0.84 402 9.99 559 51 10 9.15 245 9.15 688 0.84 312 9.99 557 bO 11 9.15 333 9.15 777 0.84 223 9.99 556 49 12 9.15 421 9.15 867 0.84 133 9.99 554 48 13 9.15 508 9.15 956 0.84 044 9.99 552 47 14 9.15 596 9.16 046 0.83 954 9.99 550 46 15 9.15 683 9.16 135 0.83 865 9.99 548 45 16 9.15 770 9.16 224 0 83 776 9.99 546 44 17 9.15 857 9.16 312 0.83 688 9.99 545 43 18 9.15 944 9.16 401 0.83 599 9.99 543 42 19 9.16 030 9.16 489 0.83 511 9.99 541 41 20 9.16 116 9.16 577 0.83 423 9.99 539 40 21 9.16 203 9.16 665 0.83 335 9.99 537 39 22 9.16 289 9.16 753 0.83 247 9.99 535 38 23 9.16 374 9.16 841 0.83 159 9.99 533 37 24 9.16 460 9.16 928 0.83 072 9.99 532 36 25 9.16 545 9.17 016 0.82 984 9.99 530 35 26 9.16 631 9.17 103 0.82 897 9.99 528 34 27 9.16 716 9.17 190 0.82 810 9.99 526 33 28 9.16 801 9.17 277 0.82 723 9.99 524 32 29 9.16 886 9.17 363 0.82 637 9.99 522 31 30 9.16 970 9.17 450 0.82 550 9.99 520 30 31 9.17 055 9.17 536 0.82 464 9.99 518 29 32 9.17 139 9.17 622 0.82 378 9.99 517 28 33 9.17 223,9.17 708 0.82 292 9.99 515 27 34 9.17 307 9.17 794 0.82 206 9.99 513 26 35 9.17 391 9.17 880 0.82 120 9.99 511 25 36 9.17 474 9.17 965 0.82 035 9.99 509 24 37 9.17 558 9.18 051 0.81 949 9.99 507 23 38 9.17 641 9.18 136 0.81 864 9.99 505 22 39 9.17 724 9.18 221 0.81 779 9.99 503 21 40 9.17 807 9.18 306 0.81 694 9.99 501 20 41 9.17 890 9.18 391 0.81 609 9.99 499 19 42 9.17 973 9.18 475 0.81 525 9.99 497 18 43 9.18 055 9.18 560 0.81 440 9.99 495 17 44 9.18 137 9.18 644 0.81 356 9.99 494 16 45 9.18 220 9.18 728 0.81 272 9.99 492 15 46 9.18 302 9.18 812 0.81 188 9.99 490 14 47 9.18 383 9.18 896 0.81 104 9.99 488 13 48 9.18 465 9.18 979 0.81 021 9.99 486 12 49 9.18 547 9.19 063 0.80 937 9.99 484 11 50 9.18 628 9.19 146 0.80 854 9.99 482 10 51 9.18 709 9.19 229 0.80 771 9.99 480 9 52 9.18 790 9.19 312 0.80 688 9.99 478 8 53 9.18 871 9.19 395 0.80 605 9.99 476 7 54 9.18 952 9.19 478 0.80 522 9.99 474 6 55 9.19 033 9.19 561 0.80 439 9.99 472 5 56 9.19 113 9.19 643 0.80 357 9.99 470 4 57 9.19 193 9.19 725 0.80 275 9.99 468 3 58 9.19 273 9.19 807 0.80 193 9.99 466 2 59 9.19 353 9.19 889 0.80 111 9.99 464 1 60 9.19 433 9.19 971 0.80 029 9.99 462 0 __..~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~i il~ L. Cos. L. Cotg. | L. Tang. 1 L. Sin. / [50] 0 90 m L. Sin. mm1 r L. Tan. T L. Cotg. r L. Cos. r I -m 0 9.19 433 9.19 971 0.80 029 9.99 462 60 1 9.19 513 9.20 053 0.79 947 9.99 460 59 2 9.19 592 9.20 134 0.79 866 9.99 458 58 3 9.19 672 9.20 216 0.79 784 9.99 456 57 4 9.19 751 9.20 297 0.79 703 9.99 454 56 5 9.19 830 9.20 378 0.79 622 9.99 452 55 6 9.19 909 9.20 459 0.79 541 9.99 450 54 7 9.19 988 9.20 540 0.79 460 9.99 448 53 8 9.20 067 9.20 621 0.79 379 9.99 446 52 9 9.20 145 9.20 701 0.79 299 9.99 444 51 10 9.20 223 9.20 782 0.79 218 9.99 442 50 11 9.20 302 9.20 862 0.79 138 9.99 440 49 12 9.20 380 9.20 942 0.79 058 9.99 438 48 13 9.20 458 9.21 022 0.78 978 9.99 436 47 14 9.20 535 9.21 102 0.78 898 9.99 434 46 15 9.20 613 9.21 182 0.78 818 9.99 432 45 16 9.20 691 9.21 261 0.78 739 9.99 429 44 17 9.20 768 9.21 341 0.78 659 9.99 427 43 18 9.20 845 9.21 420 0.78 580 9.99 425 42 19 9.20 922 9.21 499 0.78 501 9.99 423 41 20 9.20 999 9.21 578 0.78 422 9.99 421 40 21 9.21 076 9.21 657 0.78 343 9.99 419 39 22 9.21 153 9.21 736 0.78 264 9.99 417 38 23 9.21 229 9.21 814 0.78 186 9.99 415 37 24 9.21 306 9.21 893 0.78 107 9.99 413 36 25 9.21 382 9.21 971- 0.78 029 9.99 411 35 26 9.21 458 9.22 049 0.77 951 9.99 409 34 27 9.21 534 9.22 127 0.77 873 9.99 407 33 28 9.21 610 9.22 205 0.77 795 9.99 404 32 29 9.21 685 9.22 283 0.77 717 9.99 402 31 80 30 9.21 761 9.22 361 0.77 639 9.99 400 30 31 9.21 836 9.22 438 0.77 562 9.99 398 29 32 9.21 912 9.22 516 0.77 484 9.99 396 28 33 9.21 987 9.22 593 0.77 407 9.99 394 27 34 9.22 062 9.22 670 0.77 330 9.99 392 26 35 9.22 137 9.22 747 0.77 253 9.99 390 25 36 9.22 211 9.22 824 0.77 176 9.99 388 24 37 9.22 286 9.22 901 0.77 099 9.99 385 23 38 9.22 361 9.22 977 0.77 023 9.99 383 22 39 9.22 435 9.23 054 0.76 946 9.99 381 21 40 9.22 509 9.23 130 0.76 870 9.99 379 20 41 9.22 583 9.23 206 0.76 794 9.99 377 19 42 9:22 657 9.23 283 0.76 717 9.99 375 18 43 9.22 731 9.23 359 0.76 641 9.99 372 17 44 9.22 805 9.23 435 0.76 565 9.99 370 16 45 9.22 878 9.23 510 0.76 490 9.99 368 15 46 9.22 952 9.23 586 0.76 414 9.99 366 14 47 9.23 025. 9.23 661 0.76 339 9.99 364 13 48 9.23 098 9.23 737 0.76 263 9.99 362 12 49 9.23 171 9 23 812 0.76 188 9.99 359 11 50 9.23 244 9.23 887 0.76 113 9.99 357 10 51 9.23 317 9.23 962 0.76 038 9.99 355 9 52 9.23 390 9.24 037 0.75 963 9.99 353 8 53 9.23 462 9.24 112 0.75 888 9.99 351 7 54 9.23 535 9.24 186 0.75 814 9.99 348 6 55 9.23 607 9.24 261 0.75 739 9.99 346 5 56 9.23 679 9.24 335 0.75 665 9.99 344 4 57 9.23 752 9.24 410 0.75 590 9.99 342 3 58 9.23 823 9.24 484 0.75 516 9.99 340 2 59 9.23 895 9.24 558 0.75 442 9.99 337 1 60 9.23 967 9.24 632 0.75 368 9.99 335 0 _. l. _ r I 1 L. Cos. I L, Cotg. I L. Tang, I L. Sin. I l m NA [51] I 10 r m I# L. Sin. L. Tang. VW r L. Cotg. I L. Cos. r.m 0 9.23 967 9.24 632 0.75 368 9.99 335 60 1 9.24 039 9.24 706 0.75 294 9.99 333 59 2 9.24 110 9.24 779 0.75 221 9.99 331 58 3 9.24 181 9.24 853 0.75 147 9.99 328 57 4 9.24 253 9.24 926 0.75 074 9.99 326 56 5 9.24 324 9.25 000 0.75 000 9.99 324 55 6 9.24 395 9.25 073 0.74 927 9.99 322 54 7 9.24 466 9.25 146 0.74 854 9.99 319 53 8 9.24 536 9.25 219 0.74 781 9.99 317 52 9 9.24 607 9.25 292 0.74 708 9.99 315 51 o10 9.24 677 9.25 365 0.74 635 9.99 313 bO 11 9.24 748 9.25 437 0 74 563 9.99 310 49 12 9.24 818 9.25 510 0.74 490 9.99 308 48 13 9.24 888 9.25 582 0.74 418 9.99 306 47 14 9.24 958 9.25 655 0.74 345 9.99 304 46 15 9.25 028 9.25 727 0.74 273 9.99 301 45 16 9.25 098 9.25 799 0.74 201 9.99 299 44 17 9.25 168 9.25 871 0.74 129 9.99 297 43 18 9.25 237 9.25 943 0.74 057 9.99 294 42 19 9.25 307 9.26 015 0.73 985 9.99 292 41 20 9.25 376 9.26 086 0.73 914 9.99 290 40 21 9.25 445 9.26 158 0.73 842 9.99 288 39 22 9.25 514 9.26 229 0.73 771 9.99 285 38 23 9.25 583 9.26 301 0.73 699 9.99 283 37 24 9.25 652 9.26 372 0.73 628 9.99 281 36 25 9.25 721 9.26 443 0.73 557 9.99 278 35 26 9.25 790 9.26 514 0.73 486 9.99 276 34 27 9.25 858 9.26 585 0.73 415 9.99 274 33 28 9.25 927 9.26 655 0.73 345 9.99 271 32 29 9.25 995 9.26 726 0.73 274 9.99 269 31 30 9.26 063 9.26 797 0.73 203 9.99 267 30 31 9.26 131 9.26 867 0.73 133 9.99 264 29 32 9.26 199 9.26 937 0.73 063 9.99 262 28 33 9.26 267 9.27 008 0.72 992 9.99 260 27 34 9.26 335 9.27 078 0.72 922 9.99 257 26 35 9.26 403 9.27 148 0.72 852 9.99 255 25 36 9.26 470 9.27 218 0.72 782 9.99 252 24 37 9.26 538 9.27 288 0.72 712 9.99 250 23 38 9.26 605 9.27 357 0.72 643 9.99 248 22 39 9.26 672 9.27 427 0.72 573 9.99 245 21 40 9.26 739 9.27 496 0.72 504 9.99 243 20 41 9.26 806 9.27 566 0.72 434 9.99 241 19 42 9.26 873 9.27 635 0.72 365 9.99 238 18 43 9.26 940 9.27 704 0.72 296 9.99 236 17 44 9.27 007 9.27 773 0.72 227 9.99 233 16 45 9.27 073 9.27 842 0.72 158 9.99 231 15 46 9.27 140 9.27 911 0.72 089 9.99 229 14 47 9.27 206 9.27 980 0.72 020 9.99 226 13 48 9.27 273 9.28 049 0.71 951 9.99 224 12 49 9.27 339 9.28 117 0.71 883 9.99 221 11 50 9.27 405 9.28 186 0.71 814 9.99 219 10 51 9.27 471 9.28 254 0.71 746 9.99 217 9 52 9.27 537 9.28 323 0.71 677 9.99 214 8 53 9.27 602 9.28 391 0.71 609 9.99 212 7 54 9.27 668 9.28 459 0.71 541 9.99 209 6 55 9.27 734 9.28 527 0.71 473 9.99 207 5 56 9.27 799 9.28 595 0.71 405 9.99 204 4 57 9.27 864 9.28 662 0.71 338 9.99 202 3 58 9.27 930 9.28 730 0.71 270 9.99 200 2 59 9.27 995 9.28 798 0.71 202 9.99 197 1 60 9.28 060 9.28 865 0.71 135 9.99 195 0 T 9C.3 -F -I L. Cos. L. Cotg. I L. Tang. I L. Cos. I I I ML El NJ [52] I m I T m L. Sin. r L. Tang. r L. Cotg. r L. Cos. r r 0 9.28 060 9.28 865 0.71 135 9.99 195 60 1 9.28 125 9.28 933 0.71 067 9.99 192 59 2 9.28 190 9.29 000 0.71 000 9.99 190 58 3 9.28 254 9.29 067 0.70 933 9.99 187 57 4 9.28 319 9.29 134 0.70 866 9.99 185 56 5 9.28 384 9.29 201 0.70 799 9.99 182 55 6 9.28 448 9.29 268 0.70 732 9.99 180 54 7 9.28 512 9.29 335 0.70 665 9.99 177 53 8 9.28 577 9.29 402 0.70 598 9.99 175 52 9 9.28 641 9.29 468 0.70 532 9.99 172 51 10 9.28 705 9.29 535 0.70 465 9.99 170 50 11 9.28 769 9.29 601 0.70 399 9.99 167 49 12 9.28 833 9.29 668 0.70 332 9.99 165 48 13 9.28 896 9.29 734 0.70 266 9.99 162 47 14 9.28 960 9.29 800 0.70 200 9.99 160 46 15 9.29 024 9.29 866 0.70 134 9.99 157 45 16 9.29 087 9.29 932 0.70 068 9.99 155 44 17 9.29 150 9.29 998 0.70 002 9.99 152 43 18 9.29 214 9.30 064 0.69 936 9.99 150 42 19 9.29 277 9.30 130 0.69 870 9.99 147 41 20 9.29 340 9.30 195 0.69 805 9.99 145 40 21 9.29 403 9.30 261 0.69 739 9.99 142 39 22 9.29 466 9.30 326 0.69 674 9.99 140 38 23 9.29 529 9.30 391 0.69 609 9.99 137 37 24 9.29 591 9.30 457 0.69 543 9.99 135 36 25 9.29 654 9.30 522 0.69 478 9.99 132 35 26 9.29 716 9.30 587 0.69 413 9.99 130 34 27 9.29 779 9.30 652 0.69 348 9.99 127 33 28 9.29 841 9.30 717 0.69 283 9.99 124 32 29 9.29 903 9.30 782 0.69 218 9.99 122 31 30 9.29 966 9.30 846 0.69 154 - 9.99 119 30 31 9.30 028 9.30 911 0.69 089 9.99 117 29 32 9.30 090 9.30 975 0.69 025 9.99 114 28 33 9.30 151 9.31 040 0.68 960 9.99 112 27 34 9.30 213 9.31 104 0.68 896 9.99 109 26 35 9.30 275 9.31 168 0.68 832 9.99 106 25 36 9.30 336 9.31 233 0.68 767 9.99 104 24 37 9.30 398 9.31 297 0.68 703 9.99 101 23 38 9.30 459 9.31 361 0.68 639 9.99 099 22 39 9.30 521 9.31 425 0.68 575 9.99 096 21 40 9.30 582 9.31 489 0.68 511 9.99 093 20 41 9.30 643 9.31 552 0.68 448 9.99 091 19 42 9.30 704 9.31 616 0.68 384 9.99 088 18 43 9.30 765 9.31 679 0.68 321 9.99 086 17 44 9.30 826 9.31 743 0.68 257 9.99 083 16 45 9.30 887 9.31 806 0.68 194 9.99 080 15 46 9.30 947 9.31 870 0.68 130 9.99 078 14 47 9.31 008 9.31 933 0.68 067 9.99 075 13 48 9.31 068 9.31 996 0.68 004 9.99 072 12 49 9.31 129 9.32 059 0.67 941 9.99 070 11 50 9.31 189 9.32 122 0.67 878 9.99 067 10 51 9.31 250 9.32 185 0.67 815 9.99 064 9 52 9.31 310 9.32 248 0.67 752 9.99 062 8 53 9.31 370 9.32 311 0.67 689 9.99 059 7 54 9.31 430 9.32 373 0.67 627 9.99 056 6 55 9.31 490 9.32 436 0.67 564 9.99 054 5 56 9.31 549 9.32 498 0.67 502 9.99 051 4 57 9.31 609 9.32 561 0.67 439 9.99 048 3 58 9.31 669 9.32 623 0.67 377 9.99 046 2 59 9.31 728 9.32 685 0.67 315 9.99 043 1 60 9.31 788 9.32 747 0.67 253 9.99 040 0 ii u i 8' D I -W L. Cos. L. Cotg.. Tang. El L. Sin. 1 I 1 ML NI [53] 12 D m I L. Sin. T L. Tang. r L. Cotg. L. Cos. 1 0 9.31 788 9.32 747 0.67 253 9.99 040 60 1 9.31 847 9.32 810 0.67 190 9.99 038 59 2 9.31 907 9.32 872 0.67 128 9.99.035 58 3 9.31 966 9.32 933 0.67 067 9.99 032 57 4 9.32 025 9.32 995 0.67 005 9.99 030 56 5 9.32 084 9.33 057 0.66 943 9.99 027 55 6 9.32 143 9.33 119 0.66 881 9.99 024 54 7 9.32 202 9.33 180 0.66 820 9.99 022 53 8 9.o2 261 9.33 242 0.66 758 9.99 019 52 9 9.32 319 9.33 303 0.66 697 9.99 016 51 10 9.32 378 9.33 365 0.66 635 9.99 013 50 11 9.32 437 9.33 426 0.66 574 9.99 011 49 12 9.32 495 ' 9.33 487 0.66 513 9.99 008 48 13 9.32 553 9.33 548 0.66 452 9.99 005 47 14 9.32 612 9.33 609 0.66 391 9.99 002 46 15 9.32 670 9.33 670 0.66 330 9.99 000 45 16 9.32 728 9.33 731 0.66 269 9.98 997 44 17 9.32 786 9.33 792 0.66 208 9.98 994 43 18 9.32 844 9.33 853 0.66 147 9.98 991 42 19 9.32 902 9.33 913 0.66 087 9.98 989 41 20 9.32 960 9.33 974 0.66 026 9.98 986 40 21 9.33 018 9.34 034 0.65 966 9.98 983 39 22 9.33 075 9.34 095 0.65 905 9.98 980 38 23 9.33 133 9.34 155 0.65 845 9.98 978 37 24 9.33 190 9.34 215 0.65 785 9.98 975 36 25 9.33 248 9.34 276 0.65 724 9.98 972 35 26 9.33 305 9.34 336 0.65 664 9.98 969 34 27 9.33 362 9.34 396 0.65 604 9.98 967 33 28 9.33 420 9.34 456 0.65 544 9.98 964 32 29 9.33 477 9.34 516 0.65 484 9.98 961 31 30 9.33 534 9.34 576 0.65 424 9.98 958 30 31 9.33 591 9.34 635 0.65 365 9.98 955 29 32 9.33 647 9.34 695 0.65 305 9.98 953 28 33 9.33 704 9.34 755 0.65 245 9.98 950 27 34 9.33 761 9.34 814 0.65 186 9.98 947 26 35 9.33 818 9.34 874 0.65 126 9.98 944 25 36 9.33 874 9.34 933 0.65 067 9.98 941 24 37 9.33 931 9.34 992 0.65 008 9.98 938 23 38 9.33 987 9.35 051 0.64 949 9.98 936 22 39 9.34 043 9.35 111 0.64 889 9.98 933 21 40 9.34 100 9.35 170 0.64 830 9.98 930 20 41 9.34 156 9.35 229 0.64 771 9.98 927 19 42 9.34 212 9.35 288 0.64 712 9.98 924 18 43 9.34 268 9.35 347 0.64 653 9.98 921 17 44 9.34 324 9.35 405 0.64 595 9.98 919 16 45 9.34 380 9.35 464 0.64 536 9.98 916 15 46 9.34 436 9.35 523 0.64 477 9.98 913 14 47 9.34 491 9.35 581 0.64 419 9.98 910 13 48 9.34 547 9.35 640 0.64 360 9.98 907 12 49 9.34 602 9.35 698 0.64 302 9.98 904 11 50 9.34 658 9.35 757 0.64 243 9.98 901 10 51 9.34 713 9.35 815 0.64 185 9.98 898 9 52 9.34 769 9.35 873 0.64 127 9.98 896 8 53 9.34 824 9.35 931 0.64 069 9.98 893 7 54 9.34 879 9.35 989 0.64 011 9.98 890 6 55 9.34 934 9.36 047 0.63 953 9.98 887 5 56 9.34 989 9.36 105 0.63 895 9.98 884 4 57 9.35 044 9.36 163 0.63 837 9.98 881 3 58 9.35 099 9.36 221 0.63 779 9.98 878 2 59 9.35 154 9.36 279 0.63 721 9.98 875 1 60 -9.35 209 9.36 336 0.63 664 9.98 872 0 - ~~ ~~~~~ m mII I 1 [ I I L. Cos. I m L. Cotg. | L. Tang. L. Sin. I I 111 ff - I.I I~ L I - I - I...-;- -. 1 - 6wom [54] I b I - M o i ii ' 1 iiii 'I L. Sin. L. Tang. I MI L. Cotg. L. Cos. r MMOM" I * 0 9.35 209 9.36 336 0.63 664 9.98 872 60 1 9.35 263 9.36 394 0.63 606 9.98 869 59 2 9.35 318 9.36 452 0.63 548 9.98 867 58 3 9.35 373 9.36 509 0.63 491 9.98 864 57 4 9.35 427 9.36 566 0.63 434 9.98 861 56 5 9.35 481 9.36 624 0.63 376 9.98 858 55 6 9.35 536 9.36 681 0.63 319 9.98 855 54 7 9.35 590 9.36 738 0.63 262 9.98 852 53 8 9.35 644 9.36 795 0.63 205 9.98 849 52 9 9.35 698 9.36 852 0.63 148 9.98 846 51 10 9.35 752 9.36 909 0.63 091 9.98 843 50 11 9.35 806 9.36 966 0.63 034 9.98 840 49 12 9.35 860 9.37 023 0.62 977 9.98 837 48 13 9.35 914 9.37 080 0.62 920 9.98 834 47 14 9.35 968 9.37 137 0.62 863 9.98 831 46 15 9.36 022 9.37 193 0.62 807 9.98 828 45 16 9.36 075 9.37 250 0.62 750 9.98 825 44 17 9.36 129 9.37 306 0.62 694 9.98 822 43 18 9.36 182 9.37 363 0.62 637 9.98 819 42 19 9.36 236 9.37 419 0.62 581 9.98 816 41 20 9.36 289 9.37 476 0.62 524 9.98 813 40 21 9.36 342 9.37 532 0.62 468 9.98 810 39 22 9.36 395 9.37 588 0.62 412 9.98 807 38 23 9.36 449 9.37 644 0.62 356 9.98 804 37 24 9.36 502 9.37 700 0.62 300 9.98 801 36 25 9.36 555 9.37 756 0.62 244 9.98 798 35 26 9.36 608 9.37 812 0.62 188 9.98 795 34 27 9.36 660 9.37 868 0.62 132 9.98 792 33 28 9.36 713 9.37 924 0.62 076 9.98 789 32 29 9.36 766 9.37 980 0.62 020 9.98 786 31 30 9.36 819 9.38 035 0.61 965 9.98 783 30 31 9.36 871 9.38 09-1 0.61 909 9.98 780 29 32 9.36 924 9.38 147 0.61 853 9.98 777 28 33 9.36 976 9.38 202 0.61 798 9.98 774 27 34 9.37 028 9.38 257 0.61 743 9.98 771 26 35 9.37 081 9.38 313 0.61 687 9.98 768 25 36 9.37 133 9.38 368 0.61 632 9.98 765 24 37. 9.37 185 9.38 423 0.61 577 9.98 762 23 38 9.37 237 9.38 479 0.61 521 9.98 759 22 39 9.37 289 9.38 534 0.61 466 9.98 756 21 40 9.37 341 9.38 589 0.61 411 9.98 753 20 41 9.37 393 9.38 644 0.61 356 9.98 750 19 42 9.37 445 9.38 699 0.61 301 9.98 746 18 43 9.37 497 9.38 754 0.61 246 9.98 743 17 44 9.37 549 9.38 808 0.61 192 9.98 740 16 45 9.37 600 9.38 863 0.61 137 9.98 737 15 46 9.37 652 9.38 918 0.61 082 9.98 734 14 47 9.37 703 9.38 972 0.61 028 9.98 731 13 48 9.37 755 9.39 027 0.60 973 9.98 728 12 49 9.37 806 9.39 082 0.60 918 9.98 725 11 50 9.37 858 9.39 136 0.60 864 9.98 722 10 51 9.37 909 9.39 190 0.60 810 9.98 719 9 52 9.37 960 9.39 245 0.60 755 9.98 715 8 53 9.38 011 9.39 299 0.60 701 9.98 712 7 54 9.38 062 9.39 353 0.60 647 9.98 709 6 55 9.38 113 9.39 407 0.60 593 9.98 706 5 56 9.38 164 9.39 461 0.60 539 9.98 703 4 57 9.38 215 9.39 515 0.60 485 9.98 700 3 58 9.38 266 9.39 569 0.60 431 9.98 697 2 59 9.38 317 9.39 623 0.60 377 9.98 694 1 60 9.38 368 9.39 677 0.60 323 9.98 690 0 1 - 0 I I L. Cos. L. Cotg. I I L. Tang. | L. Sin. I AL __ I - & I & - a a_ ~ -I I I - - - [55] 0"" 14 C r WMMM Ir I L. Sin. m " I m I L. Tang. | L. Cotg. -— l qm Y I-LC r Z1. Cos. r M" r 0 9.38 368 9.39 677 0.60 323 9.98 690 60 1 9.38 418 9.39 731' 0.60 269 9.98 687 59 2 9.38 469 9.39 785 0.60 215 9.98 684 58 3 9.38 519 9.39 838 0.60 162 9.98 681 57 4 9.38 570 9.39 892 0.60 108 9.98 678 56 5 9.38 620 9.39 945 0.60 055 9.98 675 55 6 9.38 670 9.39 999 0.60 001 9.98 671 54 7 9.38 721 9.40 052 0.59 948 9.98 668 53 8 9.38 771 9.40 106 0.59 894 9.98 665 52 9 9.38 821 9.40 159 0.59 841 9.98 662 51 10 9.38 871 9.40 212 0.59 788 9.98 659 50 11 9.38 921 9.40 266 0.59 734 9.98 656 49 12 9.38 971 9.40 319 0.59 681 9.98 652 48 13 9.39 021 9.40 372 0.59 628 9.98 649 47 14 9.39 071 9.40 425 0.59 575 9.98 646 46 15 9.39 121 9.40 478 0.59 522 9.98 643 45 16 9.39 170 9.40 531 0.59 469 9.98 640 44 17 9.39 220 9.40 584 0.59 416 9.98 636 ' 43 18 9.39 270 9.40 636 0.59 364 9.98 633 42 19 9.39 319 9.40 689 0.59 311 9.98 630 41 20 9.39 369 9.40 742 0.59 258 9.98 627 40 21 9.39 418 9.40 795 0.59 205 9.98 623 39 22 9.39 467 9.40 847 0.59 153 9.98 620 38 23 9.39 517 9.40 900 0.59 100 9.98 617 37 24 9.39 566 9.40 952 0.59 048 9.98 614 36 25 9.39 615 9.41 005 0.58 995 9.98 610 35 26 9.39 664 9.41 057 0.58 943 9.98 607 34 27 9.39 713 9.41 109 0.58 891 9.98 604 33 28 9.39 762 9.41 161 0.58 839 9.98 601 32 29 9.39 811 9.41 214 0.58 786 9.98 597 31 30 9.39 860 9.41 266 0.58 734 9.98 594 30 31 9.39 909 9.41 318 0.58 682 9.98 591 29 32 9.39 958 9.41 370 0.58 630 9.98 588 28 33 9.40 006 9.41 422 0.58 578 9.98 584 27 34 9.40 055 9.41 474 0.58 526 9.98 581 26 35 9.40 103 9.41 526 0.58 474 9.98 578 25 36 9.40 152 9.41 578 0.58 422 9.98 574 24 37 9.40 200 9.41 629 0.58 371 9.98 571 23 38 9.40 249 9.41 681 0.58 319 9.98 568 22 39 9.40 297 9.41 733 0.58 267 9.98 565 21 40 9.40 346 9.41 784 0.58 216 9.98 561- 20 41 9.40 394 9.41 836 0.58 164 9.98 558 19 42 9.40 442 9.41 887 0.58 113 9.98 555 18 43 9.40 490 9.41 939 0.58 061 9.98 551 17 44 9.40 538 9.41 990 0.58 010 9.98 548 16 45 9.40 586 9.42 041 0.57 959 9.98 545.15 46 9.40 634 9.42 093 0.57 907 9.98 541 14 47 9.40 682 9.42 144 0.57 856 9.98 538 13 48 9.40 730 9.42 195 0.57 805 9.98 535 12 49 9.40 778 9.42 246 0.57 754 9.98 531 11,50 9.40 825 9.42 297 0.57 703 9.98 528 10 51 9.40 873 9.42 348 0.57 652 9.98 525 9 52 9.40 921 9.42 399 0.57 601 9.98 521 8 53 9.40 968 9.42 450 0.57 550 9.98 518 7 54 9.41 016 9.42 501 0.57 499 9.98 515 6 55 9.41 063 9.42 552 0.57 448 9.98 511 5 56 9.41 111 9.42 603 0.57 397 9.98 508 4 57 9.41 158 9.42 653 0.57 347 9.98 505 3 58 9.41 205 9.42 704 0.57 296 9.98 501 2 59 9.41 252 9.42 755 0.57 245 9.98 498 1 60 9.41 300 9.42 805 0.57 195 9.98 494 0 i i~~~~~~~~~~ I C I I L. Cos. 1 L. Cotg. I L. Tang. L. Sin. -1 I I NoI 0 I. C 56] I ll a I I L. Sin. I [I P [ Ill ]~['[ f[[[I m r L. Tang. L. Cotg. r - - l I L. Cos. I Edda" I 15~ 0 9.41 300 9.42 805 0.57 195 9.98 494 60 1 9.41 347 9.42 856 0.57 144 9.98 491 59 2 9.41 394 9.42 906 0.57 094 9.98 488 58 3 9.41 441 9.42 957 0.57 043 9.98 484 57 4 9.41 488 9.43 007 0.56 993 9.98 481 56 5 9.41 535 9.43 057 0.56 943 9.98 477 55 6 9.41 582 9.43 108 0.56 892 9.98 474 54 7 9.41 628 9.43 158 0.56 842 9.98 471 53 8 9.41 675 9.43 208 0.56 792 9.98 467 52 9 9.41 722 9.43 258 0.56 742 9.98 464 51 10 9.41 768 9.43 308 0.56 692 9.98 460 50 11 9.41 815 9.43 358 0.56 642 9.98 457 49 12 9.41 861 9.43 408 0.56 592 9.98 453 48 13 9.41 908 9.43 458 0.56 542 9.98 450 47 14 9.41 954 9.43 508 0.56 492 9.98 447 46 15 9.42 001 9.43 558 0.56 442 9.98 443 45 16 9.42 047 9.43 607 0.56 393 9.98 440 44 17 9.42 093 9.43 657 0.56 343 9.98 436 43 18 9.42 140 9.43 707 0.56 293 9.98 433 42 19 9.42 186 9.43 756 0.56 244 9.98 429 41 20 9.42 232 9.43 806 0.56 194 9.98 426 40 21 9.42 278 9.43 855 0.56 145 9.98 422 39 22 9.42 324 9.43 905 0.56 095 9.98 419 38 23 9.42 370 9.43 954 0.56 046 9.98 415 37 24 9.42 416 9.44 004 0.55 996 9.98 412 36 25 9.42 461 9.44 053 0.55 947 9.98 409 35 26 9.42 507 9.44 102 0.55 898 9.98 405 34 27 9.42 553 9.44 151 0.55 849 9.98 402 33 28 9.42 599 9.44 201 0.55 799 9.98 398 32 29 9.42 644 9.44 250 0.55 750 9.98 395 31 74c 30 9.42 690 9.44 299 0.55 701 9.98 391 30 31 9.42 735 9.44 348 0.55 652 9.98 388 29 32 9.42 781 9.44 397 0.55 603 9.98 384 28 33 9.42 826 9.44 446 0.55 554 9.98 381 27 34 9.42 872 9.44 495 0.55 505 9.98 377 26 35 9.42 917 9.44 544 0.55 456 9.98 373 25 36 9.42 962 9.44 592 0.55 408 9.98 370 24 37 9.43 008 9.44 641 0.55 359 9.98 366 23 38 9.43 053 9.44 690 0.55 310 9.98 363 22 39 9.43 098 9.44 738 0.55 262 9.98 359 21 40 9.43 143 9.44 787 0.55 213 9.98 356 20 41 9.43 188 9.44 836 0.55 164 9.98 352 19 42 9.43 233 9.44 884 0.55 116 9.98 349 18 43 9.43 278 9.44 933 0.55 067 9.98 345 17 44 9.43 323 9.44 981 0.55 019 9.98 342 16 45 9.43 367 9.45 029 0.54 971 9.98 338 15 46 9.43 412 9.45 078 0.54 922 9.98 334 14 47 9.43 457 9.45 126 0.54 874 9.98 331 13 48 9.43 502 9.45 174 0.54 826 9.98 327 12 49 9.43 546 9.45 222 0.54 778 9.98 324 11 50 9.43 591 9.45 271 0.54 729 9.98 320 10 51 9.43 635 9 45 319 0.54 681 9.98 317 9 52 9.43 680 9.45 367 0.54 633 9.98 313 8 53 9.43 724 9.45 414 0.54 585 9.98 309 7 54 9.43 769 9.45 463 0.54 537 9.98 306 6 55 9.43 813 9.45 511 0.54 489 9.98 302 5 56 9.43 857 9.45 559 0.54 441 9.98 299 4 57 9.43 901 9-45 606 0.54 394 9.98 295 3 58 9.43 946 9.45 654 0.54 346 -9.98 291 2 59 9.43 990 9.45 702 0.54 298 9.98 288 1 60 9.44 034 9.45 750 0.54 250 9.98 284 0 L. Cos. I m L. Cotg, L. Tang. L. Sin. I I v I - - - -;-ir.;,-^C;, L _J. - - -_I~_ _ - U~__. I UI,~~~~ I I~ __DI —l-_ -- I I L. Sin. r L. Tang. IT L. Cotg. r L. Cos. r i 16c 0 9.44 034 9.45 750 0.54 250 9.98 284 60 1 9.44 078 9.45 797 0.54 203 9.98 281 59 2 9.44 122 9.45 845 0.54 155 9.98 277 58 3 9.44 166 9.45 892 0.54 108 9.98 273 57 4 9.44 210 9.45 940 0.54 060 9.98 270 56 5 9.44 253 9.45 987 0.54 013 9.98 266 55 6 9.44 297 9.46 035 0.53 965 9.98 262 54 7 9.44 341 9.46 082 0.53 918 9.98 259 53 8 9.44 385 9.46 130 0.53 870 9.98 255 52 9 9.44 428 9.46 177 0.53 823 9.98 251 51 1 9.44 472 9.46 224 0.53 776 9.98 248 50 11 9.44 516 9.46 271 0.53 729 9.98 244 49 12 9.44 559 9.46 319 0.53 681 9.98 240 48 13 9.44 602 9.46 366 0.53 634 9.98 237 47 14 9.44 646 9.46 413 0.53 587 9.98 233 46 15 9.44 689 9.46 460 0.53 540 9.98 229 45 16 9.44 733 9.46 507 0.53 493 9.98 226 44 17 9.44 776 9.46 554 0.53 446 9.98 222 43 18 9.44 819 9.46 601 0.53 399 9.98 218 42 19 9.44 862 9.46 648 0.53 352 9.98 215 41 20 9.44 905 9.46 694 0.53 306 9.98 211 40 21 9.44 948 9.46 741 0.53 259 9.98 207 39 22 9.44 992 9.46 788 0.53 212 9.98 204 38 23 9.45 035 9.46 835 0.53 165 9.98 200 37 24 9.45 077 9.46 881 0.53 119 9.98 196 36 25 9.45 120 9.46 928 0.53 072 9.98 192 35 26 9.45 163 9.46 975 0.53 025 9.98 189 34 27 9.45 206 9.47 021 0.52 979 9.98 185 33 28 9.45 249 9.47 068 0.52 932 9.98 181 32 29 9.45 292 9.47 114 0.52 886 9.98 177 31 30 9.45 334 9.47 160 0.52 840 9.98 174 30 31 9.45 377 9.47 207 0.52 793 9.98 170 29 32 9.45 419 9.47 253 0.52 747 9.98 166 28 33 9.45 462 9.47 299 0.52 701 9.98 162 27 34 9.45 504 9.47 346 0.52 654 9.98 159 26 35 9.45 547 9.47 392 0.52 608 9.98 155 25 36 9.45 589 9.47 438 0.52 562 9.98 151 24 37 9.45 632 9.47 484 0.52 516 9.98 147 23 38 9.45 674 9.47 530 0.52 470 9.98 144 22 39 9.45 716 9.47 576 0.52 424 9.98 140 21 40 9.45 758 9.47 622 0.52 378. 9.98 136 20 41 9.45 801 9.47 668 0.52 332 9.98 132 19 42 9.45 843 9.47 714 0.52 286 9.98 129 18 43 9.45 885 9.47 760 0.52 240 9.98 125 17 44 9.45 927 9.47 806 0.52 194 9.98 121 16 45 9.45 969 9.47 852 0.52 148 9.98 117 15 46 9.46 011 9.47 897 0.52 103 9.98 113 14 47 9.46 053 9.47 943 0.52 057 9.98 110 13 48 9.46 095 9.47 989 0.52 011 9.98 106 12 49 9.46 136 9.48 035 0.51 965 9.98 102 11 50 9.46 178 9.48 080 0.51 920 9.98 098 10 51 9.46 220 9.48 126 0.51 874 9.98 094 9 52 9.46 262 9.48 171 0.51 829 9.98 090 8 53 9.46 303 9.48 217 0.51 783 9.98 087 7 54 9.46 345 9.48 262 0.51 738 9.98 083 6 55 9.46 386 9.48 307 0.51 693 9.98 079 5 56 9.46 428 9.48 353 0.51 647 9.98 075 4 57 9.46 469 9.48 398 0.51 602 9.98 071 3 58 9.46 511 9.48 443 0.51 557 9.98 067 2 59 9.46 552 9.48 489 0.51 511 9.98 063 1 60 9.46 594 9.48 534 0.51 466 9.98 060 0 L. ang |. in -I 0 L. Cos. / L. Cotg. 1 L. Tang. L. Sin. I I - I II I - 0 - 6 [58] a 17c D I L. Sin. F.m T L. Tang. L. Cotg. I a I I a II L. Cos. T 0 9.46 594 9.48 534 0.51 466 9.98 060 60 1 9.46 635 9.48 579 0.51 421 9.98 056 59 2 9.46 676 9.48 624 0.51 376 9.98 052 58 3 9.46 717 9.48 669 0.51 331 9.98 048 57 4 9.46 758 9.48 714 0.51 286 9.98 044 56 5 9.46 800 9.48 759 0.51 241 9.98 040 55 6 9.46 841 9.48 804 0'51 196 9.98 036 54 7 9.46 882 9.48 849 0.51 151 9.98 032 53 8 9.46 923 9.48 894 0.51 106 9.98 029 52 9 9.46 964 9.48 939 0.51 061 9.98 025 51 10 9.47 005 9.48 984 0.51 016 9.98 021 50 11 9.47 045 9.49 029 0.50 971 9.98 017 49 12 9.47 086 9.49 073 0.50 927 9.98 013 48 13 9.47 127 9.49 118 0.50 882 9.98 009 47 14 9.47 168 9.49 163 0.50 837 9.98 005 46 15 9.47 209 9.49 207 0.50 793 9.98 001 45 16 9.47 249 9.49 252 0.50 748 9.97 997 44 17 9.47 290 9.49 296 0.50 704 9.97 993 43 18 9'47 330 9.49 341 0.50 659 9.97 989 42 19 9.47 371 9.49 385 0.50 615 9.97 986 41 20 9.47 411 9.49 430 0.50 570 9.97 982 40 21 9.47 452 9.49 474 0.50 526 9.97 978 39 22 9.47 492 9.49 519 0.50 481 9.97 974 38 23 9.47 533 9.49 563 0.50 437 9.97 970 37 24 9.47 573 9.49 607 0.50 393 9.97 966 36 25 9.47 613 9.49 652 0.50 348 9.97 962 35 26 9.47 654 9.49 696 0.50 304 9.97 958 34 27 9.47 694 9.49 740 0.50 260 9.97 954 33 28 9.47 734 9.49 784 0.50 216 9.97 950 32 29 9.47 774 9.49 828 0.50 172 9.97 946 31 30 9.47 814 9.49 872 0.50 128 9.97 942 30 31 9.47 854 9.49 916 0.50 084 9.97 938 29 32 9.47 894 9.49 960 0.50 040 9.97 934 28 33 9.47 934 9.50 004 0.49 996 9.97 930 27 34 9.47 974 9.50 048 0.49 952 9.97 926 26 35 9.48 014 9.50 092 0.49 908 9.97 922 25 36 9.48 054 9.50 136 0.49 864 9.97 918 24 37 9.48 094 9.50 180 0.49 820 9.97 914 23 38 9.48 133 9.50 223 0.49 777 9.97 910 22 39 9.48 173 9.50 267 0.49 733 9.97 906 21 40 9.48 213 9.50 311 0.49 689 9.97 902 20 41 9.48 252 9.50 355 0.49 645 9.97 898 19 42 9.48 292 9.50 398 0.49 602 9.97 894 18 43 9.48 332 9.50 442 0.49 558 9.97 890 17 44 9.48 371 9.50 485 0.49 515 9.97 886 16 45 9.48 411 9.50 529 0.49 471 9.97 882 15 46 9.48 450 9.50 572 0.49 428 9.97 878 14 47 9.48 490 9.50 616 0.49 384 9.97 874 13 48 9.48 529 9.50 659 0.49 341 9.97 870 12 49 9.48 568 9.50 703 0.49 297 9.97 866 11 50 9.48 607 9.50 746 0.49 254 9.97 861 10 51 9.48 647 9.50 789 0.49 211 9.97 857 9 52 9.48 686 9.50 833 0.49 167 9.97 853 8 53 9.48 725 9.50 876 0.49 124 9.97 849 7 54 9.48 764 9.50 919 0.49 081 9.97 845 6 55 9.48 803 9.50 962 0.49 038 9.97 841 5 56 9.48 842 9.51 005 0.48 995 9.97 837 4 57 9.48 881 9.51 048 0.48 952 9.97 833 3 58 9.48 920 9.51 092 0.48 908 9.97 829 2 59 9.48 959 9.51 135 0.48 865 9.97 825 1 60 9.48 998 9.51 178 0.48 822 9.97 821 0 - '2C m L. Cos. L. Cotg. L. Tang. I L.Sin. I I. m 1 - I~r — I I I [59] 18c m m I 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 II L. Sin. | L. Tang. I L. Cotg. I 9.48 998 9.49 037 9.49 076 9.49 115 9.49 153 9.49 192 9.49 231 9.49 269 9.49 308 9.49 347 9.49 385 9.49 424 9.49 462 9.49 500 9.49 539 9.49 577 9.49 615 9.49 654 9.49 692 9.49 730 9.49 768 9.49 806 9.49 844 9.49 882 9.49 920 -I I i 9.51 178 9.51 221 9.51 264 9.51 306 9.51 349 r -I 1 I I r-~ 0.48 822 0.48 779 0.48 736 0.48 694 0.48 651 - I T I 9.97 821 9.97 817 9.97 812 9.97 808 9.97 804 -1 L. Cos. 1 m 60 59 58 57 56 -1 9.51 392 9.51 435 9.51 478 9.51 520 9.51 563 9.51 606 9.51 648 9.51 691 9.51 734 9.51 776 9.51 819 9.51 861 9.51 903 9.51 946 9.51 988 9.52 031 9.52 073 9.52 115 9.52 157 9.52 200 I I 0.48 608 0.48 565 0.48 522 0.48 480 0.48 437 9.97 800 9.97 796 9.97 792 9.97 788 9.97 784 55 54 53 52 51 0.48 394 9.97 779 50 0.48 352 9.97 775 49 0.48 309 9.97 771 48 0.48 266 9.97 767 47 0.48 224 9.97 763 46 0.48 181 9.97 759 45 0.48 139 9.97 754 44 0.48 097 9.97 750 43 0.48 054 9.97 746 42 0.48 012 9.97 742 41 0.47 969 0.47 927 0.47 885 0.47 843 0.47 800 I 9.97 738 9.97 734 9.97.729 9.97 725 9.97 721 I 40 39 38 37 36 I 1 5 I I 25 9.49 958 9.52 242 0.47 758 9.97 717 35 26 9.49 996 9.52 284 0.47 716 9.97 713 34 27 9.50 034 9.52 326 0.47 674 9.97 708 33 28 9.50 072 9.52 368 0.47 632 9.97 704 32 29 9.50 110 9.52 410 0.47 590 9.97 700 31 30 9.50 148 9.52 452 0.47 548 9.97 696 30 31 9.50 185 9.52 494 0.47 506 9.97 691 29 32 9.50 223 9.52 536 0.47 464 9.97 687 28 33 9.50 261 9.52 578 0.47 422 9.97 683 27 34 9.50 298 9.52 620 0.47 380 9.97 679 26 35 9.50 336 9.52 661 0.47 339 9.97 674 25 36 9.50 374 9.52 703 0.47 297 9.97 670 24 37 9.50 411 9.52 745 0.47 255 9.97 666 23 38 9.50 449 9.52 787 0.47 213 9.97 662 22 39 9.50 486 9.52 829 0.47 171 9.97 657 21 40 9.50 523 9.52 870 0.47 130 9.97 653 20 41 9.50 561 9.52 912 0.47 088 9.97 649 19 42 9.50 598 9.52 953 0.47 047 9.97 645 18 43 9.50 635 9.52 995 0.47 005 9.97 640 17 44 9.50 673 9.53 037 0.46 963 9.97 636 16 45 9.50 710 9.53 078 0.46 922 9.97 632 15 46 9.50 747 9.53 120 0.46 880 9.97 628 14 47 9.50 784 9.53 161 0.46 839 9.97 623 13 48 9.50 821 9.53 202 0.46 798 9.97 619 12 49 9.50 858 9.53 244 0.46 756 9.97 615 11 50 9.50 896 9.53 285 0.46 715 9.97 610 10 51 9.50 933 9.53 327 0.46 673 9.97 606 9 52 9.50 970 9.53 368 0.46 632 9.97 602 8 53 9.51 007 9.53 409 0.46 591 9.97 597 7 54 9.51 043 9.53 450 0.46 550 9.97 593 6 55 9.51 080 9.53 492 0.46 508 9.97 589 5 56 9.51 117 9.53 533 0.46 467 9.97 584 4 57 9.51 154 9.53 574 0.46 426 9.97 580 3 58 9.51 191 9.53 615 0.46 385 9.97 576 2 59 9.51 227 9.53 656 0.46 344 9.97 571 1 60 9.51 264 9.53 697 0.46 303 9.97 567 0 1' D I L. Cos. 1 -m m L. Cotg. L. Tang. L. Sin. I I =a. l ub. [60] -I- mu L. Sin. L. Tang. 1 Illl I II If II ] L. Cotg. L. Cos. T i- 19C 0 9.51 264 9.53 697 0.46 303 9.97 567 60 1 9.51 301 9.53 738 0.46 262 9.97 563 59 2 9.51 338 9.53 779 0.46 221 9.97 558 58 3 9.51 374 9.53 820 0.46 180 9.97 554 57 4 9.51 411 9.53 861 0.46 139 9.97 550 56 5 9.51 447 9.53 902 0.46 098 9.97 545 55 6 9.51 484 9.53 943 0.46 057 9.97 541 54 7 9.51 520 9.53 984 0.46 016 9.97 536 53 8 9'51 557 9.54 025 0.45 975 9.97 532 52 9 9.51 593 9.54 065 0.45 935 9.97 528 51 10 9.51 629 9.54 106 0.45 894 9.97 523 50 11 9.51 666 9.54 147 0.45 853 9.97 519 49 12 9.51 702 9.54 187 0.45 813 9.97 515 48 13 9.51 738 9.54 228 0.45 772 9.97 510 47 14 9.51 774 9.54 269 0.45 731 9.97 506 46 15 9.51 811 9.54 309 0.45 691 9.97 501 45 16 9.51 847 9.54 350 0.45 650 9.97 497 44 17 9.51 883 9.54 390 0.45 610 9.97 492 43 18 9.51 919 9.54 431 0.45 569 9.97 488 42 19 9.51 955 9.54 471 0.45 529 9.97 484 41 20 9.51 991 9.54 512 0.45 488 9.97 479 40 21 9.52 027 9.54 552 0.45 448 9.97 475 39 22 9.52 063 9.54 593 0.45 407 9.97 470 38 23 9.52 099 9.54 633 0.45 367 9.97 466 37 24 9.52 135 9.54 673 0.45 327 9.97 461 36 25 9.52 171 9.54 714 0.45 286 9.97 457 35 26 9.52 207 9.54 754 0.45 246 9.97 453 34 27 9.52 242 9.54 794 0.45 206 9.97 448 33 28 9.52 278 9.54 835 0.45 165 9.97 444 32 29 9.52 314 9.54 875 0.45 125 9.97 439 31 30 9.52 350 9.54 915 0.45 085 9.97 435 30 31 9.52 385 9.54 955 0.45 045 9.97 430 29 32 9.52 421 9.54 995 0.45 005 9.97 426 28 33 9.52 456 9.55 035 0.44 965 9.97 421 27 34 9.52 492 9.55 075 0.44 925 9.97 417 26 35 9.52 527 9.55 115 0.44 885 9.97 412 25 36 9.52 563 9.55 155 0.44 845 9.97 408 24 37 9.52 598 9.55 195 0.44 805 9.97 403 23 38 9.52 634 9.55 235 0.44 765 9.97 399 22 39 9.52 669 9.55 275 0.44 725 9.97 394 21 40 9.52 705 9.55 315 0.44 685 9.97 390 20 41 9.52 740 9.55 355 0.44 645 9.97 385 19 42 9.52 775 9.55 395 0.44 605 9.97 381 18 43 9.52 811 9.55 434 0.44 566 9.97 376 17 44 9.52 846 9.55 474 0.44 526 9.97 372 16 45 9.52 881 9.55 514 0.44 486 9.97 367 15 46 9.52 916 9.55 554 0.44 446 9.97 363 14 47 9.52 951 9.55 593 0.44 407 9.97 358 13 48 9.52 986 9.55 633 0.44 367 9.97 353 12 49 9.53 021 9.55 673 0.44 327 9.97 349 11 50 9.53 056 9.55 712 0.44 288 9.97 344 10 51 9.53 092 9.55 752 0.44 248 9.97 340 9 52 9.53 126 9.55 791 0.44 209 9.97 335 8 53 9.53 161 9.55 831 0.44 169 9.97 331 7 54 9.53 196 9.55 870 0.44 130 9.97 326 6 55 9.53 231 9.55 910 0.44 090 9.97 322 5 56 9.53 266 9.55 949 0.44 051 9.97 317 4 57 9.53 301 9.55 989 0.44 011 9.97 312 3 58 9.53 336 9.56 028 0.43 972 9.97 308 2 59 9.53 370 9.56 067 0.43 933 9.97 303 1 60 9.53 405 9.56 107 0.43 893 9.97 299 0 i iii I L.Cos. L. Cotg. I L. Tang. L. Sin. I I - r - I ~ _~,, ___ I_~ —. - -~- _ ~ --- - [61] 20c 0 Irl I L. Sin. L. Tang. | L. Cotg. or r L. Cos. r 0 -M 0 9.53 405 9.56 107 0.43 893 9.97 299 60 1 9.53 440 9.56 146 0.43 854 9.97 294 59 2 9.53 475 9.56 185 0.43 815 9.97 289 58 3 9.53 509 9.56 224 0.43 776 9.97 285 57 4 9.53 544 9.56 264 0.43 736 9.97 280 56 5 9.53 578 9.56 303 0.43 697 9.97 276 55 6 9.53 613 9.56 342 0.43 658 9.97 271 54 7 9.53 647 9.56 381 0.43 619 9.97 266 53 8 9.53 682 9.56 420 0.43 580 9.97 262 52 9 9.53 716 9.56 459 0.43 541 9.97 257 51 10 9.53 751 9.56 498 0.43 502 9.97 252 50 11 9.53 785 9.56 537 0.43 463 9.97 248 49 12 9.53 819 9.56 576 0.43 424 9.97 243 48 13 9.53 854 9.56 615 0.43 385 9.97 238 47 14 9.53 888 9.56 654 0.43 346 9.97 234 46 15 9.53 922 9.56 693 0.43 307 9.97 229 45 16 9.53 957 9.56 732 0.43 268 9.97 224 44 17 9.53 991 9.56 771 0.43 229 9.97 220 43 18 9.54 025 9.56 810 0.43 190 9.97 215 42 19 9.54 059 9.56 849 0.43 151 9.97 210 41 20 9.54 093 9.56 887 0.43 113 9.97 206 40 21 9.54 127 9.56 926 0.43 074 9.97 201 39 22 9.54 161 9.56 965 0.43 035 9.97 196 38 23 9.54 195 9.57 004 0.42 996 9.97 192 37 24 9.54 229 9.57 042 0.42 958 9.97 187 36 25 9.54 263 9.57 081 0.42 919 9.97 182 35 26 9.54 297 9.57 120 0.42 880 9.97 178 34 27 9.54 331 9.57 158 0.42 842 9.97 173 33 28 9.54 365 9.57 197 0.42 803 9.97 168 32 29 9.54 399 9.57 235 0.42 765 9.97 163 31 69c 30 9.54 433 9.57 274 0.42 726 9.97 159 30 31 9.54 466 9.57 312 0.42 688 9.97 154 29 32 9.54 500 9.57 351 0.42 649 9.97 149 28 33 9.54 534 9.57 389 0.42 611 9.97 145 27 34 9.54 567 9.57 428 0.42 572 9.97 140 26 35 9.54 601 9.57 466 0.42 534 9.97 135 25 36 9.54 635 9.57 504 0.42 496 9.97 130 24 37 9.54 668 9.57 543 0.42 457 9.97 126 23 38 9.54 702 9.57 581 0.42 419 9.97 121 22 39 9.54 735 9.57 619 0.42 381 9.97 116 21 40 9.54 769 9.57 658 0.42 342 9.97 111 20 41 9.54 802 9.57 696 0.42 304 9.97 107 19 42 9.54 836 9.57 734 0.42 266 9.97 102 18 43 9.54 869 9.57 772 0.42 228 9.97 097 17 44 9.54 903 9.57 810 0.42 190 9.97 092 16 45 9.54 936 9.57 849 0.42 151 9.97 087 15 46 9.54 969 9.57 887 0.42 113 9.97 083 14 47 9.55 003 9.57 925 0.42 075 9.97 078 13 48 9.55 036 9.57 963 0.42 037 9.97 073 12 49 9.55 069 9.58 001 0.41 999 9.97 068 11 50 9.55 102 9.58 039 0.41 961 9.97 063 10 51 9.55 136 9.58 077 0.41 923 9.97 059 9 52 9.55 169 9.58 115 0.41 885 9.97 054 8 53 9.55 202 9.58 153 0.41 847 9.97 049 7 54 9.55 235 9.58 191 0.41 809 9.97 044 6 55 9.55 268 9.58 229 0.41 771 9.97 039 5 56 9.55 301 9.58 267 0.41 733 9.97 035 4 57 9.55 334 9.58 304 0.41 696 9.97 030 3 58 9.55 367 9.58 342 0.41 658 9.97 025 2 59 9.55 400 9.58 380 0.41 620 9.97 020 1 60 9.55 433 9.58 418 0.41 582 9.97 015 0 l~~~~~~~~~~~~~~~ i r 0 *I *T L. Cos. L. Cotg. L. Tang. I -m L. Sin. | f I I m - I I I I 0 0 --- I I I --- -------- [62] 1 9F ' I L. Sin. T L. Tang. | L. Cotg. 1 L. Cos. 21c 0 9.55 433 9.58 418 0.41 582 9.97 015 60 1 9.55 466 9.58 455 0.41 545 9.97 010 59 2 9.55 499 9.58 493 0.41 507 9.97 005 58 3 9.55 532 9.58 531 0.41 469 9.97 001 57 4 9.55 564 9.58 569 0.41 431 9.96 996 56 5 9.55 597 9.58 606 0.41 394 9.96 991 55 6 9.55 630 9.58 644 0.41 356 9.96 986 54 7 9.55 663 9.58 681 0.41 319 9.96 981 53 8 9.55 695 9.58 719 0.41 281 9.96 976 52 9 9.55 728 9.58 757 0.41 243 9.96 971 51 10 9.55 761 9.58 794 0.41 206 9.96 966 50 11 9.55 793 9.58 832 0.41 168 9.96 962 49 12 9.55 826 9.58 869 0.41 131 9.96 957 48 13 9.55 858 9.58 907 0.41 093 9.96 952 47 14 9.55 891 9.58 944 0.41 056 9.96 947 6 15 9.55 923 9.58 981 0.41 019 9.96 942 45 16 9.55 956 9.59 019 0.40 981 9.96 937 44 17 9.55 988 9.59 056 0.40 944 9.96 932 43 18 9.56 021 9.59 094 0.40 906 9.96 927 42 19 9.56 053 9.59 131 0.40 869 9.96 922 41 20 9.56 085 9.59 168 0.40 832 9.96 917 40 21 9.56 118 9.59 205 0.40 795 9.96 912 39 22 9.56 150 9.59 243 0.40 757 9.96 907 38 23 9.56 182 9.59 280 0.40 720 9.96 903 37 24 9.56 215 9.59 317 0.40 683 9.96 898 36 25 9.56 247 9.59 354 0.40 646 9.96 893 35 26 9.56 279 9.59 391 0.40 609 9.96 888 34 27 9.56 311 9.59 429 0.40 571 9.96 883 33 28 9.56 343 9.59 466 0.40 534 9.96 878 32 29 9.56 375 9.59 503 0.40 497 9.96 873 31 30 9.56 408 9.59 540 0.40 460 9.96 868 30 31 9.56 440 9.59 577 0.40 423 9.96 863 29 32 9.56 472 9.59 614 0.40 386 9.96 858 28 33 9.56 504 9.59 651 0.40 349 9.96 853 27 34 9.56 536 9.59 688 0.40 312 9.96 848 26 35 9.56 568 9.59 725 0.40 275 9.96 843 25 36 9.56 599 9.59 762 0.40 238 9.96 838 24 37 9.56 631 9.59 799 0.40 201 9.96 833 23 38 9.56 663 9.59 835 0.40 165 9.96 828 22 39 9.56 695 9.59 872 0.40 128 9.96 823 21 40 9.56 727 9.59 909 0.40 091 9.96 818 20 41 9.56 759 9.59 946 0.40 054 9.96 813 19 42 9.56 790 9.59 983 0.40 017 9.96 808 18 43 9.56 822 9.60 019 0.39 981 9.96 803 17 44 9.56 854 9.60 056 0.39 944 9.96 798 16 45 9.56 886 9.60 093 0.39 907 9.96 793 15 46 9.56 917 9.60 130 0.39 870 9.96 788 14 47 9.56 949 9.60 166 0.39 834 9.96 783 13 48 9.56 980 9.60 203 0.39 797 9.96 778 12 49 9.57 012 9.60 240 0.39 760 9.96 772 11 50 9.57 044 9.60 276 0.39 724 9.96 767 10 51 9.57 075 9-60 313 0.39 687 9.96 762 9 52 9.57 107 9.60 349 0.39 651 9.96 757 8 53 9.57 138 9.60 386 0.39 614 9.96 752 7 54 9.57 169 9.60 422 0.39 578 9.96 747 6 55 9.57 201 9.60 459 0.39 541 9.96 742 5 56 9.57 232 9.60 495 0.39 505 9.96 737 4 57 9.57 264 9.60 532 0.39 468 9.96 732 3 58 9.57 295 9.60 568 0.39 432 9.96 727 2 59 9.57 326 9.60 605 0.39 395 9.96 722 1 60 9.57 358 9.60 641 0.39 359 9.96 717 0 ii 8' D 1 L L. Cos. I L. Cotg. L. Tang. I L. Sin. f [63] lw I I - I I L. Sin. m I L. Tang. T I WI 0 L. Cotg. L. Cos. I _ — ir 22~ 0 9.57 358 9.60 641 0.39 359 9.96 717 60 1 9.57 389 9.60 677 0.39 323 9.96 711 59 2 9.57 420 9.60 714 0.39 286 9.96 706 58 3 9.57 451 9.60 750 0.39 250 9.96 701 57 4 9.57 482 9.60 786 0.39 214 9.96 696 56 5 9.57 514 9.60 823 0.39 177 9.96 691 55 6 9.57 545 9.60 859 0.39 141 9.96 686 54 7 9.57 576 9.60 895 0.39 105 9.96 681 53 8 9.57 607 9.60 931 0.39 069 9.96 676 52 9 9.57 638 9.60 967 0.39 033 9.96 670 51 10 9.57 669 9.61 004 0.38 996 9.96 665 50 11 9.57 700 9.61 040 0.38 960 9.96 660 49 12 9.57 731 9.61 076 0.38 924 9.96 655 48 13 9.57 762 9.61 112 0.38 888 9.96 650 47 14 9.57 793 9.61 148 0.38 852 9.96 645 46 15 9.57 824 9.61 184 0.38 816 9.96 640 45 16 9.57 855 9.61 220 0.38 780 9.96 634 44 17 9.57 885 9.61 256. 0.38 744 9.96 629 43 18 9.57 916 9.61 292 ' 0.38 708 9.96 624 42 19 9.57 947 9.61 328 0.38 672 9.96 619 41 20 9.57 978 9.61 364 0.38 636 9.96 614 40 21 9.58 008 9.61 400 0.38 600 9.96 608 39 22 9.58 039 9.61 436 0.38 564,9.96 603 38 23 9.58 070 9.61 472 0.38 528 9.96 598 37 24 9.58 101 9.61 508 0.38 492 9.96 593 36 25 9.58 131 9.61 544 0.38 456 9.96 588 35 26 9.58 162 9.61 579 0.38 421 9.96 582 34 27 9.58 192 9.61 615 0.38 385 9.96 577 33 28 9.58 223 9.61 651 0.38 349 9.96 572 32 29 9.58 253 9.61 687 0.38 313 9.96 567 31 30 9.58 284 9.61 722 0.38 278 9.96 562 30 31 9.58 314 9.61 758 0.38 242 9.96 556 29 32 9.58 345 9.61 794 0.38 206 9.96 551 28 33 9.58 375 9.61 830 0.38 170 9.96 546 27 34 9.58 406 9.61 865 0.38 135 9.96 541 26 35 9.58 436 9.61 901 0.38 099 9.96 535 25 36 9.58 467 9.61 936 0.38 064 9.96 730 24 37 9.58 497 9.61 972' 0.38 028 9.96 525 23 38 9.58 527 9.62 008 0.37 992 9.96 520 22 39 9.58 557 9.62 043 0.37 957 9.96 514 21 40 9.58 588 9.62 079 0.37 921 9.96 509 20 41 9.58 618 9.62 114 0.37 886 9.96 504 19 42 9.58 648 9.62 150 0.37 850 9.96 498 18 43 9.58 678 9.62 185 0.37 815 9.96 493 17 44 9.58 709 9.62 221 0.37 779 9.96 488 16 -45 9.58 739 9.62 256 0.37 744 9.96 483 15 46 9.58 769 9.62 292 0.37 708 9.96 477 14 47 9.58 799 9.62 327 0.37 673 9.96 472 13 -48 9.58 829 9.62 362 0.37 638 9.96 467 12 49 9.58 859 9.62 398 0.37 602 9.96 461 11 50 9.58 889 9.62 433 0.37 567 9.96 456 10 51 9.58 919 9.62 468 0.37 532 9.96 451 9 52 9.58 949 9.62 504 0.37 496 9.96 445 8 53 9.58 979 9.62 539 0.37 461 9.96 440 7 54 9.59 009 9.62 574 0.37 426 9.96 435 6 55 9.59 039 9.62 609 0.37 391 9.96 429 5 56 9.59 069 9.62 645 0.37 355 9.96 424 4 57 9.59 098 9.62 680 0.37 320 9.96 419 3 58 9.59 128 9.62 715 0.37 285 9.96 413 2 59 9.59 158 9.62 750 0.37 250 9.96 408 1 60 9.59 188 9.62 785 0.37 215 9,96 403 0 r7 I -19 - L. Cos. L. Cotg. L. Tang. L. Sin. I I I m f I _I - II. c - Ir [64] 1 I r L. Sin. r 9 L. Tang. L. Cotg. r L. Cos. r I 23C 0 9.59 188 9.62 785 0.37 215 9.96 403 60 1 9.59 218 9.62 820 0.37 180 9.96 397 59 2 9.59 247 9.62 855 0.37 145 9.96 392 58 3 9.59 277 9.62 890 0.37 110 9.96 387 57 4 9.59 307 9.62 926 0.37 074 9.96 381 56 5 9.59 336 9.62 961 -0.37 039 9.96 376 55 6 9.59 366 9.62 996 0.37 004 9.96 370 54 7 9.59 396 9.63 031 0.36 969 9.96 365 53 8 9.59 425 9.63 066 0.36 934 9.96 360 52 9 9.59 455 9.63 101 0.36 899 9.96 354 51 10 9.59 484 9.63 135 0.36 865 9.96 349 50 11 9.59 514 9.63 170 0.36 830 9.96 343 49 12 9.59 543 9.63 205 0.36 795 9.96 338 48 13 9.59 573 9.63 240 0.36 760 9.96 333 47 14 9.59 602 9.63 275 0.36 725 9.96 327 46 15 9.59 632 9.63 310 0.36 690 9.96 322 45 16 9.59 661 9.63 345 0.36 655 9.96 316 44 17 9.59 690 9.63 379 0.36 621 9.96 311 43 18 9.59 720 9.63 414 0.36 586 9.96 305 42 19 9.59 749 9.63 449 0.36 551 9.96 300 41 20 9.59 778 9.63 484 0.36 516 9.96 294 40 21 9.59 808 9.63 519 0.36 481 9.96 289 39 22 9.59 837 9.63 553 0.36 447 9.96 284 38 23 9.59 866 9.63 588 0.36 412 9.96 278 37 24 9.59 895 9.63 623 0.36 377 9.96 273 36 25 9.59 924 9.63 657 0.36 343 9.96 267 35 26 9.59 954 9.63 692 0.36 308 9.96 262 34 27 9.59 983 9.63 726 0.36 274 9.96 256 33 28 9.60 012 9.63 761 0.36 239 9.96 251 32 29 9.60 041 9.63 796 0.36 204 9.96 245 31 30 9.60 070 9.63 830 0.36 170 9.96 240 30 31 9.60 099 9.63 865 0.36 135 9.96 234 29 32 9.60 128 9.63 899 0.36 101 9.96 229 28 33 9.60 157 9.63 934 0.36 066 9.96 223 27 34 9.60 186 9.63 968 0.36 032 9.96 218 26 35 9.60 215 9.64 003 0.35 997 9.96 212 25 36 9.60 244 9.64 037 0.35 963 9.96 207 24 37 9.60 273 9.64 072 0.35 928 9.96 201 23 38 9.60 302 9.64 106 0.35 894 9.96 196 22 39 9.60 331 9.64 140 0.35 860 9.96 190 21 40 9.60 359 9.64 175 0.35 825 9.96 185 20 41 9.60 388 9.64 209 0.35 791 9.96 179 19 42 9.60 417 9.64 243 0.35 757 9.96 174 18 43 9.60 446 9.64 278 0.35 722 9.96 168 17 44 9.60 474 9.64 312 0.35 688 9.96 162 16 45 9.60 503 9.64 346 0.35 654 9.96 157 15 46 9.60 532 9.64 381 0.35 619 9.96 151 14 47 9.60 561 9.64 415 0.35 585 9.96 146 13 48 9.60 589 9.64 449 0.35 551 9.96 140 12 49 9.60 618 9.64 483 0.35 517 9.96 135 11 50 9.60 646 9.64 517 0.35 483 9.96 129 10 51 9.60 675 9.64 552 0.35 448 9.96 123 9 52 9.60 704 9.64 586 0.35 414 9.96 118 8 53 9.60 732 9.64 620 0.35 380 9.96 112 7 54 9.60 761 9.64 654 0.35 346 9.96 107 6 55 9.60 789 9.64 688 0.35 312 9.96 101 5 56 9.60 818 9.64 722 0.35 278 9.96 095 4 57 9.60 846 9.64 756 0.35 244 9.96 090 3 58 9.60 875 9.64 790 0.35 210 9.96 084 2 59 9.60 903 9.64 824 0.35 176 9.96 079 1 60 9.60 931 9.64 858 0.35 142 9.96 073 0 _ _.! _l -! Atf 0 I L. Cos. I L. Cotg. L. Tang. I L. Sin. [65] 6_ T T Mr /I L. Sin. L. Tang. | L. Cotg. L. Cos. 1 -M -- a 24~ 0 9.60 931 9.64 858 0.35 142 9.96 073 60 1 9.60 960 9.64 892 0.35 108 9.96 067 59 2 9.60 988 9.64 926 0.35 074 9.96 062 58 3 9.61 016 9.64 960 0.35 040 9.96 056 57 4 9.61 045 9.64 994 0.35 006 9.96 050 56 5 9.61 073 9.65 028 0.34 972 9.96 045 55 6 9.61 101 9.65 062 0.34 938 9.96 039 54 7 9.61 129 9.65 096 0.34 904 9.96 034 53 8 9.61 158 9.65 130 0.34 870 9.96 028 52 9 9.61 186 9.65 164 0.34 836 9.96 022 51 10 9.61 214 9.65 197 0.34 803 9.96 017 50 11 9.61 242 9.65 231 0.34 769 9.96 011 49 12 9.61 270 9.65 265 0.34 735 9.96 005 48 13 9.61 298 9.65 299 0.34 701 9.96 000 47 14 9.61 326 9.65 333 0.34 667 9.95 994 46 15 9.61 354 9.65 366 0.34 634 9.95 988 45 16 9.61 382 9.65 400 0.34 600 9.95 982 44 17 9.61 411 9.65 434 0.34 566 9.95 977 43 18 9.61 438 9.65 467 0.34 533 9.95 971 42 19 9.61 466 9.65 501 0.34 499 9.95 965 41 20 9.61 494 9.65 535 0.34 465 9.95 960 40 21 9.61 522 9.65 568 0.34 432 9.95 954 39 22 9.61 550 9.65 602 0.34 398 9.95 948 38 23 9.61 578 9.65 636 0.34 364 9.95 942 37 24 9.61 606 9.65 669 0.34 331 9.95 937 36 25 9.61 634 9.65 703 0.34 297 9.95 931 35 26 9.61 662 9.65 736 0.34 264 9.95 925 34 27 9.61 689 9.65 770 0.34 230 9.95 920 33 28 9.61 717 9.65 803 0.34 197 9.95 914 32 29 9.61 745 9.65 837 0.34 163 9.95 908 31 30 9.61 773 9.65 870 0.34 130 9.95 902 30 31 9.61 800 9.65 904 0.34 096 9.95 897 29 32 9.61 828 9.65 937 0.34 063 9.95 891 28 33 9.61 856 9.65 971 0.34 029 9.95 885 27' 34 9.61 883 9.66 004 0.33 996 9.95 879 26 35 9.61 911 9.66' 038 0.33 962 9.95 873 25 36 9.61 939 9.66 071 0.33 929 9.95 868 24 37 9.61 966 9.66 104 0.33 896 9.95 862 23 38 9.61 994 9.66 138 0.33 862 9.95 856 22 39 9.62 021 9.66 171 0.33 829 9.95 850 21 40 9.62 049 9.66 204 0.33 796 9.95 844 20 41 9.62 076 9.66 238 0.33 762 9.95 839 19 42 9.62 104 9.66 271 0.33 729 9.95 833 18 43 9.62 131 9.66 304 0.33 696 9.95 827 17 44 9.62 159 9.66 337 0.33 663 9.95 821 16 45 9.62 186 9.66 371 0.33 629 9.95 815 15 46 9.62 214 9.66 404 0.33 596 9.95 810 14 47 9.62 241 9.66 437 0.33 563 9.95 804 13 48 9.62 268 9.66 470 0.33 530 9.95 798 12 49 9.62 296 9.66 503 0.33 497 9.95 792 11 50 9.62 323 9.66 537 0.33 463 9.95 786 10 51 9.62 350 9.66 570 0.33 430 9.95 780 9 52 9.62 377 9.66 603 0.33 397 9.95 775 8 53 9.62 405 9.66 636 0.33 364 9.95 769 7 54 9.62 432 9.66 669 0.33 331 9.95 763 6 55 9.62 459 9.66 702 0.33 298 9.95 757 5 56 9.62 486 9.66 735 0.33 265 9.95 751 4 57 9.62 513 9.66 768 0.33 232 9.95 745 3 58 9.62 541 9.66 801 0.33 199 9.95 739 2 59 9.62 568 9.66 834 0.33 166 9.95 733 1 60 9.62 595 9.66 867 0.33 133 9.95 728 0! ft I L. Cos. L. Cotg. I L. Tang. | L. Sin. I / i - i ii 6i [66] 2a' 3 arm mm 'I L. Sin. L. Tang. L. Cotg. WM r L. Cos. -m -a A 0 9.62 595 9.66 867 0.33 133 9.95 728 60 1 9.62 622 9.66 900 0.33 100 9.95 722 59 2 9.62 649 9.66 933 0.33 067 9.95 716 53 3 9.62 676 9.66 966 0.33 034 9.95 710 57 4 9.62 703 9.66 999 0.33 001 9.95 704 56 5 9.62 730 9.67 032 0.32 968 9.95 698 55 6 9.62 757 9.67 065 0.32 935 9.95 692 54 7 9.62 784 9.67 098 0.32 902 9.95 686 53 8 9.62 811 9.67 131 0.32'869 9.95 680 52 9 9.62 838 9.67 163 0.32 837 9.95 674 51 10 9.62 865 9.67 196 0.32 804 9.95 668 50 11 9.62 892 9.67 229 0.32 771 9.95 663 49 12 9.62 918 9.67 262 0.32 738 9.95 657 48 13 9.62 945 9.67 295 0.32 705 9.95 651 47 14 9.62 972 9.67 327 0.32 673 9 95 645 46 15 9.62 999 9.67 360 0.32 640 9.95 639 45 16 9.63 026 9.67 393 0.32 607 9.95 633 44 17 9.63 052 9.67 426 0.32 574 9.95 627 43 18 9.63 079 9.67 458 0.32 542 9.95 621 42 19 9.63 106 9.67 491 0.32 509 9.95 615 41 20 1 9.63 133 9.67 524 0.32 476 9.95 609 40 21 9.63 159 9.67 556 0.32 444 9.95 603 39 22 9.63 186 9.67 589 0.32 411 9.95 597 38 23 9.63 213 9.67 622 0.32 378 9.95 591 37 24 9.63 239 9.67 654 0.32 346 9.95 585 36 25 9.63 266 9.67 687 0.32 313 9.95 579 35 26 9.63 292 9.67 719 0.32 281 9.95 573 34 27 9.63 319 9.67 752 0.32 248 9.95 567 33 28 9.63 345 9.67 785 0.32 215 9.95 561 32 29 9.63 372 9.67 817 0.32 183 9.95 555 31 4 30 9.63 398 9.67 850 0.32 150 9.95 549 30 31 9.63 425 9.67 882 0.32 118 9.95 543 29 32 9.63 451 9.67 915 0.32 085 9.95 537 28 33 9.63 478 9.67 947 0.32 053 9.95 531 27 34 9.63 504 9.67 980 0.32 020 9.95 525 26 35 9.63 531 9.68 012 0.31 988 9.95 519 25 36 9.63 557 9.68 044 0.31 956 9.95 513 24 37 9.63 583 9.68 077 0.31 923 9.95 507 23 38 9.63 610 9.68 109 0.31 891 9.95 500 22 39 9.63 636 9.68 142 0.31 858 9.95 494 21 40 9.63 662 9.68 174 0.31 826 9.95 488 20 41 9.63 689 9.68 206 0.31 794 9.95.482 19 42 9.63 715 9.68 239 0.31 761 9.95 476 18 43 9.63 741 9.68 271 0.31 729 9.95 470 17 44 9.63 767 9.68 303 0.31 697 9.95 464 16 45 9.63 794 9.68 336 0.31 664 9.95 458 15 46 9.63 820 9.68 368 0.31 632 9.95 452 14 47 9.63 846 9.68 400 0.31 600 9.95 446 13 48 9.63 872 9.68 432 0.31 568 9.95 440 12 49 9.63 898 9.68 465 0.31 535 9.95 434 11 50 9.63 924 9.68 497 0.31 503 9.95 427 10 51 9.63 950 9.68 529 0.31 471 9.95 421 9 52 9.63 976 9.68 561 0.31 439 9.95 415 8 53 9.64 002 9.68 593 0.31 407 9.95 409 7 54 9.64 028 9.68 626 0.31 374 9.95 403 6 55 9.64 054 9.68 658 0.31 342 9.95 397 5 56 9.64 080 9.68 690 0.31 310 9.95 391 4 57 9.64 106 9.68 722 0.31 278 9.95 384 3 58 9.64 132 9.68 754 0.31 246 9.95 378 2 59 9.64 158 9.68 786 0.31 214 9.95 372 1 60 9.64 184 9.68 818 0.31 182 9.95 366 0 _ ~ T I 1 L. Cos. I L. Cotg. 1 L. Tang. I L. Sin.! I [67] 6 A 26c I L. Sin. L. Tang. T L. Cotg. L. Cos. 0 9.64 184 9.68 818 0.31 182 9.95 366 60 1 9.64 210 9.68 850 0.31 150 9.95 360 59 2 9.64 236 9.68 882 0.31 118 9.95 354 58 3 9.64 262 9.68 914 0.31 086 9.95 348 57 4 9.64 288 9.68 946 0.31 054 9.95 341 56 5 9.64 313 9.68 978 0.31 022 9.95 335 55 6 9.64 339 9.69 010 0.30 990 9.95 329 54 7 9.64 365 9.69 042 0.30 958 9.95 323 53 8 9.64 391 9.69 074 0.30 926 9.95 317 52 9 9.64 417 9.69 106 0.30 894 9.95 310 51 10 9.64 442 9.69 138 0.30 862 9.95 304 50 11 9.64 468 9.69 170 0.30 830 9.95 298 49 12 9.64 494 9.69 202 0.30 798 9.95 292 48 13 9.64 519 9.69 234 0.30 766 9.95 286 47 14 9.64 545 9.69 266 0.30 734 9.95 279 46 15 9.64 571 9.69 298 0.30 702 9.95 273 45 16 9.64 596 9.69 329 0.30 671 9.95 267 44 17 9.64 622 9.69 361 0.30 639 9.95 261 43 18 9.64 647 9.69 393 0.30 607 9.95 254 42 19 9.64 673 9.69 425 0.30 575 9.95 248 41 20 9.64 698 9.69 457 0.30 543 9.95 242 40 21 9.64 724 9.69 488 0.30 512 9.95 236 39 22 9.64 749 9.69 520 0.30 480 9.95 229 38 23 9.64 775 9.69 552 0.30 448 9.95 223 37 24 964 800 9.69 584 0.30 416 9.95 217 36 25 9.64 826 9.69 615 0.30 385 9.95 211 35 26 9.64 851 9.69 647 0.30 353 9.95 204 34 27 9.64 877 9.69 679 0.30 321 9.95 198 33 28 9.64 902 9.69 710 0.30 290 9.95 192 32 29 9.64 927 9.69 742 0.30 258 9.95 185 31 30 9.64 953 9.69 774 0.30 226 9.95 179 30 31 9.64 978 9.69 805 0.30 195 9.95 173 29 32 9.65 003 9.69 837 0.30 163 9.95 167 28 33 9.65 029 9.69 868 0.30 132 9.95 160 27 34 9.65 054 9.69 900 0.30 100 9.95 154 26 35 9.65 079 9.69 932 0.30 068 9.95 148 25 36 9.65 104 9.69 963 0.30 037 9.95 141 24 37 9.65 130 9.69 995 0.30 005 9.95 135 23 38 9.65 155 9.70 026 0.29 974 9.95 129 22 39 9.65 180 9.70 058 0.29 942 9.95 122 21 1 0 40 41 42 43 44 9.65 205 9.65 230 9.65 255 9.65 281 9.65 306 9.70 089 9.70 121 9.70 152 9.70 184 9.70 215 0.29 911 0.29 879 0.29 848 0.29 816 0.29 785 9.95 116 9.95 110 9.95 103 9.95 097 9.95 090 I 20 19 18 17 16 45 9.65 331 9.70 247 0.29 753 9.95 084 15 46 9.65 356 9.70 278 0.29 722 9.95 078 14 47 9.65 381 9.70 309 0.29 691 9.95 071 13 48 9.65 406 9.70 341 0.29 659 9.95 065 12 49 9.65 431 9.70 372 0.29 628 9.95 059 11 50 9.65 456 9.70 404 0.29 596 9.95 052 10 51 9.65 481 9.70 435 0.29 565 9.95 046 9 52 9.65 506 9.70 466 0.29 534 9.95 039 8 53 9.65 531 9.70 498 0.29 502 9.95 033 7 54 9.65 556 9.70 529 0.29 471 9.95 027 6 55 9.65 580 9.70 560 0.29 440 9.95 020 5 56 9.65 605 9.70 592 0.29 408 9.95 014 4 57 9.65 630 9.70 623 0.29 377 9.95 007 3 58 9.65 655 9.70 654 0.29 346 9.95 001 2 59 9.65 680 9.70 685 0.29 315 9 94 995 1 60 9.65 705 9.70 717 0.29 283 9.94 988 0 _~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ -m m L.Cos. L.Cotg. L. Tang. L. Sin. I 0 m I. i I m a i-r a [68] 4 M D m l MMMMM r I ' L. Sin. r li I I I V I Ir L. Tang. L. Cotg. L. Cos. m r wmm", 1 E -M 0 9.65 705 9.70 717 0.29 283 9.94 988 60 1 9.65 729 9.70 748 0.29 252 9.94 982 59 2 9.65 754 9.70 779 0.29 221 9.94 975 58 3 9.65 779 9.70 810 0.29 190 9.94 969 57 4 9.65 804 9.70 841 0.29 159 9.94 962 56 5 9.65 828 9.70 873 0.29 127 9.94 956 55 6 9.65 853 9.70 904 0.29 096 9.94 949 54 7 9.65 878 9.70 935 0.29 065 9.94 943 53 8 9.65 902 9.70 966 0.29 034 9.94 936 52 9 9.65 927 9.70 997 0.29 003 9.94 930 51 10 9.65 952 9.71 028 0.28 972 9.94 923 50 11 9.65 976 9.71 059 0.28 941 9.94 917 49 12 9.66 001 9.71 090 0.28 910 9.94 911 48 13 9.66 025 9.71 121 0.28 879 9.94 904 47 14 9.66 050 9.71 153 0.28 847 9.94 898 46 15 9.66 075 9.71 184 0.28 816 9.94 891 45 16 9.66 099 9.71 215 0.28 785 9.94 885 44 17 9.66 124 9.71 246 0.28 754 9.94 878 43 18 9.66 148 9.71 277 0.28 723 9.94 871 42 19 9.66 173 9.71 308 0.28 692 9.94 865 41 20 9.66 197- 9.71 339 0.28 661 9.94 858 40 21 9.66 221 9.71 370 0.28 630 9.94 852 39 22 9.66 246 9.71 401 0.28 599 9.94 845 38 23 9.66 270 9.71 431 0.28 569 9.94 839 37 24 9.66 295 9.71 462 0.28 538 9.94 832 36 25 9.66 319 9.71 493 0.28 507 9.94 826 35 26 9.66 343 9.71 524 0.28 476 9.94 819 34 27 9.66 368 9.71 555 0.28 445 9.94 813 33 28 9.66 392 9.71 586 0.28 414 9.94 806 32 29 9.66 416 9.71 617 0.28 383 9.94 799 31 30 9.66 441 9.71 648 0.28 352 9.94 793 30 31 9.66 465 9.71 679 0.28 321 9.94 786 29 32 9.66 489 9.71 709 0.28 291 9.94 780 28 33 9.66 513 9.71 740 0.28 260 9.94 773 27 34 9.66 537 9.71 771 0.28 229 9.94 767 26 35 9.66 562 9.71 802 0.28 198 9.94 760 25 36 9.66 586 9.71 833 0.28 167 9.94 753 24 37 9.66 610 9.71 863 0.28 137 9.94 747 23 38 9.66 634 9.71 894 0.28 106 9.94 740 22 39 9.66 658 9.71 925 0.28 075 9.94 734 21 40 9.66 682 9.71 955 0.28 045 9.94 727 20 41 9.66 706 9.71 986 0.28 014 9.94 720 19 42 9.66 731 9.72 017 0.27 983 9.94 714 18 43 9.66 755 9.72 048 0.27 952 9.94 707 17 44 9.66 779 9.72 078 0.27 922 9.94 700 16 45 9.66 803 9.72 109 0.27 891 9.94 694 15 46 9.66 827 9.72 140 0.27 860 9.94 687 14 47 9.66 851 9.72 170 0.27 830 9.94 680 13 48 9.66 875 9.72 201 0.27 799 9.94 674 12 49 9.66 899 9.72 231 0.27 769 9.94 667 11 50 9.66 922 9.72 262 0.27 738 9.94 660 10 51 9.66 946 9.72 293 0.27 707 9.94 654 9 52 9.66 970 9.72 323 0.27 677 9.94 647 8 53 9.66 994 9.72 354 0.27 646 9.94 640 7 54 9.67 018 9.72 384 0.27 616 9.94 634 6 55 9.67 042 9.72 415 0.27 585 9.94 627 5 56 9.67 066 9.72 445 0.27 555 9.94 620 4 57 9.67 090 9.72 476 0.27 524 9,94 614 3 58 9.67 113 9.72 506 0.27 494 9.94 607 2 59 9.67 137 9.72 537 0.27 463 9.94 600 1 60 9.67 161 9.72 567 0.27 433 9.94 593 0 _ I I It. 1,1~~~~~~~~! I tr9 I I L. Cos. i L. Cotg. L. Tang. 1 L. Sin. I I - =a [69] 1 4 I L.Sin. | L.Tang. 1 L. Cotg. L. Cos. 280 0 9.67 161 9.72 567 0.27 433 9.94 593 60 1 9.67 185 9.72 598 0.27 402 9.94 587 59 2 9.67 208 9.72 628 0.27 372 9.94 580 58 3 9.67 232 9.72 659 0.27 341 9.94 573 57 4 9.67 256 9.72 689 0.27 311 9.94 567 56 5 9.67 280 9.72 720 0.27 280 9.94 560 55 6 9.67 303 9.72 750 0.27 250 9.94 553 54 7 9.67 327 9.72 780 0.27 220 9.94 546 53 8 9.67 350 9.72 811 0.27 189 9.94 540 52 9 9.67 374 9.72 841 0.27 159 9.94 533 51 10 9.67 398 9.72 872 0.27 128 9.94 526 50 11 9.67 421 9.72 902 0.27 098 9.94 519 49 12 9.67 445 9.72 932 0.27 068 9.94 513 48 13 9.67 468 9.72 963 0.27 037 9.94 506 47 14 9.67 492 9 72 993 0.27 007 9.94 499 46 15 9.67 515 9.73 023 0.26 977 9.94 492 45 16 9.67 539 9.73 054 0.26 946 9.94 485 44 17 9.67 562 9.73 084 0.26 916 9.94 479 43 18 9.67 586 9.73 114 0.26 886 9.94 472 42 19 9.67 609 9.73 144 0.26 856 9.94 465 41 20 9.67 633 9.73 175 0.26 825 9.94 458 40 21 9.67 656 9.73 205 0.26 795 9.94 451 39 22 9.67 680 9.73 235 0.26 765 9.94 445 38 23 9.67 703 9.73 265 0.26 735 9.94 438 37 24 9.67 726 9.73 295 0.26 705 9.94 431 36 25 9.67 750 9.73 326 0.26 674 9.94 424 35 26 9.67 773 9.73 356 0.26 644 9.94 417 34 27 9.67 796 9.73 386 0.26 614 9.94 410 33 28 9.67 820 9.73 416 0.26 584 9.94 404 32 29 967 843 9.73 446 0.26 554 9.94 397 31 61 30 9.67 866 9.73 476 0.26 524 9.94 390 30 31 9.67 890 9.73 507 0.26 493 9.94 383 29 32 9.67 913 9.73 537. 0.26 463 9.94 376 28 33 9.67 936 9.73 567 0.26 433 9.94 369 27 34 9.67 959 9.73 597 0.26 403 9.94 362 26 35 9.67 982 9.73 627 0.26 373 9.94 355 25 36 9.68 006 9.73 657 0.26 343 9.94 349 24 37 9.68 029 9.73 687 0.26 313 9.94 342 23 38 9.68 052 9.73 717 0.26 283 9.94 335 22 39 9.68 075 9.73 747 0.26 253 9.94 328 21 40 9.68 098 9.73 777 0.26 223 9.94 321 20 41 9.68 121 9.73 807 0.26 193 9.94 314 19 42 9.68 144 9.73 837 0.26 163 9.94 307 18 43 9.68 167 9.73 867 0.26 133 9.94 300 17 44 9.68 190 9.73 897 0.26 103 9.94 293 16 45 9.68 213 9.73 927 0.26 073 9.94 286 15 46 9.68 237 9.73 957 0.26 043 9.94 279 14 47 9.68 260 9.73 987 0.26 013 9.94 273 13 48 9.68 283 9.74 017 0.25 983 9.94 266 12 49 9.68 305 9.74 047 0.25 953 9.94 259 11 50 9.68 328 9.74 077 0.25 923 9.94 252 10 51 9.68 351 9.74 107 0.25 893 9.94 245 9 52 9.68 374 9.74 137 0.25 863 9.94 238 8 53 9.68 397 9.74 166 0.25 834 9.94 231 7 54 9.68 420 9.74 196 0.25 804 9.94 224 6 55 9.68 443 9.74 226 0.25 774 9.94 217 5 56 9.68 466 9.74 256 0.25 744 9.94 210 4 57 9.68 489 9.74 286 0.25 714 9.94 203 3 58 9.68 512 9.74 316 0.25 684 9.94 196 2 59 9.68 534 9.74 345 0.25 655 9.94 189 1 60 9.68 557 9.74 375 0.25 625 9.94 182 0 __ i i.ii L. Cos. L. Cotg. 1 L. Tang. L. Sin. | [70] .f I; 4 i L. Sin. | L. Tang. r IMI L. Cotg. L. Cos. I 1 0 9.68 557 9.74 375 0.25 625 9.94 182 60 1 99.68 580 9.74 405 0.25 595" 9.94 175 59 2 9.68 603 9.74 435 0.25 565 9.94 168 58 3 9.68 625 9.74 465 0.25 535 9.94 161 57 4 9.68 648 9.74 494 0.25 506 9.94 154 56 5 9.68 671 9.74 524 0.25 476 9.94 147 55 6 9.68 694 9.74 554 0.25 446 9.94 140 54 7 9.68 716 9.74 583 0.25 417 9.94 133 53 8 9.68 739 9.74 613 0.25 387 9.94 126 52 9 9.68 762 9.74 643 0.25 357 9.94 119 51 10 9.68 784 9.74 673 0.25 327 9.94 112 50 11 9.68 807 9.74 702 0.25 298 9.94 105 49 12 9.68 829 9.74 732 0.25 268 9.94 098 48 13 9.68 852 9.74 762 0.25 238 9.94 090 47 14 9.68 875 9.74 791 0.25 209 9.94 083 46 15 9.68 897 9.74 821 0.25 179 9.94 076 45 16 9.68 920 9.74 851 0.25 149 9.94 069 44 17 9.68 942 9.74 880 0.25 120 9.94 062 43 18 9.68 965 9.74 910 0.25 090 9.94 055 42 19 9.68 987 9.74 939 0.25 061 9.94 048 41 20 9.69 010 9.74 969 0.25 031 9.94 041 40 21 9.69 032 9.74 998 0.25 002 9.94 034 39 22 9.69 055 9.75 028 0.24 972 9.94 027 38 23 9.69 077 9.75 058 0.24 942 9.94 020 37 24 9.69 100 9.75 087 0.24 913 9.94 012 36 25 9.69 122 9.75 117 0.24 883 9.94 005 35 26 9.69 144 9.75 146 0.24 854 9.93 998 34 27 9.69 167 9.75 176 0.24 824 9.93 991 33 28 9.69 189 9.75 205 0.24 795 9.93 984 32 29 9.69 212 9 75 235 0.24 765 9.93 977 31 6o 30 9.69 234 9.75 264 0.24 736 9.93 970 30 31 9.69 256 9,75 294 0.24 706 9.93 963 29 32 9.69 279 9.75 323 0.24 677 9.93 955 28 33 9.69 301 9.75 353 0.24 647 9.93 948 27 34 9.69 323 9.75 382 0.24 618 9.93 941 26 35 9.69 345 9.75 411 0.24 589 9.93 934 25 36 9.69 368 9.75 441 0.24 559 9.93 927 24 37 9.69 390 9.75 470 0.24 530 9.93 920 23 38 9.69 412 9.75 500 0.24 500 9.93 912 22 39 9.69 434 9.75 529 0.24 471 9.93 905 21 40 9.69 356 9.75 558 0.24 442 9.93 898 20 41 9.69 479 9.75 588 0.24 412 9.93 891 19 42 9.69 501 9.75 617 0.24 383 9.93 884 18 43 9.69 523 9.75 647 0.24 353 9.93 876 17 44 9.69 545 9.75 676 0.24 324 9.93 869 16 45 9.69 567 9.75 705 0.24 295 9.93 862 15 46 9.69 589 9.75 735 0.24 265 9.93 855 14 47 9.69 611 9.75 764 0.24 236 9.93 847 13 48 9.69 633 9.75 793 0.24 207 9.93 840 12 49 9.69 655 9.75 822 0.24 178 9.93 833 11 50 9.69 677 9.75 852 0.24 148 9.93 826 10 51 9.69 699 9.75 881 0.24 119 9.93 819 9 52 9.69 721 9.75 910 0.24 090 9.93 811 8 53 9.69 743 9.75 939 0.24 061 9.93 804 7 54 9.69 765 9.75 969 0.24 031 9.93 797 6 55 9.69 787 9.75 998 0.24 002 9.93 789 5 56 9.69 809 9.76 027 0.23 973 9.93 782 4 57 9.69 831 9.76 056 0.23 944 9.93 775 3 58 9.69 853 9.76 086 0.23 914 9.93 768 2 59 9.69 875 9.76 115 0.23 885 9.93 760 1 60 9.69 897 9.76 144 0.23 856 9.93 753 0 I L. Cos. L. Cotg. I L. Tang. I L. Sin. I 0 -I i -..1 a a. - I - [71] 30C I T m L. Sin. L. Tang. L. Cotg. I or r L. Cos. I 0 9.69 897 9.76 144 0.23 856 9.93 753 60 1 9.69 919 9.76 173 0.23 827 9.93 746 59 2 9.69 941 9.76 202 0.23 798 9.93 738 58 3 9.69 963 9.76 231 0.23 769 9.93 731 57 4 9.69 984 9.76 261 0.23 739 9.93 724 56 5 9.70 006 9.76 290 0.23 710 9.93 717 55 6 9.70 028 9.76 319 0.23 681 9.93 709 54 7 9.70 050 9.76 348 0.23 652 9.93 702 53 8 9.70 072 9.76 377 0.23 623 9.93 695 52 9 9.70 093 9.76 406 0.23 594 9.93 687 51 10 9.70 115 9.76 435 0.23 565 9.93 680 50 11 9.70 137 9.76 464 0.23 536 9.93 673 49 12 9.70 159 9.76 493 0.23 507 9.93 665 f8 13 9.70 180 9.76 522 0.23 478 9.93 658 47 14 9.70 202 9.76 551 0.23 449 9.93 650 46 15 9.70 224 9.76 580 0.23 420 9.93 643 45 16 9.70 245 9.76 609 0.23 391 9.93 636 44 17 9.70 267 9.76 639 0.23 361 9.93 628 43 18 9.70 288 9.76 668 0.23 332 9.93 621 42 19 9.70 310 9.76 697 0.23 303 9.93 614 41 20 9.70 332 9.76 725 0.23 275 9.93 606 40 21 9.70 353 9.76 754 0.23 246 9.93 599 39 22 9.70 375 9.76 783 0.23 217 9.93 591 38 23 9.70 396 9.76 812 0.23 188 9.93 584 37 24 9.70 418 9.76 841 0.23 159 9.93 577 36 25 9.70 439 9.76 870 0.23 130 9.93 569 35 26 9.70 461 9.76 899 0.23 101 9.93 562 34 27 9.70 482 9.76 928 0.23 072 9.93 554 33 28 9.70 504 9.76 957 0.23 043 9.93 547 32 29 9.70 525 9.76 986 0.23 014 9:93 539 31 59 30 9.70 547 9.77 015 0.22 985 9.93 532 30 31 9.70 568 9.77 044 0 22 956 9.93 525 29 32 9.70 590 9.77 073 0.22 927 9.93 517 28 33 9.70 611 9.77 101 0.22 899 9.93 510 27 34 9.70 633 9.77 130 0.22 870 9.93 502 26 35 - 9.70 654 9.77 159 0.22 841 9.93 495 25 36 9.70 675 9.77 188 0.22 812 9.93 487 24 37 9.70 697 9.77 217 0.22 783 9.93 480 23 38 9.70 718 9.77 246 0.22 754 9.93 472 22 39 9.70 739 9.77 274 0.22 726 9.93 465 21 40 9.70 761 9.77 303 0.22 697 9.93 457 20 41 9.70 782 9.77 332 0.22 668 9.93 450 19 42 9.70 803 9.77 361 0.22 639 9.93 442 18 43 9.70 824 9.77 390 0.22 610 9.93 435 17 44 9.70 846 9.77 418 0.22 582 9.93 427 16 45 9.70 867 9.77 447 0.22 553 9.93^420 15 46 9.70 888 9.77 476 0.22 524 9.93 412 14 47 9.70 909 9.77 505 0.22 495 9.93 405 13 48 9.70 931 9.77 533 0.22 467 9.93 397 12 49 9.70 952 9.77 562 0.22 438 9.93 390 11 50 9.70 973 9.77 591 0.22 409 9.93 382 10 51 9.70 994 9.77 619 0.22 381 9.93 375 9 52 9.71 015 9.77 648 0.22 352 9.93 367 8 53 9.71 036 9.77 677 0.22 323 9.93 360 7 54 9.71 058 9.77 706 0.22 294 9.93 352 6 55 9.71 079 9.77 734 0.22 266 9.93 344 5 56 9.71 100 9.77 763 0.22 237 9.93 337 4 57 9.71 121 9.77 791 0.22 209 9.93 329 3 58 9.71 142 9.77 820 0.22 180 9.93 322 2 59 9.71 163 9.77 849 0.22 151 9.93 314 1 60 9.71 184 9.77 877 0.22 123 9.93 307 0 i i~ i- i1~ a. r I I L. Cos. L. Cotg. L. Tang. L. Sin. I w m -a -M 6 I a - -— [72 C 72] I! l T L. Sin. | L. Tang. L. Cotg. I L. Cos. I r 0 9.71 184 9.77 877 0.22 123 9.93 307 60 1 9.71 205 9.77 906 0.22 094 9.93 299 59 2 9.71 226 9.77 935 0.22 065 9.93 291 58 3 9.71 247 9.77 963 0.22 037 9.93 284 57 4 9.71 268 9.77 992 0.22 008 9.93 276 56 5 9.71 289 9.78 020 0.21 980 9.93 269 55 6 9.71 310 9.78 049 0.21 951 9.93 261 54 7 9.71 331 9.78 077 0.21 923 9.93 253 53 8 9.71 352 9.78 106 0.21 894 9.93 246 52 9 9.71 373 9.78 135 0.21 865 9.93 238 51 10 9.71 393 9.78 163 0.21 837 9.93 230 50 11 9.71 414 9.78 192 0.21 808 9.93 223 49 12 9.71 435 9.78 220 0.21 780 9.93 215 48 13 9.71 456 9.78 249 0.21 751 9.93 207 47 14 9.71 477 9.78 277 0.21 723 9.93 200 46 15 9.71 498 9.78 306 0.21 694 9.93 192 45 16 9.71 519 9.78 334 0.21 666 9.93 184 44 17 9.71 539 9.78 363 0.21 637 9.93 177 43 18 9.71 560 9.78 391 0.21 609 9.93 169 42 19 9.71 581 9.78 419 0.21 581 9.93 161 41 20 9.71 602 9.78 448 0.21 552 9.93 154 40 21 9.71 622 9.78 476 0.21 524 9.93 146 39 22 9.71 643 9.78 505 0.21 495 9.93 138 38 23 9.71 664 9.78 533 0.21 467 9.93 131 37 24 9.71 685 9.78 562 0.21 438 9.93 123 36 25 9.71 705 9.78 590 0.21 410 9.93 115 35 26 9.71 726 9.78 618 0.21 382 9.93 108 34 27 9.71 747 9.78 647 0.21 353 9.93 100 33 28 9.71 767 9.78 675 0.21 325 9.93 092 32 29 9.71 788 9.78 704 0.21 296 9.93 084 31 30 9.71 809 9.78 732 0.21 268 9.93 077 30 31 9.71 829 9.78 760 0.21 240 9.93 069 29 32 9.71 850 9.78 789 0.21 211 9.93 061 28 33 9.71 870 9.78 817 0.21 183 9.93 053 27 34 9.71 891 9.78 845 0.21 155 9.93 046 26 35 9.71 911 9.78 874 0.21 126 9.93 038 25 36 9.71 932 9.78 902 0.21 098 9.93 030 24 37 9.71 952 9.78 930 0.21 070 9.93 022 23 38 9.71 973 9.78 959 0.21 041 9.93 014 22 39 9.71 994 9.78 987 0.21 013 9.93 007 21 40 9.72 014 9.79 015 0.20 985 9.92 999 20 41 9.72 034 9.79 043 0.20 957 9.92 991 19 42 9.72 055 9.79 072 0.20 928 9.92 983 18 43 9.72 075 9.79 100 0.20 900 9.92 976 17 44 9.72 096 9.79 128 0.20 872 9.92 968 16 45 9.72 116 9.79 156 0.20 844 9.92 960 15 46 9.72 137 9.79 185 0.20 815 9.92 952 14 47 9.72 157 9.79 213 0.20 787 9.92 944 13 48 9.72 177 9.79 241 0.20 759 9.92 936 12 49 9.72 198 9.79 269 0.20 731 9.92 929 11 50 9.72 218 9.79 297 0.20 703 9.92 921 10 51 9.72 238 9.79 326 0.20 674 9.92 913 9 52 9.72 259 9.79 354 0.20 646 9.92 905 8 53 9.72 279 9.79 382 0.20 618 9.92 897 7 54 9.72 299 9.79 410 0.20 590 9.92 889 6 55 9.72 320 9.79 438 0.20 562 9.92 881 5 56 9.72 340 9.79 466 0.20 534 9.92 874 4 57 9.72 360 9.79 495 0.20 505 9.92 866 3 58 9.72 381 9.79 523 0.20 477 9.92 858 2 59 9.72 401 9.79 551 0.20 449 9.92 850 1 60 9.72 421 9.79 579 0.20 421 9.92 842 0 D 1 L. Cos. L. Cotg. L. Tang. L. Sin. I I I -I ow [78] 19 l L. Sin. L. Tang. L. Cotg. I L. Cos. T -1 32c 0 9.72 421 9.79 579 0.20 421 9.92 842 60 1 9.72 441 9.79 607 0.20 393 9.92 834 59 2 9.72 461 9.79 635 0.20 365 9.92 826 58 3 9.72 482 9.79 663 0.20 337 9.92 818 57 4 9.72 502 9.79 691 0.20 309 9.92 810 56 5 9.72 522 9.79 719 0.20 281 9.92 803 55 6 9.72 542 9.79 747 0.20 253 9.92 795 54 7 9.72 562 9.79 776 0.20 224 9.92 787 53 8 9.72 582 9.79 804 0.20 196 9.92 779 52 9 9.72 602 9.79 832 0.20 168 9.92 771 51 10 9.72 622 9.79 860 0.20 140 9.92 763 50 11 9.72 643 9.79 888 0.20 112 9.92 755 49 12 9.72 663 9.79 916 0.20 084 9.92 747 48 13 9.72 683 9.79 944 0.20 056 9.92 739 47 14 9.72 703 9.79 972 0.20 028 9.92 731 46 15 9.72 723 9.80 000 0.20 000 9.92 723 45 16 9.72 743 9.80 028 0.19 972 9.92 715 44 17 9.72 763 9.80 056 0.19 944 9.92 707 43 18 9.72 783 9.80 084 0.19 916 9.92 699 42 19 9.72 803 9.80 112 0.19 888 9.92 691 41 20 9.72 823 9.80 140 0.19 860 9.92 683 40 21 9.72 843 9.80 168 0.19 832 9.92 675 39 22 9.72 863 9.80 195 0.19 805 9.92 667 38 23 9.72 883 9.80 223 0.19 777 9.92 659 37 24 9.72 902 9.80 251 0.19 749 9.92 651 36 25 9.72 922 9.80 279 0.19 721 9.92 643 35 26 9.72 942 9.80 307 0.19 693 9.92 635 34 27 9.72 962 9.80 335 0.19 665 9.92 627 33 28 9.72 982 9.80 363 0.19 637 9.92 619 32 29 9.73 002 9.80 391 0.19 609 9.92 611 31 30 9.73 022 9.80 419 0.19 581 9.92 603 30 31 9.73 041 9.80 447 0.19 553 9.92 595 29 32 9.73 061 9.80 474 0.19 526 9.92 587 28 33 9.73 081 9.80 502 0.19 498 9.92 579 27 34 9.73 101 9.80 530 0.19 470 9.92 571 26 35 9.73 121 9.80 558 0.19 442 9.92 563 25 36 9.73 140 9.80 586 0.19 414 9.92 555 24 37 9.73 160 9.80 614 0.19 386 9.92 546 23 38 9.73 180 9.80 642 0.19 358 9.92 538 22 39 9.73 200 9.80 669 0.19 331 9.92 530 21 40 9.73 219 9.80 697 0.19 303 9.92 522 20 41 9.73 239 9.80 725 0.19 275 9.92 514 19 42 9.73 259 9.80 753 0.19 247 9.92 506 18 43 9.73 278 9.80 781 0.19 219 9.92 498 17 44 9.73 298 9.80 808 0.19 192 9.92 490 16 45 9.73 318 9.80 836 0.19 164 9.92 482 15 46 9.73 337 9.80 864 0.19 136 9.92 473 14 47 9.73 357 9.80 892 0.19 108 9.92 465 13 48 9.73 377 9.80 919 0.19 081 9.92 457 12 49 9.73 396 9.80 947 0.19 053 9.92 449 11 50 9.73 416 9.80 975 0.19 025 9.92 441 10 51 9.73 435 9.81 003 0.18 997 9.92 433 9 52 9.73 455 9.81 030 0.18 970 9.92 425 8 53 9.73 474 9.81 058 0.18 942 9.92 416 7 54 9.73 494 9.81 086 0.18 914 9.92 408 6 55 9.73 513 9.81 113 0.18 887 9.92 400 5 56 9.73 533 9.81 141 0.18 859 9.92 392 4 57 9.73 552 9.81 169 0.18 831 9.92 384 3 58 9.73 572 9.81 196 0.18 804 9.92 376 2 59 9.73 591 9.81 224 0.18 776 9.92 367 1 60 9.73 611 9.81 252 0.18 748 9.92 359 0 I~~~~__ L.Cos. L. Cotg. L. Tang. I L. Sin. I [74] li -- 330 I I T L. Sin. I L. Tang. L. Cotg. | L. Cos. I I r"MMMM19 T - 0 9.73 611 9.81 252 0.18 748 9.92 359 60 1 9.73 630 9.81 279 0.18 721 9.92 351 59 2 9.73 650 9.81 307 0.18 693 9.92 343 58 3 9.73 669 9.81 335 0.18 665 9.92 335 57 4 9.73 689 9.81 362 0.18 638 9.92 326 56 5 9.73 708 9.81 390 0.18 610 9.92 318 55 6 9.73 727 9.81 418 0.18 582 9.92 310 54 7 9.73 747 9.81 445 0.18 555 9.92 302 53 8 9.73 766 9.81 473 0.18 527 9.92 293 52 9 9.73 785 9.81 500 0.18 500 9.92 285 51 10 9.73 805 9.81 528 0.18 472 9.92 277 50 11 9.73 824 9.81 556 0.18 444 9.92 269 49 12 9.73 843 9.81 583 0.18 417 9.92 260 48 13 9.73 863 9.81 611 0.18 389 9.92 252 47 14 9.73 882 9.81 638 0.18 362 9.92 244 46 15 9.73 901 9.81 666 0.18 334 9.92 235 45 16 9.73 921 9.81 693 0.18 307 9.92 227 44 17 9.73 940 9.81 721 0.18 279 9.92 219 43 18 9.73 959 9.81 748 0.18 252 9.92 211 42 19 9.73 978 9.81 776 0.18 224 9.92 202 41 20 9.73 997 9.81 803 0.18 197 9.92 194 40 21 9.74 017 9.81 831 0.18 169 9.92 186 39 22 9.74 036 9.81 858 0.18 142 9.92 177 38 23 9.74 055 9.81 886 0.18 114 9.92 169 37 24 9.74 074 9.81 913 0.18 087 9.92 161 36 25 9.74 093 9.81 941 0.18 059 9.92 152 35 26 9.74 113 9.81 968 0.18 032 9.92 144 34 27 9.74 132 9.81 996 0.18 004 9.92 136 33 28 9.74 151 9.82 023 0.17 977 9.92 127 32 29 9.74 170 9.82 051 0.17 949 9.92 119 31 56o 30 9.74 189 9.82 078 0.17 922 9.92 111 30 31 9.74 208 9.82 106 0.17 894 9.92 102 29 32 9.74 227 9.82 133 0.17 867 9.92 094 28 33 9.74 246 9.82 161 0.17 839 9.92 086 27 34 9.74 265 9.82 188 0.17 812 9.92 077 26 35 9.74 284 9.82 215 0.17 785 9.92 069 25 36 9.74 303 9.82 243 0.17 757 9.92 060 24 37 9.74 322 9.82 270 0.17 730 9.92 052 23 38 9.74 341 9.82 298 0.17 702 9.92 044 22 39 9.74 360 9.82 325 0.17 675 9.92 035 21 40 9.74 379 9.82 352 0.17 648 9.92 027 20 41 9.74 398 9.82 380 0.17 620 9.92 018 19 42 9.74 417 9.82 407 0.17 593 9.92 010 18 43 9.74 436 9.82 435 0.17 565 9.92 002 17 44 9.74 455 9.82 462 0.17 538 9.91 993 16 45 9.74 474 9.82 489 0.17 511 9.91 985 15 46 9.74 493 9.82 517 0.17 483 9.91 976 14 47 9.74 512 9.82 544 0.17 456 9.91 968 13 48 9.74 531 9.82 571 0.17 429 9.91 959 12 49 9.74 549 9.82 599 0.17 401 9.91 951 11 50 9.74 568 9.82 626 0.17 374 9.91 942 10 51 9.74 587 9.82 653 0.17 347 9.91 934 9 52 9.74 606 9.82 681 0.17 319 9.91 925 8 53 9.74 625 9.82 708 0.17 292 9.91 917 7 54 9.74 644 9.82 735 0.17 265 9.91 908 6 55 9.74 662 9.82 762 0.17 238 9.91 900 5 56 9.74 681 9.82 790 0.17 210 9.91 891 4 57 9.74 700 9.82 817 0.17 183 9.91 883 3 58 9.74 719 9.82 844 0.17 156 9.91 874 2 59 9.74 737 9.82 871 0.17 129 9.91 866 1 60 9.74 756 9.82 899 0.17 101 9.91 857 0 7 I L. Cos. I L. Cotg. L. Tang. L. Sin. I I I J I -Yi I0 i- w -i - 0 v I i [75] I L. Sin. L. Tang. L. Cotg. i L. Cos. 34~ 0 9.74 756 9.82 899 0.17 101 9.91 857 60 1 9.74 775 9.82 926 0.17 074 9.91 849 59 2 9.74 794 9.82 953 0.17 047 9.91 840 58 3 9.74 812 9.82 980 0.17 020 9.91 832 57 4 9.74 831 9.83 008 0.16 992 9.91 823 56 5 9.74 850 9.83 035 0.16 965 9.91 815 55 6 9.74 868 9.83 062 0.16 938 9.91 806 54 7 9.74 887 9.83 089 0.16 911 9.91 798 53 8 9.74 906 9.83 117 0.16 883 9.91 789 52 9 9.74 924 9.83 144 0.16 856 9.91 781 51 10 9.74 943 9.83 171 0.16 829 9.91 772 50 11 9.74 961 9.83 198 0.16 802 9.91 763 49 12 9.74 980 9.83 225 0.16 775 9.91 755 48 13 9.74 999 9.83 252 0.16 748 9.91 746 47 14 9.75 017 9.83 280 0.16 720 9.91 738 46 15 9.75 036.9.83 307 0.16 693 9.91 729 45 16 9.75 054 9.83 334 0.16 666 9.91 720 44 17 9.75 073 9.83 361 0.16 639 9.91 712 43 18 9.75 091 9.83 388 0.16 612 9.91 703 42 19 9.75 110 9.83 415 0.16 585 9.91 695 41 20 9.75 128 9.83 442 0.16 558 9.91 686 40 21 9.75 147 9.83 470 0.16 530 9.91 677 39 22 9.75 165 9.83 497 0.16 503 9.91 669 38 23 9.75 184 9.83 524 0.16 476 9.91 660 37 24 9.75 202 9.83 551 0.16 449 9.91 651 36 25 9.75 221 9.83 578 0.16 422 9.91 643 35 26 9.75 239 9.83 605 0.16 395 9.91 634 34 27 9.75 258 9.83 632 0.16 368 9.91 625 33 28 9.75 276 9.83 659 0.16 341 9.91 617 32 29 9.75 294 9.83 686 0.16 314 9.91 608 31 5ac 30 9.75 313 9.83 713 0.16 287 9.91 599 30 31 9.75 331 9.83 740 0.16 260 9.91 591 29 32 9.75 350 9.83 768 0.16 232 9.91 582 28 33 9.75 368 9.83 795 0.16 205 9.91 573 27 34 9.75 386 9.83 822 0.16 178 9.91 565 26 35 9.75 405 9.83 849 0.16 151 9.91 556 25 36 9.75 423 9.83 876 0.16 124 9.91 547 24 37 9.75 441 9.83 903 0.16 097 9.91 538 23 38 9.75 459 9.83 930 0.16 070 9.91 530 22 39 9.75 478 9.83 957 0.16 043 9.91 521 21 40 9.75 496 9.83 984 0.16 016 9.91 512 20 41 9.75 514 9.84 011 0.15 989 9.91 504 19 42 9.75 533 9.84 038 0.15 962 9.91 495 18 43 9.75 551 9.84 065 0.15 935 9.91 486 17 44 9.75 569 9.84 092 0.15 908 9.91 477 16 45 9.75 587 9.84 119 0.15 881 9.91 469 15 46 9.75 605 9.84 146 0.15 854 9.91 460 14 47 9.75 624 9.84 173 0.15 827 9.91 451 13 48 9.75 642 9.84 200 0.15 800 9.91 442 12 49 9.75 660 9.84 227 0.15 773 9.91 433 11 50 9.75 678 9.84 254 0.15 746 9.91 425 10 51 9.75 696 9.84 280 0.15 720 9.91 416 9 52 9.75 714 9.84 307 0.15 693 9.91 407 8 53 9.75 733 9.84 334 0.15 666. 9.91 398 7 54 9.75 751 9.84 361 0.15 639 9.91 389 6 55 9.75 769 9.84 388 0.15 612 9.91 381 5 56 9.75 787 9.84 415 0.15 585 9.91 372 4 57 9.75 805 9.84 442 0.15 558 9.91 363 3 58 9.75 823 9.84 469 0.15 531 9.91 354 2 59 9.75 841 9.84 496 0.15 504 9.91 345 1 60 9.75 859 9.84 523 0.15 477 9.91 336 0 L. Cos. L. Cotg. L. Tang. I L. Sin. I 0 n I I. ~ [76] 6 f I L. Sin. L. Tang. T o L. Cotg. L. Cos. 1 35c 0 9.75 859 9.84 523 0.15 477 9.91 336 60 1 9.75 877 9.84 550 0.15 450 9.91 328 59 2 9.75 895 9.84 576 0.15 424 9.91 319 58 3 9.75 913 9.84 603 0.15 397 9.91 310 57 4 9.75 931 9.84 630 0.15 370 9.91 301 56 5 9.75 949 9.84 657 0.15 343 9.91 292 55 6 9.75 967 9.84 684 0.15 316 9.91 283 54 7 9.75 985 9.84 711 0.15 289 9.91 274 53 8 9.76 003 9.84 738 0.15 262 9.91 266 52 9 9.76 021 9.84 764 0.15 236 9.91 257 51 10 9.76 039 9.84 791 0.15 209 9.91 248 50 11 9.76 057 9.84 818 0.15 182 9.91 239 49 12 9.76 075 9.84 845 0.15 155 9.91 230 48 13 9.76 093 9.84 872 0.15 128 9.91 221 47 14 9.76 111 9.84 899 0.15 101 9.91 212 46 15 9.76 129 9.84 925 0.15 075 9.91 203 45 16 9.76 146 9.84 952 0.15 048 9.91 194 44 17 9.76 164 9.84 979 0.15 021 9.91 185 43 18 9.76 182 9.85 006 0.14 994 9.91 176 42 19 9.76 200 9.85 033 0.14 967 9.91 167 41 20 9.76 218 9.85 059 0.14 941 9.91 158 40 21 9.76 236 9.85 086 0.14 914 9.91 149 39 22 9.76 253 9.85 113 0.14 887 9.91 141 38 23 9.76 271 9.85 140 0.14 860 9.91 132 37 24 9.76 289 9.85 166 0.14 834 9.91 123 36 25 9.76 307 9.85 193 0.14 807 9.91 114 35 26 9.76 324 9.85 220 0.14 780 9.91 105 34 27 9.76 342 9.85 247 0.14 753 9.91 096 33 28 9.76 360 9.85 273 0.14 727 9.91 087 32 29 9.76 378 9.85 300 0.14 700 9.91 078 31 30 9.76 395 9.85 327 0.14 673 9.91 069 30 31 9.76 413 9.85 354 0.14 646 9.91 060 29 32 9.76 431 9.85 380 0.14 620 9.91 051 28 33 9.76 448 9.85 407 0.14 593 9.91 042 27 34 9.76 466 9.85 434 0.14 566 9.91 033 26 35 9.76 484 9.85 460 0.14 540 9.91 023 25 36 9.76 501 9.85 487 0.14 513 9.91 014 24 37 9.76 519 9.85 514 0.14 486 9.91 005 23 38 9.76 537 9.85 540 0.14 460 9.90 996 22 39 9.76 554 9.85 567 0.14 433 9.90 987 21 40 9.76 572 9.85 594 0.14 406 9.90 978 20 41 9.76 590 9.85 620 0.14 380 9.90 969 19 42 9.76 607 9.85 647 0.14 353 9.90 960 18 43 9.76 625 9.85 674 0.14 326 9.90 951 17 44 9.76 642 9.85 700 0.14 300 9.90 942 16 45 9.76 660 9.85 727 0.14 273 9.90 933 15 46 9.76 677 9.85 754 0.14 246 9.90 924 14 47 9.76 695 9.85 780 0.14 220 9.90 915 13 48 9.76 712 9.85 807 0.14 193 9.90 906 12 49 9.76 730 9.85 834 0.14 166 9.90 896 11 50 9.76 747 9.85 860 0.14 140 9.90 887 10 51 9.76 765 9.85 887 0.14 113 9.90 878 9 52 9.76 782 9.85 913 0.14 087 9.90 869 8 53 9.76 800 9.85 940 0.14 060 9.90 860 7 54 9.76 817 9.85 967 0.14 033 9.90 851 6 55 9.76 835 9.85 993 0.14 007 9.90 842 5 56 9.76 852 9.86 020 0.13 980 9.90 832 4 57 9.76 870 9.86 046 0.13 954 9.90 823 3 58 9.76 887 9.86 073 0.13 927 9.90 814 2 59 9.76 904 9.86 100 0.13 900 9.90 805 1 60 9.76 922 9.86 126 0.13 874 9.90 796 0 -i AC L. Cos. L. Cotg. L. Tang. I L. Sin. I EL I L. Sin. T L. Tang. | L. Cotg. L. Cos. 36~ 0 9.76 922 9.86 126 0.13 874 9.90 796 60 1 9.76 939 9.86 153 0.13 847 9.90 787 59 2 9.76 957 9.86 179 0.13 821 9.90 777 58 3 9.76 974 9.86 206 0.13 794 9.90 768 57 4 9.76 991 9.86 232 0.13 768 9.90 759 56 5 9.77 009 9.86 259 0.13 741 9.90 750 55 6 9.77 026 9.86 285 0.13 715 9.90 741 54 7 9.77 043 9.86 312 0.13 688 9.90 731 53 8 9.77 061 9.86 338 0.13 662 9.90 722 52 9 9.77 078 9.86 365 0.13 635 9.90 713 51 10 9.77 095 9.86 392 0.13 608 9.90 704 50 11 9.77 112 9.86 418 0.13 582 9.90 694 49 12 9.77 130 9.86 445 0.13 555 9.90 685 48 13 9.77 147 9.86 471 0.13 529 9.90 676 47 14 9.77 164 9.86 498 0.13 502 9.90 667 46 15 9.77 181 9.86 524 0.13 476 9.90 657 45 16 9.77 199 9.86 551 0.13 449 9.90 648 44 17 9.77 216 9.86 577 0.13 423 9.90 639 43 18 9.77 233 9.86 603 0.13 397 9.90 630 42 19 9.77 250 9.86 630 0.13 370 9.90 620 41 20 9.77 268 9.86 656 0.13 344 9.90 611 40 21 9.77 285 9.86 683 0.13 317 9.90 602 39 22 9.77 302 9.86 709 0.13 291 9.90 592 38 23 9.77 319 9.86 736 0.13 264 9.90 583 37 24 9.77 336 9.86 762 0.13 238 9.90 574 36 25 9.77 353 9.86 789 0.13 211 9.90 565 35 26 9.77 370 9.86 815 0.13 185 9.90 555 34 27 9.77 387 9.86 842 0.13 158 9.90 546 33 28 9.77 405 9.86 868 0.13 132 9.90 537 32 29 9.77 422 9.86 894 0.13 106 9.90 527 31 53c 30 9.77 439 9.86 921 0.13 079 9.90 518 30 31 9.77 456 9.86 947 0.13 053 9.90 509 29 32 9.77 473 9.86 974 0.13 026 9.90 499 28 33 9.77 490 9.87 000 0.13 000 9.90 490 27 34 9.77 507 9.87 027 0.12 973 9.90 480 26 35 9.77 524 9.87 053 0.12 947 9.90 471 25 36 9.77 541 9.87 079 0.12 921 9.90 462 24 37 9.77 558 9.87 106 0.12 894 9.90 452 23 38 9.77 575 9.87 132 0.12 868 9.90 443 22 39 9.77 592 9.87 158 0.12 842 9.90 434 21 40 9.77 609 9.87 185 0.12 815 9.90 424 20 41 9.77 626 9.87 211 0.12 789 9.90 415 19 42 9.77 643 9.87 238 0.12 762 9.90 405 18 43 9.77 660 9.87 264 0.12 736 9.90 396 17 44 9.77 677 9.87 290 0.12 710 9.90 386 16 45 9.77 694 9.87 317 0.12 683 9.90 377 15 46 9.77 711 9.87 343 0.12 657 9.90 368 14 47 9.77 728 9.87 369 0.12 631 9.90 358 13 48 9.77 744 9.87 396 0.12 604 9.90 349 12 49 9.77 761 9.87 422 0.12 578 9.90 339 11 50 9.77 778 9.87 448 0.12 552 9.90 330 10 51 9.77 795 9.87 475 0.12 525 9.90 320 9 52 9.77 812 9.87 501 0.12 499 9.90 311 8 53 9.77 829 9.87 527 0.12 473 9.90 301 7 54 9.77 846 9.87 554 0.12 446 9.90 292 6 55 9.77 862 9.87 580 0.12 420 9.90 282 5 56 9.77 879 9.87 606 0.12 394 9.90 273 4 57 9.77 896 9.87 633 0.12 367 9.90 263 3 58 9.77 913 9.87 659 0.12 341 9.90 254 2 59 9.77 930 9.87 685 0.12 315 9.90 244 1 60 977 946 9.87 711 0.12 289 9.90 235 0 D L. Cos. L. Cotg. L. Tang. L. Sin. | h1 [78] 37c ) L. Sin. L. Tang. T L. Cotg. L. Cos. 0 9.77 946 9.87 711 0.12 289 9.90 235 60 1 9.77 963 9.87 738 0.12 262 9.90 225 59 2 9.77 980 9.87 764 0.12 236 9.90 216 58 3 9.77 997 9.87 790 0.12 210 9.90 206 57 4 9.78 013 9.87 817 0.12 183 9.90 197 56 5 9.78 030 9.87 843 0.12 157 9.90 187 55 6 9.78 047 9.87 869 0.12 131 9.90 178 54 7 9.78 063 9.87 895 0.12 105 9.90 168 53 8 9.78 080 9.87 922 0.12 078 9.90 159 52 9 9.78 097 9.87 948 0.12 052 9.90 149 51 10 9.78 113 9.87 974 0.12 026 9.90 139 50 11 9.78 130 9.88 000 0.12 000 9.90 130 49 12 9.78 147 9.88 027 0.11 973 9.90 120 48 13 9.78 163 9.88 053 0.11 947 9.90 111 47 14 9.78 180 9.88 079 0.11 921 9.90 101 46 15 9.78 197 9.88 105 0.11 895 9.90 091 45 16 9.78 213 9.88 131 0.11 869 9.90 082 44 17 9.78 230 9.88 158 0.11 842 9.90 072 43 18 9.78 246 9.88 184 0.11 816 9.90 063 42 19 9.78 263 9.88 210 0.11 790 9.90 053 41 20 9.78 280 9.88 236 0.11 764 9.90 043 40 21 9.78 296 9.88 262 0.11 738 9.90 034 39 22 9.78 313 9.88 289 0.11 711 9.90 024 38 23 9.78 329 9.88 315 0.11 685 9.90 014 37 24 9.78 346 9.88 341 0.11 659 9.90 005 36 25 9.78 362 9.88 367 0.11 633 9.89 995 35 26 9.78 379 9.88 393 0.11 607 9.89 985 34 27 9.78 395 9.88 420 0.11 580 9.89 976 33 28 9.78 412 9.88 446 0.11 554 9.89 966 32 29 9.78 428 9.88 472 0.11 528 9.89 956 31 30 9.78 445 9.88 498 0.11 502 9.89 947 30 31 9.78 461 9.88 524 0.11 476 9.89 937 29 32 9.78 478 9.88 550 0.11 450 9.89 927 28 33 9.78 494 9.88 577 0.11 423 9.89 918 27 34 9.78 510 9.88 603 0.11 397 9.89 908 26 35 9.78 527 9.88 629 0.11 371 9.89 898 25 36 9.78 543 9.88 655 0.11 345 9.89 888 24 37 9.78 560 9.88 681 0.11 319 9.89 879 23 38 9.78 576 9.88 707 0.11 293 9.89 869 22 39 9.78 592 9.88 733 0.11 267 9.89 859 21 40 9.78 609 9.88 759 0.11 241 9.89 849 20 41 9.78 625 9.88 786 0.11 214 9.89 840 19 42 9.78 642 9.88 812 0.11 188 9.89 830 18 43 9.78 658 9.88 838 0.11 162 9.89 820 17 44 9.78 674 9.88 864 0.11 136 9.89 810 16 45 9.78 691 9.88 890 0.11 110 9.89 801 15 46 9.78 707 9.88 916 0.11 084 9.89 791 14 47 9.78 723 9.88 942 0.11 058 9.89 781 13 48 9.78 739 9.88 968 0.11 032 9.89 771 12 49 9.78 756 9.88 994 0.11 006 9.89 761 11 50 9.78 772 9.89 020 0.10 980 9.89 752 10 51 9.78 788 9.89 046 0.10 954 9.89 742 9 52 9.78 805 9.89 073 0.10 927 9.89 732 8 53 9.78 821 9.89 099 0.10 901 9.89 722 7 54 9.78 837 9.89 125 0.10 875 9.89 712 6 55 9.78 853 9.89 151 0.10 849 9.89 702 5 56 9.78 869 9.89 177 0.10 823 9.89 693 4 57 9.78 886 9.89 203 0.10 797 9.89 683 3 58 9.78 902 9.89 229 0.10 771 9.89 673 2 59 9.78 918 9.89 255 0.10 745 9.89 663 1 60 9.78 934 9.89 281 0.10 719 9.89 653 0 P9c r D I L. Cos. L. Cotg. I L. Tang. L. Sin. I - II ii I K.- a [79] 4 r r L. Sin. r L. Tang. I r L. Cotg. r L. Cos. r r 38c 0 9.78 934 9.89 281 0.10 719 9.89 653 60 1 9.78 950 9.89 307 0.10 693 9.89 643 59 2 9.78 967 9.89 333 0.10 667 9.89 633 58 3 9.78 983 9.89 359 0.10 641 9.89 624 57 4 9.78 999 9.89 385 0.10 615 9.89 614 56 5 9.79 015 9.89 411 0.10 589 9.89 604 55 6 9.79 031 9.89 437 0.10 563 9.89 594 54 7 9.79 047 9.89 463 0.10 537 9.89 584 53 8 9.79 063 9.89 489 0.10 511 9.89 574 52 9 9.79 079 9.89 515 0.10 485 9.89 564 51 10 9.79 095 9.89 541 0.10 459 9.89 554 50 11 9.79 111 9.89 567 0.10 433 9.89 544 49 12 9.79 128 9.89 593 0.10 407 9.89 534 48 13 9.79 144 9.89 619 0.10 381 9.89 524 47 14 9.79 160 9.89 645 0.10 355 9.89 514 46 15 9.79 176 9.89 671 0.10 329 9.89 504 45 16 9.79 192 9.89 697 0.10 303 9.89 495 44 17 9.79,208 9.89 723 0.10 277 9.89 485 43 18 9.79 224 9.89 749 0.10 251. 9.89 475 42 19 9.79 240 9.89 775 0.10 225 9.89 465 41 20 9.79 256 9.89 801 0.10 199 9.89 455 40 21 9.79 272 9.89 827 0.10 173 9.89 445 39 22 9.79 288 9.89 853 0.10 147 9.89 435 38 23 9.79 304 9.89 879 0.10 121 9.89 425 37 24 9.79 319 9.89 905 0.10 095 9.89 415 36 25 9.79 335 9.89 931 0.10 069 9.89 405 35 26 9.79 351 9.89 957 0.10 043 9.89 395 34 27 9.79 367 9.89 983 0.10 017 9.89 385 33 28 9.79 383 9.90 009 0.09 991 9.89 375 32 29 9.79 399 9.90 035 0.09 965 9.89 364 31 5 30 9.79 415 9.90 061 0.09 939 9.89 354 30 31 9.79 431 9.90 086 0.09 914 9.89 344 29 32 9.79 447 9.90 112 0.09 888 9.89 334 28 33 9.79 463 9.90 138 0.09 862 9.89 324 27 34 9.79 478 9.90 164 0.09 836 9.89 314 26 35 9.79 494 9.90 190 0.09 810 9.89 304 25 36 9.79 510 9.90 216 0.09 784 9.89 294 24 37 9.79 526 9.90 242 0.09 758 9.89 284 23 38 9.79 542 9.90 268 0.09 732 9.89 274 22 39 9.79 558 9.90 294 0.09 706 9.89 264 21 40 9.79 573 9.90 320 0.09 680 9.89 254 20 41 9.79 589 9.90 346 0.09 654 9.39 244 19 42 9.79 605 9.90 371 0.09 629 9.89 233 18 43 9.79 621 9.90 397 0.09 603 9.89 223 17 44 9.79 636 9.90 423 0.09 577 9.89 213 16 45 9.79 652 9.90 449 0.09 551 9.89 203 15 46 9.79 668 9.90 475 0.09 525 9.89 193 14 47 9.79 684 9.90 501 0.09 499 9.89 183 13 48 9.79 699 9.90 527 0.09 473 9.89 173 12 49 9.79 715 9.90 553 0.09 447 9.89 162 11 50 - 9.79 731 9.90 578 0.09 422 9.89 152 10 51 9.79 746 9.90 604 0.09 396 9.89 142 9 52 9.79 762 9.90 630 0.09 370 9.89 132 8 53 9.79 778 9.90 656 0.09 344 9.89 122 7 54 9.79 793 9.90 682 0.09 318 9.89 112 6 55 9.79 809 9.90 708 0.09 292 9.89 101 5 56 9.79 825 9.90 734 0.09 266 9.89 091 4 57 9.79 840 9.90 759 0.09 241 9.89 081 3 58 9.79 856 9.90 785 0.09 215 9.89 071 2 59 9.79 872 9.90 811 0.09 189 9.89 060 1 60 9.79 887 9.90 837 0.09 163 9.89 050 0 D I I L. Cos. I L. Cotg. L. Tang. I L. Sin. 1 I [80] or I L. Sin. L. Tang. L. Cotg. T L. Cos. 1 39~ 0 9.79 887 9.90 837 0.09 163 9.89 050 60 1 9.79 903 9.90 863 0.09 137 9.89 040 59 2 9.79 918 9.90 889 0.09 111 9.89 030 58 3 9.79 934 9.90 914 0.09 086 9.89 020 57 4 9.70 950 9.90 940 0.09 060 9.89 009 56 5 9.79 965 9.90 966 0.09 034 9.88 999 55 6 9.79 981 9.90 992 0.09 008 9.88 989 54 7 9.79 996 9.91 018 0.08 982 9.88 978 53 8 9.80 012 9.91 043 0.08 957 9.88 968 52 9 9.80 027 9.91 069 0.08 931 9.88 958 51 10 9.80 043 9.91 095 0.08 905 9.88 948 50 11 9.80 058 9.91 121 0.08 879 9.88 937 49 12 9.80 074 9.91 147 0.08 853 9.88 927 48 13 9.80 089 9.91 172 0.08 828 9.88 917 47 14 9.80 105 9.91 198 0.08 802 9.88 906 46 15 9.80 120 9.91 224 0.08 776 9.88 896 45 16 9.80 136 9.91 250 0.08 750 9.88 886 44 17 9.80 151 9.91 276 0.08 724 9.88 875 43 18 9.80 166 9.91 301 0.08 699 9.88 865 42 19 9.80 182 9.91 327 0.08 673 9.88 855 41 20 9.80 197 9.91 353 0.08 647 9.88 844 40 21 9.80 213 9.91 379 0.08 621 9.88 834 39 22 9.80 228 9.91 404 0.08 596 9.88 824 38 23 9.80 244 9.91 430 0.08 570 9.88 813 37 24 9.80 259 9.91 456 0.08 544 9.88 803 36 25 9.80 274 9.91 482 0.08 518 9.88 793 35 26 9.80 290 9.91 507 0.08 493 9.88 782 34 27 9.80 305 9.91 533 0.08 467 9.88 772 33 28 9.80 320 9.91 559 0.08 441 9.88 761 32 29 9.80 336 9.91 585 0.08 415 9.88 751 31 30 9.80 351 9.91 610 0.08 390 9.88 741 30 31 9.80 366 9.91 636 0.08 364 9.88 730 29 32 9.80 382 9.91 662 0.08 338 9.88 720 28 33 9.80 397 9.91 688 0.08 312 9.88 709 27 34 9.80 412 9.91 713 0.08 287 9.88 699 26 35 9.80 428 9.91 739 0.08 261 9.88 688 25 36 9.80 443 9.91 765 0.08 235 9.88 678 24 37 9.80 458 9.91 791 0.08 209 9.88 668 23 38 9.80 473 9.91 816 0.08 184 9.88 657 22 39 9.80 489 9.91 842 0.08 158 9.88 647 21 40 9.80 504 9.91 868 0.08 132 9.88 636 20 41 9.80 519 9.91 893 0.08 107 9.88 626 19 42 9.80 534 9.91 919 0.08 081 9.88 615 18 43 9.80 550 9.91 945 0.08 055 9.88 605 17 44 9.80 565 9.91 971 0.08 029 9.88 594 16 45 9.80 580 9.91 996 0.08 004 9.88 584 15 46 9.80 595 9.92 022 0.07 978 9.88 573 14 47 9.80 610 9.92 048 0.07 952 9.88 563 13 48 9.80 625 9.92 073 0.07 927 9.88 552 12 49 9.80 641 9.92 099 0.07 901 9.88 542 11 50 9.80 656 9.92 125 0.07 875 9.88 531 10 51 9.80 671 9.92 150 0.07 850 9.88 521 9 52 9.80 686 9.92 176 0.07 824 9.88 510 8 53 9.80 701 9.92 202 0.07 798 9.88 499 7' 54 9.80 716 9.92 227 0.07 773 9.88 489 6 55 9.80 731 9.92 253 0.07 747 9.88 478 5 56 9.80 746 9.92 279 0.07 721 9.88 468 4 57 9.80 762 9.92 304 0.07 696 9.88 457 3 58 9.80 777 9.92 330 0.07 670 9.88 447 2 59 9.80 792 9.92 356 0.07 644 9.88 436 1 60 9.80 807 9.92 381 0.07 619 9.88 425 0 i~._ I D L. Cos. L. Cotg. L. Tang. L. Sin. 0 0 0~~ I I.5 IIN [ 81] L. Sin. L. Tang. | L. Cotg. L. Cos. 40~ 0 9.80 807 9.92 381 0.07 619 9.88 425 60 1 9.80 822 9.92 407 0.07 593 9.88 415 59 2 9.80 837 9.92 433 0.07 567 9.88 404 58 3 9.80 852 9.92 458 0.07 542 9.88 394 57 4 9.80 867 9.92 484 0.07 516 9.88 383 56 5 9.80 882 9.92 510 0.07 490 9.88 372 55 6 9.80 897 9.92 535 0.07 465 9.88 362 54 7 9.80 912 9.92 561 0.07 439 9.88 351 53 8 9.80 927 9.92 587 0.07 413 9.88 340 52 9 9.80 942 9.92 612 0.07 388 9.88 330 51 10 9.80 957 9.92 638 0.07 362 9.88 319 50 11 9.80 972 9.92 663 0.07 337 9.88 308 49 12 9.80 987 9.92 689 0.07 311 9.88 298 48 13 9.81 002 9.92 715 0.07 285 9.88 287 47 14 9.81 017 9.92 740 0.07 260 9.88 276 46 15 9.81 032 9.92 766 0.07 234 9.88 266 45 16 9.81 047 9.92 792 0.07 208 9.88 255 44 17 9.81 061 9.92 817 0.07 183 9.88 244 43 18 9.81 076 9.92 843 0.07 157 9.88 234 42 19 9.81 091 9.92 868 0.07 132 9.88 223 41 20 9.81 106 9.92 894 0.07 106 9.88 212 40 21 9.81 121 9.92 920 0.07 080 9.88 201 39 22 9.81 136 9.92 945 0.07 055 9.88 191 38 23 9.81 151 9.92 971 0.07 029 9.88 180 37 24 9.81 166 9.92 996 0.07 004 9.88 169 36 25 9.81 180 9.93 022 0.06 978 9.88 158 35 26 9.81 195 9.93 048 0.06 952 ' 9.88 148 34 27 9.81 210 9.93 073 0.06 927 9.88 137 33 28 9.81 225 9.93 099 0.06 901 9.88 126 32 29 9.81 240 9.93 124 0.06 876 9.88 115 31 49 30 9.81 254 9.93 150 0.06 850 9.88 105 30 31 9.81 269 9.93 175 0.06 825 9.88 094 29 32 9.81 284 9.93 201 0.06 799 9.88 083 28 33 9.81 299 9.93 227 0.06 773 9.88 072 27 34 9.81 314 9.93 252 0.06 748 9.88 061 26 35 9.81 328 9.93 278 0.06 722 9.88 051 25 36 9.81 343 9.93 303 0.06 697 9.88 040 24 37 9.81 358 9.93 329 0.06 671 9.88 029 23 38 9.81 372 9.93 354 0.06 646 9.88 018 22 39 9.81 387 9.93 380 0.06 620 9.88 007 21 40 9.81 402 9.93 406 0.06 594 9.87 996 20 41 9.81 417 9.93 431 0.06 569 9.87 985 19 42 9.81 431 9.93 457 0.06 543 9.87 975 18 43 9.81 446 9.93 482 0.06 518 9.87 964 17 44 9.81 461 9.93 508 0.06 492 9.87 953 16 45 9.81 475 9.93 533 0.06 467 9.87 942 15 46 9.81 490 9.93 559 0.06 441 9.87 931 14 47 9.81 505 9.93 584 0 06 416 9.87 920 13 48 9.81 519 9.93 610 0.06 390 9.87 909 12 49 9.81 534 9.93 636 0.06 364 9.87 898 11 50 9.81 549 9.93 661 0.06 339 9.87 887 10 51 9.81 563 9.93 687 0.06 313 9.87 877 9 52 9.81 578 9.93 712 0.06 288 9.87 866 8 53 9.81 592 9.93 738 0.06 262 9.87 855 7 54 9.81 607 9.93 763 0.06 237 9.87 844 6 55 9.81 622 9.93 789 0.06 211 9.87 833 5 56 9.81 636 9.93 814 0.06 186 9.87 822 4 57 9.81 651 9.93 840 0.06 160 9.87 811 3 58 9.81 665 9.93 865 0.06 135 9.87 800 2 59 9.81 680 9.93 891 0.06 109 9.87 789 1 60 9.81 694 9.93 916 0.06 084 9 87 778 0 L. Cos. L. Cotg. L. Tang. L. Sin. I * 2 * - [82] r I T L. Sin. L. Tang. = T T' L. Cotg. L. Cos. [ 1 41c 0 9.81 694 9.93 916 0.06 084 9.87 778 60 1 9.81 709 9.93 942 0.06 058 9.87 767 59 2 9.81 723 9.93 967 0.06 033 9.87 756 58 3 9.81 738 9.93 993 0.06 007 9.87 745 57 4 9.81 752 9.94 018 0.05 982 9.87 734 56 5 9.81 767 9.94 044 0.05 956 9.87 723 55 6 9.81 781 9.94 069 0.05 931 9.87 712 54 7 9.81 796 9.94 095 0.05 905 9.87 701 53 8 9.81 810 9.94 120 0.05 880 9.87 690. 52 9 9.81 825 9.94 146 0.05 854 9.87 679 51 10 9.81 839 9.94 171 0.05 829 9.87 668 50 11 9.81 854 9.94 197 0.05 803 9.87 657 49 12 9.81 868 9.94 222 0.05 778 9.87 646 48 13 9.81 882 9.94 248 0.05 752 9.87 635 47 14 9.81 897 9.94 273 0.05 727 9.87 624 46 15 9.81 911 9.94 299 0.05 701 9.87 613 45 16 9.81 926 9.94 324 0.05 676 9.87 601 44 17 9.81 940 9.94 350 0.05 650 9.87 590 43 18 9-81 955 9.94 375 0.05 625 9.87 579 42 19 9.81 969 9.94 401 0.05 599 9.87 568 41 20 9.81 983 9.94 426 0.05 574 9.87 557 40 21 9.81 998 9.94 452 0.05 548 9.87 546 39 22 9.82 012 9.94 477 0.05 523 9.87 535 38 23 9.82 026 9.94 503 0.05 497 9.87 524 37 24 9.82 041 9.94 528 0.05 472 9.87 513 36 25 9.82 055 9.94 554 0.05 446 9.87 501 35 26 9.82 069 9.94 579 0.05 421 9.87 490 34 27 9.82 084 9.94 604 0.05 396 9.87 479 33 28 9.82 098 9.94 630 0.05 370 9.87 468 32 29 9.82 112 9.94 655 0.05 345 9.87 457 31 30 9.82 126 9.94 681 0.05 319 9.87 446 30 31 9.82 141 9.94 706 0.05 294 9.87 434 29 32 9.82 155 9.94 732 0.05 268 9.87 423 28 33 9.82 169 9.94 757 0.05 243 9.87 412 27 34 9.82 184 9.94 783 0.05 217 9.87 401 26 35 9.82 198 9.94 808 0.05 192 9.87 390 25 36 9.82 212 9.94 834 0.05 166 9.87 378 24 37 9.82 226 9.94 859 0.05 141 9.87 367 23 38 9.82 240 9.94 884 0.05 116 9.87 356 22 39 9.82 255 9.94 910 0.05 090 9.87 345 21 40 9.82 269 9.94 935 0.05 065 9.87 334 20 41 9.82 283 9.94 961 0.05 039 9.87 322 19 42 9.82 297 9.94 986 0.05 014 9.87 311 18 43 9.82 311 9.95 012 0.04 988 9.87 300 17 44 9.82 326 9.95 037 0.04 963 9.87 288 16 45 9.82 340 9.95.062 0.04 938 9.87 277 15 46 9.82 354 9.95 088 0.04 912 9.87 266 14 47 9.82 368 9.95 113 0.04 887 9.87 255 13 48 9.82 382 9.95 139 0.04 861 9.87 243 12 49 9.82 396 9.95 164 0.04 836 9.87 232 11 50 9.82 410 9.95 190 0.04 810 9.87 221 10 51 9.82 424 9.95 215 0.04 785 9.87 209 9 52 9.82 439 9.95 240 0.04 760 9.87 198 8 53 9.82 453 9.95 266 0.04 734 9.87 187 7 54 9.82 467 9.95 291 0.04 709 9.87 175 6 55 9.82 481 9.95 317 0.04 683 9.87 164 5 56 9.82 495 9.95 342 0.04 658 9.87 153 4 57 9.82 509 9.95 368 0.04 632 9.87 141 3 58 9.82 523 9.95 393 0.04 607 9.87 130 2 59 9.82 537 9.95 418 0.04 582 9.87 119 1 60 9.82 551 9.95 444 0.04 556 9.87 107 0 ii _" 1QC I I m m L. Cos. L. Cotg. L. Tang. I L. Sin. I I m m I * i - m [ sr`-~ r ~ -~ r I8 6 - 4 9 o f L. Sin. L.Tang. L. Cotg. L. Cos. T 0 42~ 0 9.82 551 9.95 444 0.04 556 9.87 107 60 1 9.82 565 9.95 469 0.04 531 9.87 096 59 2 9.82 579 9.95 495 0.04 505 9.87 085 58 3 9.82 593 9.95 520 0.04 480 9.87 073 57 4 9.82 607 9.95 545 0.04 455 9.87 062 56 5 9.82 621 9.95 571 0.04 429 9.87 050 55 6 9.82 635 9.95 596 0.04 404 9.87 039 54 7 9.82 649 9.95 622 0.04 378 9.87 028 53, 8 9.82 663 9.95 647 0.04 353 9.87 016 52 9 9.82 677 9.95 672 0.04 328 9.87 005 51 10 9.82 691 9.95 698 0.04 302 9.86 993 50 11 9.82 705 9.95 723 0.04 277 9.86 982 49 12 9.82 719 9.95 748 0.04 252 9.86 970 48 13 9.82 733 9.95 774 0.04 226 9.86 959 47 14 9.82 747 9.95 799 0.04 201 9.86 947 46 15 9.82 761 9.95 825 0.04 175 9.86 936 45 16 9.82 775 9.95 850 0.04 150 9.86 924 44 17 9.82 788 9.95 875 0.04 125 9.86 913 43 18 9.82 802 9.95 901 0.04 099 9.86 902 42 19 9.82 816 9.95 926 0.04 074 9.86 890 41 20 9.82 830 9.95 952 0.04 048 9.86 879 40 21 9.82 844 9.95 977 0.04 023 9.86 867 39 22 9.82 858 9.96 002 0.03 998 9.86 855 38 23 9.82 872 9.96 028 0.03 972 9.86 844 37 24 9.82 885 9.96 053 0.03 947 9.86 832 36 25 9.82 899 -9.96 078 0.03 922 9.86 821 35 26 9.82 913 9.96 104 0.03 896 9.86 809 34 27 9.82 927 9.96 129 0.03 871 9.86 798 33 28 9.82 941 9.96 155 0.03 845 9.86 786 32 29 9.82 955 9.96 180 0.03 820 9.86 775 31 4 30 9.82 968 9.96 205 0.03 795 9.86 763 30 31 9.82 982 9.96 231 0.03 769 9.86 752 29 32 9.82 996 9.96 256 0.03 744 9.86 740 28 33 9.83 010 9.96 281 0.03 719 9.86 728 27 34 9.83 023 9.96 307 0.03 693 9.86 717 26 35 9.83 037 9.96 332 0.03 668 9.86 705 25 36 9.83 051 9.96 357 0.03 643 9.86 694 24 37 9.83 065 9.96 383 0.03 617 9.86 682 23 38 9.83 078 9.96 408 0.03 592 9.86 670 22 39 9.83 092 9.96 433 0.03 567 9.86 659 21 40 9.83 106 9.96 459 0.03 541 9.86 647 20 41 9.83 120 9.96 484 0.03 516 9.86 635 19 42 9.83 133 9.96 510 0.03 490 9.86 624 18 43 9.83 147 9.96 535 0.03 465 9.86 612 17 44 9.83 161 9.96 560 0.b3 440 9.86 600 16 45 9.83 174 9.96 586 0.03 414 9.86 589 15 46 9.83 188 9.96 611 0.03 389 9.86 577 14 47 9.83 202 9.96 636 0.03 364 9.86 565 13 48 9.83 215 9.96 662 0.03 338 9.86 554 12 49 9.83 229 9.96 687 0.03 313 9.86 542 11 50- 9.83 242 9.96 712 0.03 288 9.86 530 10 51 9.83 256 9.96 738 0.03 262 9.86 518 9 52 9.83 270 9.96 763 0.03 237 9.86 507 8 53 9.83 283 9.96 788 0.03 212 9.86 495 7 54 9.83 297 9.96 814 0.03 186 9.86 483 6 55 9.83 310 9.96 839 0.03 161 9'86 472 5 56 9.83 324 9.96 864 0.03 136 9.86 460 4 57 9.83 338 9.96 890 0.03 110 9.86 448 3 58 9.83 351 9.96 915 0.03 085 9.86 436 2 59 9.83 365 9.96 940 0.03 060 9.86 425 1 60 9.83 378 9.96 966 0.03 034 9.86 413 0 IL. Cos. I m L. Cotg. L. Tang. L.Sin. / * 1184j - - I T l Er I r III L. Sin. L. Tang.,. L. Cotg. I L. Cos. T mm W 0 9.83 378 9.96 966 0.03 034 9.86 413 60 1 9.83 392 9.96 991 0.03 009 9.86 401 59 2 9.83 405 9.97 016 0.02 984 9.86 389 58 3 9.83 419 9.97 042 0.02 958 9.86 377 57 4 9.83 432 9.97 067 0.02 933 9.86 366 56 5 9.83 446 9.97 092 0.02 908 9.86 354 55 6 9.83 459 9.97 118 0.02 882 9.86 342 54 7 9.83 473 9.97 143 0.02 857 9.86 330 53 8 9.83 486 9.97 168 0.02 832 9.86 318 52 9 9.83 500 9.97 193 0.02 807 9.86 306 51 10 9.83 513 9.97 219 0.02 781 9.86 295 50 11 9.83 527 9.97 244 0.02 756 9.86 283 49 12 9.83 540 9.97 269 0.02 731 9.86 271 48 13 9.83 554 9.97 295 0.02 705 9.86 259 47 14 9.83 567 9.97 320 0.02 680 9.86 247 46 15 9.83 581 9.97 345 0.02 655 9.86 235 45 16 9.83 594 9.97 371 0.02 629 9.86 223 44 17 9.83 608 9.97 396 0.02 604 9.86 211 43 18 9.83 621 9.97 421 0.02 579 9.86 200 42 19 9.83 634 9.97 447 0.02 553 9.86 188 41...... 20 21 22 23 24 i I -1 9.83 648 9.83 661 9.83 674 9.83 688 9.83 701 9.97 472 9.97 497 9.97 523 9.97 548 9.97 573 0.02 528 0.02 503 0.02 477 0.02 452 0.02 427 9.86 176 9.86 164 9.86 152 9.86 140 9.86 128 40 39 38 37 36 I -1 25 9.83 715 9.97 598 0.02 402 9.86 116 35 26 9.83 728 9.97 624 0.02 376 9.86 104 34 27 9.83 741 9.97 649 0.02 351 9.86 092 33 28 9.83 755 9.97 674 0.02 326 9.86 080 32 29 9.83 768 9.97 700 0.02 300 9.85 068 31 30 9.83 781 9.97 725 0.02 275 9.86 056 30 31 9.83 595 9.97 750 0.02 250 9.86 044 29 32 9.83 808 9.97 776 0.02 224 9.86 032 28 33 9.83 821 9.97 801 0.02 199 9.86 020 27 34 9.83 834 9.97 826 0.02 174 9.86 008 26 35 9.83 848 9.97 851 0.02 149 9.85 996 25 36 9.83 861 9.97 877 0.02 123 9.85 984 24 37 9.83 874 9.97 902 0.02 098 9.85 972 23 38 9.83 887 9.97 927 0.02 073 9.85 960 22 39 9.83 901 9.97 953 0.02 047 9.85 948 21 40 9.83 914 9.97 978 0.02 022 9.85 936 20 41 9.83 927 9.98 003 0.01 997 9.85 924 19 42 9.83 940 9.98 029 0.01 971 9.85 912 18 43 9.83 954 9.98 054 0.01 946 9.85 900 17 44 9.83 967 -9.98 079 0.01 921 9.85 888 16 45 9.83 980 9.98 104 0.01 896 9.85 876 15 46 9.83 993 9.98 130 0.01 870 9.85 864 14 47 9.84 006 9.98 155 0.01 845 9.85 851 13 48 9.84 020 9.98 180 0.01 820 9.85 839 12 49 9.84 033 9.98 206 0.01 794 9.85 827 11 50 9.84 046 9.98 231 0.01 769 9.85 815 10 51 9.84 059 9.98 256 0.01 744 9.85 803 9 52 9.84 072 9.98 281 0.01 719 9.85 791 8 53 9.84 085 9.98 307 0.01 693 9.85 779 7 54 9.84 098 9.98 332 0.01 668 9.85 766 6 55 9.84 112 9.98 357 0.01 643 9.85 754 5 56 9.84 125 9.98 383 0.00 617 9.85 742 4 57 9.84 138 9.98 408 0.01 592 9.85 730 3 58 9.84 151 9.98 433 0.01 567 9.85 718 2 59 9.84 164 9.98 458 0.01 542 9.85 706 1 60 9.84 177 9.98 484 0.01 516 9.85 693 0 16c I L. Cos. I L. Cotg. L. Tang. I L. Sin. I I - - I i - [85] 44c L. Sin. r Mr L. Tang. | L. Cotg. L. Cos. 0 9.84 177 9.98 484 0.01 516 9.85 693 60 1 9.84 190 9.98 509 0.01 491 9.85 681 59 2 9.84 203 9.98 534 0.01 466 9.85 669 58 3 9.84 216 9.98 560 0.01 440 9.85 657 57 4 9.84 229 9.98 585 0.01 415 9.85 645 56 5 9.84 242 9.98 610 0.01 390 9.85 632 55 6 9.84 255 9.98 635 0.01 365 9.85 620 54 7 9.84 269 9.98 661 0.01 339 9.85 608 53 8 9.84 282 9.98 686 0.01 314 9.85 596 52 9 9.84 295 9.98 711 0.01 289 9.85 583 51 10 9.84 308 9.98 737 0.01 263 9.85 571 50 11 9.84 321 9.98 762 0.01 238 9.85 559 49 12 9.84 334 9.98 787 0.01 213 9.85 547 48 13 9.84 347 9.98 812 0.01 188 9.85 534 47 14 9.84 360 9.98 838 0.01 162 9.85 522 46 15 9.84 373 9.98 863 0.01 137 9.85 510 45 16 9.84 385 9.98 888 0.01 112 9.85 497 44 17 9.84 398 9.98 913 0.01 087 9.85 485 43 18 9.84 411 9.98 939 0.01 061 9.85 473 42 19 9.84 424 9.98 964 0.01 036 9.85 460 41 20 9.84 437 9.98 989 0.01 011 9.85 448 40 21 9.84 450 9.99 015 0.00 985 9.85 436 39 22 9.84 463 9.99 040 0.00 960 9.85 423 38 23 9.84 476 9.99 065 0.00 935 9.85 411 37 24 9.84 489 9.99 090 0.00 910 9.85 399 36 25 9.84 502 9.99 116 0.00 884 9.85 386 35 26 9.84 515 9.99 141 0.00 859 9.85 374 34 27 9.84 528 9.99 166 0.00 834 9.85 361 33 28 9.84 540 9.99 191 0.00 809 9.85 349 32 29 9.84 553 9.99 217 0.00 783 9.85 337 31 4 30 9.84 566 9.99 242 0.00 758 9.85 324 30 31 9.84 579 9.99 267 0.00 733 9.85 312 29 32 9.84 592 9.99 293 0.00 707 9.85 299 28 33 9.84 605 9.99 318 0.00 682 9.85 287 27 34 9.84 618 9.99 343 0.00 657 9.85 274 26 35 9.84 630 9.99 368 0.00 632 9.85 262 25 36 9.84 643 9.99 394 0.00 606 9.85 250 24 37 9.84 656 9.99 419 0.00 581 9.85 237 23 38 9.84 669 9.99 444 0.00 556 9.85 225 22 39 9.84 682 9.99 469 0.00 531 9.85 212 21 40 9.84 694 9.99 495 0.00 505 9.85 200 20 41 9.84 707 9.99 520 0.00 480 9.85 187 19 42 9.84 720 9.99 545 0.00 455 9.85 175 18 43 9.84 733 9.99 570 0.00 430 9.85 162 17 44 9.84 745 9.99 596 0.00 404 9.85 150 16 45 9.84 758 9.99 621 0.00 379 9.85 137 15 46 9.84 771 9.99 646 0.00 354 9.85 125 14 47 9.84 784 9.99 672 0.00 328 9.85 112 13 48 9.84 796 9.99 697 0.00 303 9.85 100 12 49 9.84 809 9.99 722 0.00 278 9.85 087 11 50 9.84 822 9.99 747 0.00 253 9.85 074 10 51 9.84 835 9.99 773 0.00 227 9.85 062 9 52 9.84 847 9.99 798 0.00 202 9.85 049 8 53 9.84 860 9.99 823 0.00 177 9.85 037 7 54 9.84 873 9.99 848 0.00 152 9.85 024 6 55 9.84 885 9.99 874 0.00 126 9.85 012 5 56 9.84 898 9.99 899 0.00 101 9.84 999 4 57 9.84 911 9.99 924 0.00 076 9.84 986 3 58 9.84 923 9.99 949 0.00 051 9.84 974 2 59 9.84 936 9.99 975 0.00 025 9.84 961 1 60 9.84 949 0.00 000 0.00 000 9.84 949 0 P~~~~~~~~ i T D I L. Cos. I L. Cotg. 1 -w L.Tang. | L. Sin. | ' -A I I - d [86] I I I L ~ -I ~ - L I — ~e TABLE IV AUXILIARY FIVE-PLACE TABLE FOR SMALL ANGLES [87,] r I. or -~~~~~ l II 'I s T I -- - - - -I......I T' | L. Sin. m I - I iI - - - 0 0 4.68557 4.68557 5.31443 5.31443 60 1.68557.68557.31443.31443 6.46373 120 2.68557.68557.31443.31443.76476 180 3.68557.68557.31443.31443.94085 240 4.68557.68558.31443.31442 7.06579 300 5 4.68557 4.68558 5.31443 5.31442 7.16270 360 6..68557.68558.31443.31442.24188 420 7.68557.68558.31443.31442.30882 480 8.68557.68558.31443.31442.36682 540 9.68557.68558.31443.31442.41797 600 10 4.68557 4.68558 5.31443 5.31442 7.46373 660 11.68557.68558.31443.31442.50512 720 12.68557.68558.31443.31442.54291 780 13.68557.68558.31443.31442.57767 840 14.68557.68558.31443.31442.60985 900 15 4.68557 4.68558 5.31443 5.31442 7.63982 960 16.68557.68558.31443.31442.66784 1020 17.68557.68558.31443.31442.69417 1080 18.68557.68558.31443.31442.71900 1140 19.68557.68558.31443.31442.74248 1200 20 4.68557 4.68558 5.31443 5.31442 7.76475 1260 21.68557.68558.31443.31442.78594 1320 22.68557.68558.31443.31442.80615 1380 23.68557.68558.31443.31442.82545 1440 24.68557.68558.31443.31442.84393 1500 25 4.68557 4.68558 5.31443 5.31442 7.86166 1560 26.68557.68558.31443.31442.87870 1620 27.68557.68558.31443.31442.89509 1680 28.68557.68558.31443.31442.91088 1740 29.68557.68559.31443.31441.92612 1800 30 4.68557 4.68559 5.31443 5.31441 7.94084 1860 31.68557.68559.31443.31441.95508 1920 32.68557.68559.31443.31441.96887 1980 33.68557.68559.31443.31441.98223 2040 34.68557.68559.31443.31441.99520 2100 35 4.68557 4.68559 5.31443 5.31441 8.00779 2160 36.68557.68559.31443.31441.02002 2220 37.68557.68559.31443.31441.03192 2280 38.68557.68559.31443.31441.04350 2340 39.68557.68559.31443.31441.05478 2400 40 4.68557 4.68559 5.31443 5.31441 8.06578 2460 41.68556.68560.31444.31440.07650 2520 42.68556.68560.31444.31440.08696 2580 43.68556.68560.31444.31440.09718 2640 44.68556.68560-.31444.31440.10717 2700 45 4.68556 4.68560 5.31444 5.31440 8.11693 2760 46.68556.68560.31444.31440.12647 2820 47.68556.68560.31444.31440.13581 2880 48.68556.68560.31444.31440.14495 2940 49.68556.68560.31444.31440.15391 3000 50 4.68556 4.68561 5.31444 5.31439 8.16268 3060 51.68556.68561.31444.31439.17128 3120 52.68556.68561.31444.31439.17971 3180 53.68556.68561.31444.31439.18798 3240 54.68556.68561.31444.31439.19610 3300 55 4.68556 4.68561 5.31444 5.31439 8.20407 3360 56.68556.68561.31444.31439.21189 3420 57.68555.68561.31445.31439.21958 3480 58.68555.68562.31445.31438.22713 3540 59.68555.68562.31445.31438.23456 3600 60 4.68555 4.68562 5.31445 5.31438 8.24186 [88] I I II a /l I S T S' T' T L. Sin. ] 1~0 3600 0 4.68555 4.68562 5.31445 5.31438 8.24186 3660 1.68555.68562.31445.31438.24903 3720 2.68555.68562.31445.31438.25609,. 3780 3.68555.68562.31445.31438.26304 3840 4.68555.68563.31445.31437.26988 3900 5 4.68555 4.68563 5.31 5.5.31437 8.27661 3960 6.68555.68563.31445.31437.28324 4020 7.68555.68563.31445.31437.28977 4080 8.68555.68563.31445.31437.29621 4140 9.68555.68563.31445.31437.30255 4200 10 4.68554 4.68563 5.31446 5.31437 8.30879 4260 11.68554.68564.31446.31436.31495 4320 12.68554.68564.31446.31436.32103 4380 13.68554.68564.31446.31436.32702 4440 14.68554.68564.31446.31436.33292 4500 15 4.68554 4.68564 5.31446 5.31436 8.33875 4560 16.68554.68565.31446.31435.34450 4620 17.68554.68565.31446.31435.35018 4680 18.68554.68565.31446.31435.35578 4740 19.68554.68565.31446.31435.36131 4800 20 4.68554 4.68565 5.31446 5.31435 8.36678 4860 21.68553.68566.31447.31434.37217 4920 22.68553.68566.31447.31434.37750 4980 23.68553.68566.31447.31434.38276 5040 24.68553.68566.31447.31434.38796 5100 25 4.68553 4.68566 5.31447 5.31434 8.39310 5160 26.68553.68567.31447.31433.39818 5220 27.68553.68567.31447.31433.40320 5280 28.68553.68567.31447.31433.40816 5340 29.68553.68567.31447.31433.41307 5400 30 4.68553 4.68567 5.31447 5.31433 8.41792 5460 31.68552.68568.31448.31432.42272 5520 32.68552.68568.31448.31432.42746 5580 33.68552.68568.31448.31432.43216 5640 34.68552.68568.31448.31432.43680 5700 35 4.68552 4.68569 5.31448 5.31431 8.44139 5760 36.68552.68569.31448.31431.44594 5820 37.68552.68569.31448.31431.45044 5880 38.68552.68569.31448.31431.45489 5940 39.68551.68569.31449.31431.45930 6000 40 4.68551 4.68570 5.31449 5.31430 8.46366 6060 41.68551.68570.31449.31430.46799 6120 42.68551.68570.31449.31430.47226 6180 43.68551.68570.31449.31430.47650 6240 44.68551.68571.31449.31429.48069 6300 45 4.68551 4.68571 5.31449 5.31429 8.48485 6360 46.68551.68571.31449.31429.48896 6420 47.68550.68572.31450.31428.49304 6480 48.68550.68572.31450.31428.49708 6540 49.68550.68572.31450.31428.50108 6600 50 4.68550 4.68572 5.31450 5.31428 8.50504 6660 51.68550.68573.31450.31427.50897 6720 52.68550.68573.31450.31427.51287 6780 53.68550.68573.31450.31427.51673 6840 54.68550.68573.31450.31427.52055 6900 55 4.68549 4.68574 5.31451 5.31426 8.52434 6960 56.68549.68574.31451.31426.52810 7020 57.68549.68574.31451.31426.53183 7080 58.68549.68575.31451.31425.53552 7140 59.68549.68575.31451.31425.53919 7200 60 1 4.68549 4.68575 1 5.31451 5.31425 8.54282 — A!- ~ ~ ~ ~ ~ ~ -~ —_ - L — [89] TABLE V FOUR-PLACE TABLE OF THE NATURAL SINE, COSINE, TANGENT, AND COTANGENT FOR EVERY 10' OF THE QUADRANT [91] N. Sin. N. Tan. N. Cot. N. Cos. 0 00 -..0000.0000 1.0000 00 90 10.0029.0029 343.77 1.0000 50 20.0058.0058 171.89 1.0000 40 30.0087.0087 114.59 1.0000 30 40.0116.0116 85.940.9999 20 50.0145.0145 68.750.9999 10 1 00.0175.0175 57.290.9998 00 89 -10.0204.0204 49.104.9998 50 20.0233.0233 42.964.9997 40 30.0262.0262 38.188.9997 30 40.0291.0291 34.368.9996 20 50.0320.0320 31,242.9995 10 2 00.0349.0349 28.636.9994 00 88 10.0378.0378 26.432.9993 50 20.0407.0407 24.542.9992, 40 30.0436.0437 22.904.9990 30 40.0465.0466 21.470.9989 20 50.0494.0495 20.206.9988 10 3 00.0523.0524 19.081.9986 00 87 10.0552.0553 18.075.9985 50 20.0581.0582 17.169.9983 40 30.0610.0612 16.350.9981 30 40.0640.0641 15.605.9980 20 50.0669.0670 14.924.9978 10 4 00.0698.0699 14.301.9976 00 86 10.0727.0729 13.727.9974 50 20.0756.0758 13.197.9971 40 30.0785.0787 12.706.9969 30 40.0814.0816 12.251.9967 20 50.0843 0846 11.826.9964 10 5 00,0872.0875 11.430.9962 00 85 10.0901.0904 11.059.9959 50 20.0929.0934 10.712.9957 40 30.0958.0963 10.385.9954 30 40.0987.0992 10.078.9951 20 50.1016.1022 9.7882.9948 10 6 00.1045.1051 9.5144.9945 00 84 10.1074.1080 9.2553.9942 50 20.1103.1110 9.0098.9939 40 30.1132.1139 8.7769.9936 30 40.1161.1169 8.5555.9932 20 50.1190.1198 8.3450.9929 10 7 00.1219.1228 8.1443.9925 00 83 10.1248.1257 7.9530.9922 50 20.1276.1287 7.7704.9918 40 30.1305.1317 7.5958.9914 30 40.1334.1346 7.4287.9911 20 50.1363.1376 7.2687.9907 10 8 00.1392.1405 7.1154.9903 00 82 10.1421.1435 6.9682.9899 50 20.1449.1465 6.8269.9894 40 30.1478.1495 6.6912.9890 30 40.1507.1524 6.5606.9886 20 50.1536.1554 6.4348.9881 10 9 00.1564.1584 6.3138.9877 00 81 N. Cos. N. Cot. N. Tan. N. Sin. ~ [92] oN. Sin. N. Tan. N. Cot. N. Cos. 9 00.1564.1584 6.3138.9877 00 81 10.1593.1614 6.1970.9872 50 20.1622.1644 6.0844.9868 40 30.1650.1673 5.9758.9863 30 40.1679.1703 5.8708.9858 20 50.1708.1733 5.7694.9853 10 10 00.1736.1763 5.6713.9848 00 80 10.1765.1793 5.57U4.9843 50 20.1794.1823 5.4845.9838 40 30.1822.1853 5.3955.9833 30 40.1851.1883 5.3093.9827 20 50.1880.1914 5.2257.9822 10 11 00.1908.1944 5.1446.9816 00 79 10.1937.1974 5.0658.9811 50 20.1965.2004 4.9894.9805 40 30.1994.2035 4.9152.9799 30 40.2022.2065 4.8430.9793 20 50.2051.2095 4.7729.9787 10 12 00.2079.2126 4.7046.9781 00 78 10.2108.2156 4.6382.9775 50 20.2136.2186 4.5736.9769 40 30.2164.2217 4.5107.9763 30 40.2193.2247 4.4494.9757 20 50.2221.2278 4.3897.9750 10 13 00.2250.2309 4.3315.9744 00 77 10.2278.2339 4.2747.9737 50 20.2306.2370 4.2193.9730 40 30.2334.2401 4.1653.9724 30 40.2363.2432 4.1126.9717 20 50.2391.2462 4.0611.9710 10 14 00.2419.2493 4.0108.9703 00 76 10.2447.2524 3.9617.9696 50 20.2476.2555 3.9136.9689 40 30.2504.2586 3.8667.9681 30 40.2532.2617 3.8208.9674 20 50.2560.2G48 3.7760.9667 10 15 00.2588.2679 3.7321.9659 00 75 10.2616.2711 3.6891.9G52 50 20.2644.2742 3.6470.9644 40 30.2672.2773 3.6059.9636 30 40.2700.2805 3.5656.9628 20 50.2728.2836 3.5261.9621 10 16 00.2756.2867 3.4874.9613 00 74 10.2784.2899 3.4495.9605 50 20.2812.2931 3.4124.9596 40 30.2840.2962 3.3759.9588 30 40.2868.2994 3 3402.9580 20 50.2896.3026 3.3052.9572 10 17 00.2924.3057 3.2709.9563 00 73 10.2952.3089 3.2371.9555 50 20.2979.3121 3.2041.9546 40 30.3007.3153 3.1716.9537 30 40.3035.3185 3.1397.9528 '20 50.3062.3217 3.1084.9520 10 18 00.3090.3249 3.0777.9511 00 72 N. Cos. N. Cot. N. Tan. N. Sin. [93] o I N. Sin. N. Tan. N. Cot. N. Cos. 18 00.3090.3249 3.0777.9511 00 72 10.3118.3281 3.0475.9502 50 20.3145.3314 3.0178.9492 40 30,3173,3346 2.9887.9483 30 40.3201.3378 2.9600.9474 20 50.3228.3411 2.9319.9465 10 19 00.3256.3443 2.9042.9455 00 71 10.3283.3476 2.8770.9446 50 20-.3311.3508 2.8502.9436 40 30.3338.3541 2.8239.9426 30 40.3365.3574 2.7980.9417 20 50.3393.3607 2.7725.9407 10 20 00.3420.3640 2.7475.9397 00 70 10.3448.3673 2.7228.9387 50 20.3475.3706 2.6985.9377 40 30.3502.3739 2.6746.9367 30 40.3529.3772 2.6511.9356 20 50.3557.3805 26279.9346 10 21 00.3584.3839 2.6051.9336 00 69 10.3611.3872 2.5826.9325 50 20.3638.3906 2.5605.9315 40 30.3665.3939 2.5386.9304 30 40.3692.3973 2.5172.9293 20 50.3719.4006 2.4960.9283 10 22 00.3746.4040 2.4751.9272 00 68 10.3773.4074 2.4545.9261 50 20.3800.4108 2.4342.9250 40 30.3827.4142 2.4142.9239 30 40.3854.4176 2.3945.9228 20 50.3881.4210 2.3750.9216 10 23 00.3907.4245 2.3559.9205 00 67 10.3934.4279 2.3369.9194 50 20.3961.4314 2.3183.9182 40 30.3987.4348 2.2998.9171 30 40.4014.4383 2.2817.9159 20 50.4041.4417 2.2637.9147 10 24 00.4067.4452 2.2460.9135 00 66 10.4094.4487 2.2286.9124 50 20.4120.4522 2.2113.9112 40 30.4147.4557 2.1943.9100 30 40.4173.4592 2.1775.9088 20 50.4200.4628 2.1609.9075 10 25 00.4226.4663 2.1445.9063 00 65 10.4253.4699 2.1283.9051 50 20.4279.4734 2.1123.9038 40 30.4305 4770 2.0965.9026 30 40.4331.4806 2.0809.9013 20 50.4358.4841 2.0655.9001 10 26 00.4384.4877 2.0503.8988 00 64 10.4410.4913 2.0353.8975 50 20.4436.4950 2.0204.8962 40 30.4462.4986 2.0057.8949 30 40.4488.5022 1.9912.8936 20 50.4514.5059 1.9768.8923 10 27 00.4540.5095 1.9626.8910 00 63 N. Cos. N. Cot. N. Tan. N. Sin. ' [94] 0 N. Sin. N. Tan. - N. Cot. N. Cos. 27 00.4540.5095 1.9626.8910 00 63 10.4566.5132 1.9486.8897 50 20.4592.5169 1.9347.8884 40 30.4617.5206 1.9210.8870 30 40.4643.5243 1.9074.8857 20 50.4669.5280 1.8940.8843 10 28 00.4695.5317 1.8807.8829 00 62 10.4720.5354 1.8676.8816 50 20.4746.5392 1.8546.8802 40 30.4772.5430 1.8418.8788 30 40.4797.5467 1.8291.8774 20 50.4823.5505 1.8165.8760 10 29 00.4848.5543 1.8040.8746 00 61 10.4874.5581 1.7917.8732 50 20.4899.5619 1.7796.8718 40 30.4924.5658 1.7675.8704 30 40.4950.5696 1.7556.8689 20 50.4975.5735 1.7437.8675 10 30 00.5000.5774 1.7321.8660 00 60 10.5025.5812 1.7205.8646 50 20.5050.5851 1.7090.8631 40 30.5075.5890 1.6977.8616 30 40.5100.5930 1.6864.8601 20 50.5125.5969 1.6753.8587 10 31 00.5150.6009 1.6643.8572 00 59 10.5175.6048 1.6534.8557 50 20.5200.6088 1.6426.8542 40 30.5225.6128 1.6319.8526 30 40.5250.6168 1.6212.8511 20 50.5275.6208 1.6107.8496 10 32 00.5299.6249 1.6003.8480 00 58 10.5324.6289 1.5900.8465 50 20.5348.6330 1.5798.8450 40 30.5373.6371 1.5697.8434 30 40.5398.6412 1.5597.8418 20 50.5422.6453 1.5497.8403 10 33 00.5446.6494 1.5399.8387 00 57 10.5471.6536 1.5301.8371 50 20.5495.6577 1.5204.8355 40 30.5519.6619 1.5108.8339 30 40.5544.6661 1.5013.8323 20 50.5568.6703 1.4919.8307 10 34 00.5592.6745 1.4826.8290 00 56 10.5616.6787 1.4733.8274 50 20.5640.6830 1.4641.8258 40 30.5664.6873 1.4550.8241 30 40.5688.6916 1.4460.8225 20 50.5712.6959 1.4370.8208 10 35 00.5736.7002 1.4281.8192 00 55 10.5760.7046 1.4193.8175 50 20.5783.7089 1.4106.8158 40 30.5807.7133 1.4019.8141 30 40.5831.7177 1.3934.8124 20 50.5854.7221 1.3848.8107 10 36 00.5878.7265 1.3764.8090 00 54 N. Cos, N. Cot. N. Tan. N. Sin. ' [95] o0 N. Sin. N. Tan. N. ot. N. Cot36 00.5878.7265 1.3764.8090 00 54 10.5901.7310 1.3680.8073 50 20.5925.7355 1.3597.8056 40 30.5948.7400 1.3514.8039 30 40.5972.7445 1.3432.8021 20 50.5995.7490 1.3351.8004 10 37 00.6018.7536 1.3270.7986 00 53 10.6041.7581 1.3190.7969 50 20.6065.7627 1.3111.7951 40 30.6088.7673 1.3032.7934 30 40.6111.7720 1.2954.7916 20 50.6134.7766 1.2876.7898 10 38 00.6157.7813 1.2799.7880 00 52 10.6180.7860 1.2723.7862 50 20.6202.7907 1.2647.7844 40 30.6225.7954 1.2572.7826 30 40.6248.8002 1.2497.7808 20 50.6271.8050 1.2423.7790 10 39 00.6293.8098 1.2349.7771 00 51 10.6316.8146 1.2276.7753 50 20.6338.8195 1.2203.7735 40 30.6361.8243 1.2131.7716 30 40.6383.8292 1.2059.7698 20 50.6406.8342 1.1988.7679 10 40 00.6428.8391 1.1918.7660 00 50 10.6450.8441 1.1847.7642 50 20.6472.8491 1.1778.7623 40 30.6494.8541 1.1708.7604 30 40.6517.8591 1.1640.7585 20 50.6539.8642 1.1571.7566 10 41 00.6561.8693 1.1504.7547 00 49 10.6583.8744 1.1436.7528 50 20.6604.8796 1.1369.709 40 30.6626.8847 1.1303.7490 30 40.6648.8899 1.1237.7470 20 50.6670.8952 1.1171.7451 10 42 00.6691.9004 1.1106.7431 00 48 10.6713.9057 1.1041.7412 50 20.6734.9110 1.0977.7392 40 30.6756.9163 1.0913.7373 30 40.6777.9217 1.0850.7353 20 50.6799.9271 1.0786.7333 10 43 00.6820.9325 1.0724.7314 00 47 10.6841.9380 1.0661.7294 50 20.6862.9435 100599.7274 40 30.6884.9490 1.0538.7254 30 40.6905.9545 1.0477.7234 20 50.6926.9601 1.0416.7214 10 44 00.6947.9657 1.0355.7193 00 46 10.6967.9713 1.0295.7173 50 20.6988.9770 1.0235.7153 40 30.7009.9827 1.0176.7133 30 40.7030.9884 1.0117.7112 20 50.7050.9942 1.0058.7092 10 45 00.7071 1.0000 1.0000.7071 00 45 N. Cos. N. Cot. N. Tan.., Sin. 0 [96] TABLE VI FOUR-PLACE LOGARITHMS OF NUMBERS 1-2000 [ 97] N. 0 1 2 3 4 5 6 7 8 19 ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~m~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~i 0 0000 0000 3010 4771 6021 6990 7782 8451 9031 9542 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 0000 3010 4771 6021 6990 7782 8451 9031 9542 0414 3222 4914 6128 7076 7853 8513 9085 9590 0792 3424 5051 6232 7160 7924 8573 9138 9638 1139 3617 5185 6335 7243 7993 8633 9191 9685 1461 3802 5315 6435 7324 8062 8692 9243 9731 1761 3979 5441 6532 7404 8129 8751 9294 9777 2041 4150 5563 6628 7482 8195 8808 9345 9823 2304 4314 5682 6721 7559 8261 8865 9395 9868 2553 4472 5798 6812 7634 8325 8921 9445 9912 2788 4624 5911 6902 7709 8388 8976 9494 9956 0000 0043 0086 0128 0170 0212 0253 0294 0334 0374 0414 0453 0492 0531 0569 0607 0645 0682 0719 0755 0792 0828 0864 0899 0934 0969 1004 1038 1072 1106 1139 1173 1206 1239 1271 1303 1335 1367 1399 1430 1461 1492 1523 1553 1584 1614 1644 1673 1703 1732 1761 1790 1818 1847 1875 1903 1931 1959 1987 2014 2041 2068 2095 2122 2148 2175 2201 2227 2253 2279 2304 2330 2355 2380 2405 2430 2455 2480 2504 2529 2553 2577 2601 2625 2648 2672 2695 2718 2742 2765 2788 2810 2833 2856 2878 2900 2923 2945 2967 2989 3010 3032 3054 3075 3096 3118 3139 3160 3181 3201 3222 3243 3263 3284 3304 3324 3345 3365 3385 3404 3424 3444 3464 3483 3502 3522 3541 3560 3579 3598 3617 3636 3655 3674 3692 3711 3729 3747 3766 3784 3802 3820 3838 3856 3874 3892 3909 3927 3945 3962 3979 3997 4014 4031 4048 4065 4082 4099 4116 4133 4150 4166 4183 4200 4216 4232 4249 4265 4281 4298 4314 4330 4346 4362 4378 4393 4409 4425 4440 4456 4472 4487 4502 4518 4533 4548 4564 4579 4594 4609 4624 4639 4654 4669 4683 4698 4713 4728 4742 4757 4771 4786 4800 4814 4829 4843 4857 4871 4886 4900 4914 4928 4942 4955 4969 4983 4997 5011 5024 5038 5051 5065 5079 5092 5105 5119 5132 5145 5159 5172 5185 5198 5211 5224 5237 5250 5263 5276 5289 5302 5315 5328 5340 5353 5366 5378 5391 5403 5416 5428 5441 5453 5465 5478 5490 5502 5514 5527 5539 5551 5563 5575 5587 5599 5611 5623 5635 5647 5658 5670 5682 5694 5705 5717 5729 5740 5752 5763 5775 5786 5798 5809 5821 5832 5843 5855 5866 5877 5888 5900 5911 5922 5933 5944 5955 5966 5977 5988 5999 6010 6021 6031 6042 6053 6064 6075 6085 6096 6107 6117 --. I...... 6128 6232 6335 6435 6532 6628 6721 6812 6902 6138 6243 6345 6444 6542 6637 6730 6821 6911 6149 6253 6355 6454 6551 6646 6739 6830 6160 6263 6365 6464 6561 6656 6749 6839 6928 6170 6274 6375 6474 6571 6665 6758 6848 6937 6180 6284 6385 6484 6580 6675 6767 6857 6946 6191 6294 6395 6493 6590 6684 6776 6866 6955 6201 6304 6405 6503 6599 6693 6785 6875 6964 6212 6314 6415 6513 6609 6702 6794 6884 6972 6222 6325 6425 6522 6618 6712 6803 6893 N. 0 1 2 3 4 5 6 7 8 9 - m m - -_ - I - - 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 6990 6998 7007 7016 7024 7033 7042 7050 7059 7067 7076 7160 7243 7324 7404 7482 7559 7634 7709 7084 7168 7251 7332 7412 7490 7566 7642 7716 7093 7177 7259 7340 7419 7497 7574 7649 7723 7101 7185 7267 7348 7427 7505 7582 7657 7731 7110 7193 7275 7356 7435 7513 7589 7664 7738 7118 7202 7284 7364 7443 7520 7597 7672 7745 7126 7210 7292 7372 7451 7528 7604 7679 7752 7135 7218 7300 7380 7459 7536 7612 7686 7760 7143 7226 7308 7388 7466 7543 7619 7694 7767 7152 7235 7316 7396 7474 7551 7627 7701 7774 7782 7853 7924 7993 8062 8129 8195 8261 8325 8388 7789 7796 7803 7810 7818 7825 7832 7839 7846 7860 7931 8000 8069 8136 8202 8267 8331 8395 7868 7938 8007 8075 8142 8209 8274 8338 8401 7875 7945 8014 8082 8149 8215 8280 8344 8407 7882 7952 8021 8089 8156 8222 8287 8351 8414 7889 7959 8028 8096 8162 8228 8293 8357 8420 7896 7966 8035 8102 8169 8235 8299 8363 8426 7903 7973 8041 8109 8176 8241 8306 8370 8432 7910 7980 8048 8116 8182 8248 8312 8376 8439 7917 7987 8055 8122 8189 8254 8319 8382 8445 -1 8451 8513 8573 8633 8692 8751 8808 8865 8921 8976 8457 8519 8579 8639 8698 8756 8814 8871 8927 8982 8463 8525 8585 8645 8704 8762 8820 8876 8932 8987 8470 8531 8591 8651 8710 8768 8825 8882 8938 8993 8476 8537 8597 8657 8716 8774 8831 8887 8943 8998 8482 8543 8603 8663 8722 8779 8837 8893 8949 9004 8488 8549 8609 8669 8727 8785 8842 8899 8954 9009 8494 8555 8615 8675 8733 8791 8848 8904 8960 9015 8500 8561 8621 8681 8739 8797 8854 8910 8965 9020 8506 8567 8627 8686 8745 8802 8859 8915 8971 9025 9031 9036 9042 9047 9053 9058 9063 9069 9074 9079 9085 9090 9096 9101 9106 9112 9117 9122 9128 9133 9138 9143 9149 9154 9159 9165 9170 9175 9180 9186 9191 9196 9201 9206 9212 9217 9222 9227 9232 9238 9243 9248 9253 9258 9263 9269 9274 9279 9284 9289 9294 9299 9304 9309 9315 9320 9325 9330 9335 9340 9345 9350 9355 9360 9365 9370 9375 9380 9385 9390 9395 9400 9405 9410 9415 9420 9425 9430 9435 9440 9445 9450 9455 9460 9465 9469 9474 9479 9484 9489 9494 9499 9504 9509 9513 9518 9523 9528 9533 9538 9542 9590 9638 9685 9731 9777 9823 9868 9912 9956 9547 9595 9643 9689 9736 9782 9827 9872 9917 9961 9552 9600 9647 9694 9741 9786 9832 9877 9921 9557 9605 9652 9699 9745 9791 9836 9881 9926 9562 9609 9657 9703 9750 9795 9841 9886 9930 9974 9566 9614 9661 9708 9754 9800 9845 9890 9934 9978 9571 9619 9666 9713 9759 9805 9850 9894 9939 9983 0026 9576 9624 9671 9717 9763 9809 9854 9899 9943 9987 0030 9581 9628 9675 9722 9768 9814 9859 9903 9948 9991 0035 9586 9633 9680 9727 9773 9818 9863 9908 9952 9996 0039 9965 9969 00000 0004 0009 0013 10017 1 0022 N. 0 1 2 3 4 1 6 7 8 9 m~~ m m E m E m E m Em m E m E m E Eiiii i [99] N. 0 1 2 3 4 5 6 7 8 9,,~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ll 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150.N. 0000 0004 0009 0013 0017 0043 0086 0128 0170 0212 0253 0294 0334 0374 0048 0090 0133 0175 0216 0257 0298 0338 0378 0052 0095 0137 0179 0220 0261 0302 0342 0382 0056 0099 0141 0183 0224 0265 0306 0346 0386 0060 0103 0145 0187 0228 0269 0310 0350 0390 0022 0026 0065 0107 0149 0191 0233 0273 0314 0354 0394 0069 0111 0154 0195 0237 0278 0318 0358 0398 -. - 0030 0035 0039 0073 0116 0158 0199 0241 0282 0322 0362 0402 0077 0120 0162 0204 0245 0286 0326 0366 0406 0082 0124 0166 0208 0249 0290 0330 0370 0410 0414 0418 0422 0426- 0430 0434 0438 0441 0445 0449 0453 0457 0461 0465' 0469 0473 0477 0481 0484 0488 0492 0496 0500 0504, 0508 0512 0515 0519 0523 0527 0531 0535 0538 0542 0546 0550 0554 0558 0561 0565 0569 0573 0577 0580 0584 0588 0592 0596 0599 0603 0607 0611 0615 0618 0622 0626 0630 0633 0637 0641 0645 0648 0652 0656 0660 0663 0667 0671 0674 0678 0682 0686 0689 0693 0697 0700 0704 0708 0711 0715 0719 0722 0726 0730 0734 0737 0741 0745 0748 0752 0755 0759 0763 0766 0770 0774 0777 0781 0785 0788 0792 0795 0799 0803 0806 0810 0813 0817 0821 0824 0828 0831 0835 0839 0842 0846 0849 0853 0856 0860 0864 0867 0871 0874 0878 0881 0885 0888 0892 0896 0899 0903 0906 0910 0913 0917 0920 0924 0927 0931 0934 0938 0941 0945 0948 0952 0955 0959 0962 0966 0969 0973 0976 0980 0983 0986 0990 0993 0997 1000 1004 1007 1011 1014 1017 1021 1024 1028 1031 1035 1038 1041 1045 1048 1052 1055 1059 1062 1065 1069 1072 1075 1079 1082 1086 1089 1092 1096 1099 1103 1106 1109 1113 1116 1119 1123 1126 1129 1133 1136 1139 1143 1146 1149 1153 1156 1159 1163 1166 1169 1173 1176 1179 1183 1186 1189 1193 1196 1199 1202 1206 1209 1212 1216 1219 1222 1225 1229 1232 1235 1239 1242 1245 1248 1252 1255 1258 1261 1265 1268 1271 1274 1278 1281 1284 1287 1290 1294 1297 1300 1303 1307 1310 1313 1316 1319 1323 1326 1329 1332 1335 1339 1342 1345 1348 1351 1355 1358 1361 1364 1367 1370 1374 1377 1380 1383 1386 1389 1392 1396 1399 1402 1405 1408 1411 1414 1418 1421 1424 1427 1430 1433 1436 1440 1443 1446 1449 1452 1455 1458 1461 1464 1467 1471 1474 1477 1480 1483 1486 1489 1492 1495 1498 1501 1504 1508 1511 1514 1517 1520 1523 1526 1529 1532 1535 1538 1541 1544 1547 1550 1553 1556 1559 1562 1565 1569 1572 1575 1578 1581 1584 1587 1590 1593 1596 1599 1602 1605 1608 1611 1614 1617 1620 1623 1626 1629 1632 1635 1638 1641 1644 1647 1649 1652 1655 1658 1661 1664 1667 1670 1673 1676 1679 1682 1685 1688 1691 1694 1697 1700 1703 1706 1708 1711 1714 1717 1720 1723 1726 1729 1732 1735 1738 1741 1744 1746 1749 1752 1755 1758 ~~~~~~~~~~~~~~~~~_... 1761 1764 1767 1770 1772 1775 - I - 1778 1781 1784 1787 I I 0 1 2 3 4; 6 7 8 9 UI- ur U - ______I 1 [100] N. O 1 2 3 4 5 6 7 8 9 _ - I m m N 0 -1 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 1761 1764 1 767 I 1770 1772 1775 1778 1781 1 1784 1787 1790 1818 1847 1875 1903 1931 1959 1987 2014 1793 1821 1850 1878 1906 1934 1962 1989 2017 1796 1824 1853 1881 1909 1937 1965 1992 2019 1798 1827 1855 1884 1912 1940 1967 1995 2022 1801 1830 1858 1886 1915 1942 1970 1998 2025 I I 1804 1833 1861 1889 1917 1945 1973 2000 2028 1807 1836 1864 1892 1920 1948 1976 2003 2030 1810 1838 1867 1895.1923 1951 1978 2006 2033 1813 1841 1870 1898 1926 1953 1981 2009 2036 1816 1844 1872 1901 1928 1956 1984 2011 2038 -1 2041 2044 2047 2049 2052 2055 2057 2060 2063 2066 2068 2071 2074 2076 2079 2082 2084 2087 2090 2092 2095 2098 2101 2103 2106 2109 2111 2114 2117 2119 2122 2125 2127 2130 2133 2135 2138 2140 2143 2146 2148 2151 2154 2156 2159 2162 2164 2167 2170 2172 2175 2177 2180 2183 2185 2188 2191 2193 2196 2198 2201 2204 2206 2209 2212 2214 2217 2219 2222 2225 2227 2230 2232 2235 2238 2240 2243 2245 2248 2251 2253 2256 2258 2261 2263 2266 2269 2271 2274 2276 2279 2281 2284 2287 2289 2292 2294 2297 2299 2302 2304 2307 2310 2312 2315 2317 2320 2322 2325 2327 2330 2333 2335 2338 2340 2343 2345 2348 2350 2353 2355 2358 2360 2363 2365 2368 2370 2373 2375 2378 2380 2383 2385 2388 2390 2393 2395 2398 2400 2403 2405 2408 2410 2413 2415 2418 2420 2423 2425 2428 2430 2433 2435 2438 2440 2443 2445 2448 2450 2453 2455 2458 2460 2463 2465 2467 2470 2472 2475 2477 2480 2482 2485 2487 2490 2492 2494 2497 2499 2502 2504 2507 2509 2512 2514 2516 2519 2521 2524 2526 2529 2531 2533 2536 2538 2541 2543 2545 2548 2550 2553 2555 2558 2560 2562 2565 2567 2570 2572 2574 2577 2579 2582 2584 2586 2589 2591 2594 2596 2598 2601 2603 2605 2608 2610 2613 2615 2617 2620 2622 2625 2627 2629 2632 2634 2636 2639 2641 2643 2646 2648 2651 2653 2655 2658 2660 2662 2665 2667 2669 2672 2674 2676 2679 2681 2683 2686 2688 2690 2693 2695 2697 2700 2702 2704 2707 2709 2711 2714 2716 2718 2721 2723 2725 2728 2730 2732 2735 2737 2739 2742 2744 2746 2749 2751 2753 2755 2758 2760 2762 2765 2767 2769 2772 2774 2776 2778 2781 2783 2785 2788 2790 2792 2794 2797 2799 2801 2804 2806 2808 2810 2813 2815 2817 2819 2822 2824 2826 2828 2831 2833 2835 2838 2840 2842 2844 2847 2849 2851 2853 2856 2858 2860 2862 2865 2867 2869 2871 2874 2876 2878 2880 2883 2885 2887 2889 2891 2894 2896 2898 2900 2903 2905 2907 2909 2911 2914 2916 2918 2920 2923 2925 2927 2929 2931 2934 2936 2938 2940 2942 2945 2947 2949 2951 2953 2956 2958 2960 2962 2964 2967 2969 2971 2973 2975 2978 2980 2982 2984 2986 2989 2991 2993 2995 2997 2999 3002 3004 3006 3008 3010 3012 3015 3017 3019 I 3021 3023 3025 3028 3030 m 1 I I I N. 0 1 2 3 4 5 6 7 8 9 II. I....... p [l101 ] TABLE VII FOUR-PLACE LOGARITHMS OF THE TRIGONOMETRIC FUNCTIONS FOR THE DECIMALLY DIVIDED DEGREE L. Sin. o I 2 3 4 5 6 7 8 9 00.0 0.1 0.2 0.3 0.4 - 00 7.2419 7.5429 7.7190 7.8439 6.2419 2833 5641 7332 8547 5429 3211 5843 7470 8651 7190 3558 6036 7604 8753 8439 3880 6221 7734 8853 9408 4180 6398 7859 8951 *0200 4460 6568 7982 9046 *0870 4723 6732 8101 9140 *1450 4971 6890 8217 9231 *1961 5206 7043 8329 9321 *2419 5429 7190 8439 9408 89.9 89.8 89.7 89.6 89.5 0.5 7.9408 9494 9579 9661 9743 9822 9901 9977 *0053 *0127 *0200 89.4 0.6 8.0200 0272 0343 0412 0480 0548 0614 0679 0744 0807 0870 89.3 0.7 8.0870 0931 0992 1052 1111 1169 1227 1284 1340 1395 1450 89.2 0.8 8.1450 1503 1557 1609 1661 1713 1764 1814 1863 1912 1961 89.1 0.9 8.1961 2009 2056 2103 2150 2196 2241 2286 2331 2375 2419 89~.0 1~.0 8.2419 2462 2505 2547 2589 2630 2672 2712 2753 2793 2832 88.9 1.1 8.2832 2872 2911 2949 2988 3025 3063 3100 3137 3174 3210 88.8 1.2 8.3210 3246 3282 3317 3353 3388 3422 3456 3491 3524 3558 88.7 1.3 8.3558 3591 3624 3657 3689 3722 3754 3786 3817 3848 3880 88.6 1.4 8.3880 3911 3941 3972 4002 4032 4062 4091 4121 4150 4179 88.5 1.5 8.4179 4208 4237 4265 4293 4322 4349 4377 4405 4432 4459 88.4 1.6 8.4459 4486 4513 4540 4567 4593 4619 4645 4671 4697 4723 88.3 1.7 8.4723 4748 4773 4799 4824 4848 4873 4898 4922 4947 4971 88.2 1.8 8.4971 4995 5019 5043 5066 5090 5113 5136 5160 5183 5206 88.1 1.9 8.5206 5228 5251 5274 5296 5318 5340 5363 5385 5406 5428 88~.0 2~.0 8.5428 5450 5471 5493 5514 5535 5557 5578 5598 5619 5640 87.9 2.1 8.5640 5661 5681 5702 5722 5742 5762 5782 5802 5822 5842 87.8 2.2 8.5842 5862 5881 5901 5920 5939 5959 5978 5997 6016 6035 87.7 2.3 8.6035 6054 6072 6091 6110 6128 6147 6165 6183 6201 6220 87.6 2.4 8.6220 6238 6256 6274 6291 6309 6327 6344 6362 6379 6397 87.5 2.5 8.6397 6414 6431 6449 6466 6483 6500 6517 6534 6550 6567 87.4 2.6 8.6567 6584 6600 6617 6633 6650 6666 6682 6699 6715 6731 87.3 2.7 8.6731 6747 6763 6779 6795 6810 6826 6842 6858 6873 6889 87.2 2.8 8.6889 6904 6920 6935 6950 6965 6981 6996 7011 7026 7041 87.1 2.9 8.7041 7056 7071 7086 7100 7115 7130 7144 7159 7174 7188 87~.0 3~.0 8.7188 7202 7217 7231 7245 7260 7274 7288 7302 7316 7330 86.9 3.1 8.7330 7344 7358 7372 7386 7400 7413 7427 7441 7454 7468 86.8 3.2 8.7468 7482 7495 7508 7522 7535 7549 7562 7575 7588 7602 86.7 3.3 8.7602 7615 7628 7641 7654 7667 7680 7693 7705 7718 7731 86.6 3.4 8.7731 7744 7756 7769 7782 7794 7807 7819 7832 7844 7857 86.5 3.5 8.7857 7869 7881 7894 7906 7918 7930 7943 7955 7967 7979 86.4 3.6 8.7979 7991 8003 8015 8027 8039 8051 8062 8074 8086 8098 86.3 3.7 8.8098 8109 8121 8133 8144 8156 8168 8179 8191 8202 8213 86.2 3.8 8.8213 8225 8236 8248 8259 8270 8281 8293 8304 8315 8326 86.1 3.9 8.8326 8337 8348 8359 8370 8381 8392 8403 8414 8425 8436 86~.0 4~.0 8.8436 8447 8457 8468 8479 8490 8500 8511 8522 8532 8543 85.9 4.1 8.8543 8553 8564 8575 8585 8595 8606 8616 8627 8637 8647 85.8 4.2 8.8647 8658 8668 8678 8688 8699 8709 8719 8729 8739 8749 85.7 4.3 8.8749 8759 8769 8780 8790 8799 8809 8819 8829 8839 8849 85.6 4.4 8.8849 8859 8869 8878 8888 8898 8908 8917 8927 8937 8946 85.5 4.5 8.8946 8956 8966 8975 8985 8994 9004 9013 9023 9032 9042 85.4 4.6 8.9042 9051 9060 9070 9079 9089 9098 9107 9116 9126 9135 85.3 4.7 8.9135 9144 9153 9162 9172 9181 9190 9199 9208 9217 9226 85.2 4.8 8.9226 9235 9244 9253 9262 9271 9280 9289 9298 9307 9315 85.1 4.9 8.9315 9324 9333 9342 9351 9359 9368 9377 9386 9394 9403 85~.0 9 8 7 6 5 4 3 2 1 0 L. Cos. - i [104] L. Sin. 0 1 2 3 4 5 6 7 8 9 5~.0 8.9403 9412 9420 9429 9437 9446 9455 9463 9472 9480 9489 84.9 5.1 8.9489 9497 9506 9514 9523 9531 9539 9548 9556 9565 9573 84.8 5.2 8.9573 9581 9589 9598 9606 9614 9623 9631 9639 9647 9655 84.7 5.3 8.9655 9664 9672 9680 9688 9696 9704 9712 9720 9728 9736 84.6 5.4 8.9736 9744 9752 9760 9768 9776 9784 9792 9800 9808 9816 84.5 5.5 8.9816 9824 9831 9839 9847 9855 9863 9870 9878 9886 9894 84.4 5.6 8.9894 9901 9909 9917 9925 9932 9940 9948 9955 9963 9970 84.3 5.7 8.9970 9978 9986 9993 * *000100080016 *0023 *0031 *0038*0046 84.2 5.8 9.0046 0053 0061 0068 0075 0083 0090 0098 0105 0112 0120 84.1 5.9 9.0120 0127 0134 0142 0149 0156 0163 0171 0178 0185 0192 84~.0 6~.0 9.0192 0200 0207 0214 0221 0228 0235 0243 0250 0257 0264 83.9 6.1 9.0264 0271 0278 0285 0292 0299 0306 0313 0320 0327 0334 83.8 6.2 9.0334 0341 0348 0355 0362 0369 0376 0383 0390 0397 0403 83.7 6.3 9.0403 0410 0417 0424 0431 0438 0444 0451 0458 0465 0472 83.6 6.4 9.0472 0478 0485 0492 0498 0505 0512 0519 0525 0532 0539 83.5 6.5 9.0539 0545 0552 0558 0565 0572 0578 0585 0591 0598 0605 83.4 6.6 9.0605 0611 0618 0624 0631 0637 0644 0650 0657 0663 0670 83.3 6.7 9.0670 0676 0683 0689 0695 0702 0708 0715 0721 0727 0734 83.2 6.8 9.0734 0740 0746 0753 0759 0765 0772 0778 0784 0790 0797 83.1 6.9 9.0797 0803 0809 0816 0822 0828 0834 0840 0847 0853 0859 830.0 70.0 9.0859 0865 0871 0877 0884 0890 0896 0902 0908 0914 0920 82.9 7.1 9.0920 0926 0932 0938 0945 0951 0957 0963 0969 0975 0981 82.8 7.2 9.0981 0987 0993 0999 1005 1011 1017 1022 1028 1034 1040 82.7 7.3 9.1040 1046 1052 1058 1064 1070 1076 1081 1087 1093 1099 82.6 7.4 9.1099 1105 1111 1116 1122 1128 1134 1140 1145 1151 1157 82.5 7.5 9.1157 1163 1168 1174 1180 1186 1191 1197 1203 1208 1214 82.4 7.6 9.1214 1220 1226 1231 1237 1242 1248 1254 1259 1265 1271 82.3 7.7 9.1271 1276 1282 1287 1293 1299 1304 1310 1315 1321 1326 82.2 7.8 9.1326 1332 1337 1343 1348 1354 1359 1365 1370 1376 1381 82.1 7.9 9.1381 1387 1392 1398 1403 1409 1414 1419 1425 1430 1436 82~.0 8~.0 9.1436 1441 1446 1452 1457 1462 1468 1473 1478 1484 1489 81.9 8.1 9.1489 1494 1500 1505 1510 1516 1521 1526 1532 1537 1542 81.8 8.2 9.1542 1547 1553 1558 1563 1568 1574 1579 1584 1589 1594 81.7 8.3 9.1594 1600 1605 1610 1615 1620 1625 1631 1636 1641 1646 81.6 8.4 9.1646 1651 1656 1661 1666 1672 1677 1682 1687 1692 1697 81.5 8.5 9.1697 1702 1707 1712 1717 1722 1727 1732 1737 1742 1747 81.4 8.6 9.1747 1752 1757 1762 1767 1772 1777 1782 1787 1792 1797 81.3 8.7 9.1797 1802 1807 1812 1817 1822 1827 1832 1837 1842 1847 81.2 8.8 9.1847 1851 1856 1861 1866 1871 1876 1881 1886 1890 1895 81.1 8.9 9.1895 1900 1905 1910 1915 1919 1924 1929 1934 1939 1943 81~.0 9~.0 9.1943 1948 1953 1958 1962 1967 1972 1977 1981 1986 1991 80.9 9.1 9.1991 1996 2000 2005 2010 2015 2019 2024 2029 2033 2038 80.8 9.2 9.2038 2043 2047 2052 2057 2061 2066 2071 2075 2080 2085 80.7 9.3 9.2085 2089 2094 2098 2103 2108 2112 2117 2121 2126 2131 80.6 9.4 9.2131 2135 2140 2144 2149 2153 2158 2162 2167 2172 2176 80.5 9.5 9.2176 2181 2185 2190 2194 2199 2203 2208 2212 2217 2221 80.4 9.6 9.2221 2226 2230 2235 2239 2243 2248 2252 2257 2261 2266 80.3 9.7 9.2266 2270 2275 2279 2283 2288 2292 2297 2301 2305 2310 80.2 9.8 9.2310 2314 2319 2323 2327 2332 2336 2340 2345 2349 2353 80.1 9.9 9.2353 2358 2362 2367 2371 2375 2379 2384 2388 2392 2397 80~.0 9 8 7 6 5 4 3 2 1 0 L. Cos. [105 L. Sin. 0 1 2 3 4 5 6 7 8 9 -al 90~ 0~ -0 7.2419 5429 7190 8439 9408 *0200 *0870 *1450 *1961 *2419 89 1 8.2419 2832 3210 3558 3880 4179 4459 4723 4971 5206 5428 88 2 8.5428 5640 5842 6035 6220 6397 6567 6731 6889 7041 7188 87 3 8.7188 7330 7468 7602 7731 7857 7979 8098 8213 8326 8436 86 4 8.8436 8543 8647 8749 8849 8946 9042 9135 9226 9315 9403 85 5 8.9403 9489 9573 9655 9736 9816 9894 9970 *0046 *0120 *0192 84 6 9.0192 0264 0334 0403 0472 0539 0605 0670 0734 0797 0859 83 7 9.0859 0920 0981 1040 1099 1157 1214 1271 1326 1381 1436 82 8 9.1436 1489 1542 1594 1646 1697 1747 1797 1847 1895 1943 81 9 9.1943 1991 2038 2085 2131 2176 2221 2266 2310 2353 2397 80~ 100 9.2397 2439 2482 2524 2565 2606 2647 2687 2727 2767 2806 79 11 9.2806 2845 2883 2921 2959 2997 3034 3070 3107 3143 3179 78 12 9.3179 3214 3250 3284 3319 3353 3387 3421 3455 3488 3521 77 13 9.3521 3554 3586 3618 3650 3682 3713 3745 3775 3806 3837 76 14 9.3837 3867 3897 3927 3957 3986 4015 4044 4073 4102 4130 75 15 9.4130 4158 4186 4214 4242 4269 4296 4323 4350 4377 4403 74 16 9.4403 4430 4456 4482 4508 4533 4559 4584 4609 4634 4659 73 17 9.4659 4684 4709 4733 4757 4781 4805 4829 4853 4876 4900 72 18 9.4900 4923 4946 4969 4992 5015 5037 5060 5082 5104 5126 71 19 9.5126 5148 5170 5192 5213 5235 5256 5278 5299 5320 5341 70~ 200 9.5341 5361 5382 5402 5423 5443 5463 5484 5504 5523 5543 69 21 9.5543 5563 5583 5602 5621 5641 5660 5679 5698 5717 5736 68 22 9.5736 5754 5773 5792 5810 5828 5847 5865 5883 5901 5919 67 23 9.5919 5937 5954 5972 5990 6007 6024 6042 6059 6076 6093 66 24 9.6093 6110 6127 6144 6161 6177 6194 6210 6227 6243 6259 65 25 9.6259 6276 6292 6308 6324 6340 6356 6371 6387 6403 6418 64 26 9.6418 6434 6449 6465 6480 6495 6510 6526 6541 6556 6570 63 27 9.6570 '6585 6600 6615 6629 6644 6659 6673 6687 6702 6716 62 28 9.6716 6730 6744 6759 6773 6787 6801 6814 6828 6842 6856 61 29 9.6856 6869 6883 6896 6910 6923 6937 6950 6963 6977 6990 60~ 30~ 9.6990 7003 7016 7029 7042 7055 7068 7080 7093 7106 7118 59 31 9.7118 7131 7144 7156 7168 7181 7193 7205 7218 7230 7242 58 32 9.7242 7254 7266 7278 7290 7302 7314 7326 7338 7349 7361 57 33 9.7361 7373 7384 7396 7407 7419 7430 7442 7453 7464 7476 56 34 9.7476 7487 7498 7509 7520 7531 7542 7553 7564 7575 7586 55 35 9.7586 7597 7607 7618 7629 7640 7650 7661 7671 7682 7692 54 36 9.7692 7703 7713 7723 7734 7744 7754 7764 7774 7785 7795 53 37 9.7795 7805 7815 7825 7835 7844 7854 7864 7874 7884 7893 52 38 9.7893 7903 7913 7922 7932 7941 7951 7960 7970 7979 7989 51 39 9.7989 7998 8007 8017 8026 8035 8044 8053 8063 8072 8081 50~ 40~ 9.8081 8090 8099 8108 8117 8125 8134 8143 8152 8161 8169 49 41 9.8169 8178 8187 8195 8204 8213 8221 8230 8238 8247 8255 48 42 9.8255 8264 8272 8280 8289 8297 8305 8313 8322 8330 8338 47 43 9.8338 8346 8354 8362 8370 8378 8386 8394 8402 8410 8418 46 44 9.8418 8426 8433 8441 8449 8457 8464 8472 8480 8487 8495 45~ 450 9.8495 9 8 7 6 5 4 3 2 i 0 L. Cos. [106] L. Sin. 0 1 2 3 4 5 6 7 8 9 9.8495 450 450 9.8495 8502 8510 8517 8525 8532 8540 8547 8555 8562 8569 44 46 9.8569 8577 8584 8591 8598 8606 8613 8620 8627 8634 8641 43 47 9.8641 8648 8655 8662 8669 8676 8683 8690 8697 8704 8711 42 48 9.8711 8718 8724 8731 8738 8745 8751 8758 8765 8771 8778 41 49 9.8778 8784 8791 8797 8804 8810 8817 8823 8830 8836 8843 400 50~ 9.8843 8849 8855 8862 8868 8874 8880 8887 8893 8899 8905 39 51 9.8905 8911 8917 8923 8929 8935 8941 8947 8953 8959 8965 38 52 9.8965 8971 8977 8983 8989 8995 9000 9006 9012 9018 9023 37 53 9.9023 9029 9035 9041 9046 9052 9057 9063 9069 9074 9080 36 54 9.9080 9085 9091 9096 9101 9107 9112 9118 9123 9128 9134 35 55 9.9134 9139 9144 9149 9155 9160 9165 9170 9175 9181 9186 34 56 9.9186 9191 9196 9201 9206 9211 9216 9221 9226 9231 9236 33 57 9.9236 9241 9246 9251 9255 9260 9265 9270 9275 9279 9284 32 58 9.9284 9289 9294 9298 9303 9308 9312 9317 9322 9326 9331 31 59 9.9331 9335 9340 9344 9349 9353 9358 9362 9367 9371 9375 30~ 600 9.9375 9380 9384 9388 9393 9397 9401 9406 9410 9414 9418 29 61 9.9418 9422 9427 9431 9435 9439 9443 9447 9451 9455 9459 28 62 9.9459 9463. 9467 9471 9475 9479 9483 9487 9491 9495 9499 27 63 9.9499 9503 9506 9510 9514 9518 9522 9525 9529 9533 9537 26, r 9.9537 9540 9544 9548 9551 9555 9558 9562 9566 9569 9573 25 65 9.9573 9576 9580 9583 9587 9590 9594 9597 9601 9604 9607 24 66 9.9607 9611 9614 9617 9621 9624 9627 9631 9634 9637 9640 23 67 9.9640 9643 9647 9650 9653 9656 9659 9662 9666 9669 9672 22 68 9.9672 9675 9678 9681 9684 9687 9690 9693 9696 9699 9702 21 69 9.9702 9704 9707 9710 9713 9716 9719 9722 9724 9727 9730 20~ 700 9.9730 9733 9735 9738 9741 9743 9746 9749 9751 9754 9757 19 71 9.9757 9759 9762 9764 9767 9770 9772 9775 9777 9780 9782 18 72 9.9782 9785 9787 9789 9792 9794 9797 9799 9801 9804 9806 17 73 9.9806 9808 9811 9813 9815 9817 9820 9822 9824 9826 9828 16 74 9.9828 9831 9833 9835 9837 9839 9841 9843 9845 9847 9849 15 75 9.9849 9851 9853 9855 9857 9859 9861 9863 9865 9867 9869 14 76 9.9869 9871 9873 9875 9876 9878 9880 9882 9884 9885 9887 13 77 9.9887 9889 9891 9892 9894 9896 9897 9899 9901 9902 9904 12 78 9.9904 9906 9907 9909 9910 9912 9913 9915 9916 9918 9919 11 79 9.9919 9921 9922 9924 9925 9927 9928 9929 9931 9932 9934 10~ 800 9.9934 9935 9936 9937 9939 9940 9941 9943 9944 9945 9946 9 81 9.9946 9947 9949 9950 9951 9952 9953 9954 9955 9956 9958 8 82 9.9958 9959 9960 9961 9962 9963 9964 9965 9966 9967 9968 7 83 9.9968 9968 9969 9970 9971 9972 9973 9974 9975 9975 9976 6 84 9.9976 9977 9978 9978 9979 9980 9981 9981 9982 9983 9983 5 85 9.9983 9984 9985 9985 9986 9987 9987 9988 9988 9989 9989 4 86 9.9989 9990 9990 9991 9991 9992 9992 9993 9993 9994 9994 3 87 9.9994 9994 9995 9995 9996 9996 9996 9996 9997 9997 9997 2 88 9.9997 9998 9998 9998 9998 9999 9999 9999 9999 9999 9999 1 89 9.9999 9999 *0000 0000 0000 0000 0000 0000 0000 *0000 0000 900 0.0000 9 8 7 6 5 4 3 2 1 0 L. Cos. [ 107] .Tang. 0 1 2 3 4 5 6 7 8 9 0~.0 -oo 6.2419 5429 7190 8439 9408 *0200 *0870 *1450 *1961 *2419 89.9 0.1 7.2419 2833 3211 3558 3880 4180 4460 4723 4972 5206 5429 89.8 0.2 7.5429 5641 5843 6036 6221 6398 6569 6732 6890 7043 7190 89.7 0.3 7.7190 7332 7470 7604 7734 7860 7982 8101 8217 8329 8439 89.6 0.4 7.8439 8547 8651 8754 8853 8951 9046 9140 9231 9321 9409 89.5 0.5 7.9409 9495 9579 9662 9743 9823 9901 9978 *0053 *0127 *0200 89.4 0.6 8.0200 0272 043 0412 0481 0548 0614 0680 0744 0807 0870 89.3 0.7 8.0870 0932 0992 1052 1111 1170 1227 1284 1340 1395 1450 89.2 0.8 8.1450 1504 1557 1610 1662 1713 1764 1814 1864 1913 1962 89.1 0.9 8.1962 2010 2057 2104 2150 2196 2242 2287 2331 2376 2419 89~.0 1~.0 8.2419 2462 2505 2548 2590 2631 2672 2713 2754 2794 2833 88.9 1.1 8.2833 2873 2912 2950 2988 3026 3064 3101 3138 3175 3211 88.8 1.2 8.3211 3247 3283 3318 3354 3389 3423 3458 3492 3525 3559 88.7 1.3 8.3559 3592 3625 3658 3691 3723 3755 3787 3818 3850 3881 88.6 1.4 8.3881 3912 3943 3973 4003 4033 4063 4093 4122 4152 4181 88.5 1.5 8.4181 4210 4238 4267 4295 4323 4351 4379 4406 4434 4461 88.4 1.6 8.4461 4488 4515 4542 4568 4595 4621 4647 4673 4699 4725 88.3 1.7 8.4725 4750 4775 4801 4826 4851 4875 4900 4924 4949 4973 88.2 1.8 8.4973 4997 5021 5045 5068 5092 5115 5139 5162 5185 5208 88.1 1.9 8.5208 5231 5253 5276 5298 5321 5343 5365 5387 5409 5431 880.0 20.0 8.5431 5453 5474 5496 5517 5538 5559 5580 5601 5622 5643 87.9 2.1 8.5643 5664 5684 5705 5725 5745 5765 5785 5805 5825 5845 87.8 2.2 8.5845 5865 5884 5904 5923 5943 5962 5981 6000 6019 6038 87.7 2.3 8.6038 6057 6076 6095 6113 6132 6150 6169 6187 6205 6223 87.6 2.4 8.6223 6242 6260 6277 6295 6313 6331 6348 6366 6384 6401 87.5 2.5 8.6401 6418 6436 6453 6470 6487 6504 6521 6538 6555 6571 87.4 2.6 8.6571 6588 6605 6621 6638 6654 6671 6687 6703 6719 6736 87.3 2.7 8.6736 6752 6768 6784 6800 6815 6831 6847 6863 6878 6894 87.2 2.8 8.6894 6909 6925 6940 6956 6971 6986 7001 7016 7031 7046 87.1 2.9 8.7046 7061 7076 7091 7106 7121 7136 7150 7165 7179 7194 87~.0 30.0 8.7194 7208 7223 7237 7252 7266 7280 7294 7308 7323 7337 86.9 3.1 8.7337 7351 7365 7379 7392 7406 7420 7434 7448 7461 7475 86.8 3.2 8.7475 7488 7502 7515 7529 7542 7556 7569 7582 7596.7609 86.7 3.3 8.7609 7622 7635 7648 7661 7674 7687 7700 7713 7726 7739 86.6 3.4 8.7739 7751 7764 7777 7790 7802 7815 7827 7840 7852 7865 86.5 3.5 8.7865 7877 7890 7902 7914 7927 7939 7951 7963 7975 7988 86.4 3.6 8.7988 8000 8012 8024 8036 8048 8059 8071 8083 8095 8107 86.3 3.7 8.8107 8119 8130 8142 8154 8165 8177 8188 8200 8212 8223 86.2 3.8 8.8223 8234 8246 8257 8269 8280 8291 8302 8314 8325 8336 86.1 3.9 8.8336 8347 8358 8370 8381 8392 8403 8414 8425 8436 8446 86~.0 4~.0 8.8446 8457 8468 8479 8490 8501 8511 8522 8533 8543 8554 85.9 4.1 8.8554 8565 8575 8586 8596 8607 8617 8628 8638 8649 8659 85.8 4.2 8.8659 8669 8680 8690 8700 8711 8721 8731 8741 8751 8762 85.7 4.3 8.8762 8772 8782 8792 8802 8812 8822 8832 8842 8852 8862 85.6 4.4 8.8862 8872 8882 8891 8901 8911 8921 8931 8940 8950 8960 85.5 4.5 8.8960 8970 8979 8989 8998 9008 9018 9027 9037 9046 9056 85.4 4.6 8.9056 9065 9075 9084 9093 9103 9112 9122 9131 9140 9150 85.3 4.7 8.9150 9159 9168 9177 9186 9196 9205 9214 9223 9232 9241 85.2 4.8 8.9241 9250 9260 9269 9278 9287 9296 9305 9313 9322 9331 85.1 4.9 8.9331 9340 9349 9358 9367 9376 9384 9393 9402 9411 9420 85~.0 9 8 7 6 5 4 3 2 1 0 L. Cot. [108] L. Tang. 0 1 2 3 4 5 6 7 8 9 5~.0 8.9420 9428 9437 9446 9454 9463 9472 9480 9489 9497 9506 84.9 5.1 8.9506 9515 9523 9532 9540 9549 9557 9565 9574 9582 9591 84.8 5.2 8.9591 9599 9608 9616 9624 9633 9641 9649 9657 9666 9674 84.7 5.3 8.9674 9682 9690 9699 9707 9715 9723 9731 9739 9747 9756 84.6 5.4 8.9756 9764 9772 9780 9788 9796 9804 9812 9820 9828 9836 84.5 5.5 8.9836 9844 9852 9860 9867 9875 9883 9891 9899 9907 9915 84.4 5.6 8.9915 9922 9930 9938 9946 9953 9961 9969 9977 9984 9992 84.3 5.7 8.9992 *0000 *0007 *0015 *0022 *0030 *0038 *0045 *0053 *0060 *0068 84.2 5.8 9.0068 0075 0083 0090 0098 0105 0113 0120 0128 0135 0143 84.1 5.9 9.0143 0150 0157 0165 0172 0180 0187 0194 0202 0209 0216 84~.0 6~.0 9.0216 0223 0231 0238 0245 0253 0260 0267 0274 0281 0289 83.9 6.1 9.0289 0296 0303 0310 0317 0324 0331 0338 0346 0353 0360 83.8 6.2 9.0360 0367 0374 0381 0388 0395 0402 0409 0416 0423 0430 83.7 6.3 9.0430 0437 0444 0451 0457 0464 0471 0478 0485 0492 0499 83.6 6.4 9.0499 0506 0512 0519 0526 0533 0540 0546 0553 0560 0567 83.5 6.5 9.0567 0573 0580 0587 0593 0600 0607 0614 0620 0627 0633 83.4 6.6 9.0633 0640 0647 0653 0660 0667 0673 0680 0686 0693 0699 83.3 6.7 9.0699 0706 0712 0719 0725 0732 0738 0745 0751 0758 0764 83.2 6.8 9.0764 0771 0777 0784 0790 0796 0803 0809 0816 0822 0828 83.1 6.9 9 0828 0835 0841 0847 0854 0860 0866 0873 0879 0885 0891 83~.0 7~.0 9.0891 0898 0904 0910 0916 0923 0929 0935 0941 0947 0954 82.9 7.1 9.0954 0960 0966 0972 0978 0984 0991 0997 1003 1009 1015 82.8 7.2 9.1015 1021 1027 1033 1039 1045 1051 1058 1064 1070 1076 82.7 7.3 9.1076 1082 1088 1094 1100 1106 1112 1117 1123 1129 1135 82.6 7.4 9.1135 1141 1147 1153 1159 1165 1171 1177 1183 1188 1194 82.5 7.5 9.1194 1200 1206 1212 1218 1223 1229 1235 1241 1247 1252 82.4 7.6 9.1252 1258 1264 1270 1276 1281 1287 1293 1299 1304 1310 82.3 7.7 9.1310 1316 1321 1327 1333 1338 1344 1350 1355 1361 1367 82.2 7.8 9.1367 1372 1378 1384 1389 1395 1400 1406 1412 1417 1423 82.1 7.9 9.1423 1428 1434 1439 1445 1450 1456 1461 1467 1473 1478 82~.0 80.0 9.1478 1484 1489 1494 1500 1505 1511 1516 1522 1527 1533 81.9 8.1 9.1533 1538 1544 1549 1554 1560 1565 1571 1576 1581 1587 81.8 8.2 9.1587 1592 1597 1603 1608 1613 1619 1624 1629 1635 1640 81.7 8.3 9.1640 1645 1651 1656 1661 1667 1672 1677 1682 1688 1693 81.6 8.4 9.1693 1698 1703 1709 1714 1719 1724 1729 1735 1740 1745 81.5 8.5 9.1745 1750 1755 1761 1766 1771 1776 1781 1786 1791 1797 81.4 8.6 9.1797 1802 1807 1812 1817 1822 1827 1832 1837 1842 1848 81.3 8.7 9.1848 1853 1858 1863 1868 1873 1878 1883 1888 1893 1898 81.2 8.8 9.1898 1903 1908 1913 1918 1923 1928 1933 1938 1943 1948 81.1 8.9 9.1948 1953 1958 1963 1968 1973 1977 1982 1987 1992 1997 81~.0 9~.0 9.1997 2002 2007 2012 2017 2022 2026 2031 2036 2041 2046 80.9 9.1 9.2046 2051 2056 2060 2065 2070 2075 2080 2085 2089 2094 80.8 9.2 9.2094 2099 2104 2109 2113 2118 2123 2128 2132 2137 2142 80.7 9.3 9.2142 2147 2151 2156 2161 2166 2170 2175 2180 2185 2189 80.6 9.4 9.2189 2194 2199 2203 2208 2213 2217 2222 2227 2231 2236 80.5 9.5 9.2236 2241 2245 2250 2255 2259 2264 2269 2273 2278 2282 80.4 9.6 9.2282 2287 2292 2296 2301 2305 2310 2315 2319 2324 2328 80.3 9.7 9.2328 2333 2337 2342 2346 2351 2356 2360 2365 2369 2374 80.2 9.8 9.2374 2378 2383 2387 2392 2396 2401 2405 2410 2414 2419 80.1 9.9 9.2419 2423 2428 2432 2437 2441 2445 2450 2454 2459 2463 80~.0 9 8 7 6 5 4 3 2 1 0 L. Cot. [ 109 ] L.Tang. 0 1 2 3 4 5 6 7 8 9 — 03 90~ 00 -o 7.2419 5429 7190 8439 9409 *0200 *0870 *1450*1962 *2419 89 1 8.2419 2833 3211 3559 3881 4181 4461 4725 4973 5208 5431 88 2 8.5431 5643 5845 6038 6223 6401 6571 6736 6894 7046 7194 87 3 8.7194 7337 7475 7609 7739 7865 7988 8107 8223 8336 8446 86 4 8.8446 8554 8659 8762 8862 8960 9056 9150 9241 9331 9420 85 5 8.9420 9506 9591 9674 9756 9836 9915 9992 *0068 *0143 *0216 84 6 9.0216 0289 0360 0430 0499 0567 0633 0699 0764 0828 0891 83 7 9.0891 0954 1015 1076 1135 1194 1252 1310 1367 1423 1478 82 8 9.1478 1533 1587 1640 1693 1745 1797 1848 1898 1948 1997 81 9 9.1997 2046 2094 2142 2189 2236 2282 2328 2374 2419 2463 80~ 10~ 9.2463 2507 2551 2594 2637 2680 2722 2764 2805 2846 2887 79 11 9.2887 2927 2967 3006 3046 3085 3123 3162 3200 3237 3275 78 12 9.3275 3312 3349 3385 3422 3458 3493 3529 3564 3599 3634 77 13 9.3634 3668 3702 3736 3770 3804 3837 3870 3903 3935 3968 76 14 9.3968 4000 4032 4064 4095 4127 4158 4189 4220 4250 4281 75 15 9.4281 4311 4341 4371 4400 4430 4459 4488 4517 4546 4575 74 16 9.4575 4603 4632 4660 4688 4716 4744 4771 4799 4826 4853 73 17 9.4853 4880 4907 4934 4961 4987 5014 5040 5066 5092 5118 72 18 9.5118 5143 5169 5195 5220 5245 5270 5295 5320 5345 5370 71 19 9.5370 5394 5419 5443 5467 5491 5516 5539 5563 5587 5611 70~ 200 9.5611 5634 5658 5681 5704 5727 5750 5773 5796 5819 5842 69 21 9.5842 5864 5887 5909 5932 5954 5976 5998 6020 6042 6064 68 22 9.6064 6086 6108 6129 6151 6172 6194 6215 6236 6257 6279 67 23 9.6279 6300 6321 6341 6362 6383 6404 6424 6445 6465 6486 66 24 9.6486 6506 6527 6547 6567 6587 6607 6627 6647 6667 6687 65 25 9.6687 6706 6726 6746 6765 6785 6804 6824 6843 6863 6882 64 26 9.6882 6901 6920 6939 6958 6977 6996 7015 7034 7053 7072 63 27 9.7072 7090 7109 7128 7146 7165 7183 7202 7220 7238 7257 62 28 9.7257 7275 7293 7311 7330 7348 7366 7384 7402 7420 7438 61 29 9.7438 7455 7473 7491 7509 7526 7544 7562 7579 7597 7614 60~ 300 9.7614 7632 7649 7667 7684 7701 7719 7736 7753 7771 7788 59 31 9.7788 7805 7822 7839 7856 7873 7890 7907 7924 7941 7958 58 32 9.7958 7975 7992 8008 8025 8042 8059 8075 8092 8109 8125 57 33 9.8125 8142 8158 8175 8191 8208 8224 8241 8257 8274 8290 56 34 9.8290 8306 8323 8339 8355 8371 8388 8404 8420 8436 8452 55 35 9.8452 8468 8484 8501 8517 8533 8549 8565 8581 8597 8613 54 36 9.8613 8629 8644 8660 8676 8692 8708 8724 8740 8755 8771 53 37 9.8771 8787 8803 8818 8834 8850 8865 8881 8897 8912 8928 52 38 9.8928 8944 8959 8975 8990 9006 9022 9037 9053 9068 9084 51 39 9.9084 9099 9115 9130 9146 9161 9176 9192 9207 9223 9238 50~ 400 9.9238 9254 9269 9284 9300 9315 9330 9346 9361 9376 9392 49 41 9.9392 9407 9422 9438 9453 9468 9483 9499 9514 9529 9544 48 42 9.9544 9560 9575 9590 9605 9621 9636 9651 9666 9681 9697 47 43 9.9697 9712 9727 9742 9757 9772 9788 9803 9818 9833 9848 46 44 9.9848 9864 9879 9894 9909 9924 9939 9955 9970 9985 *0000 450 450 0.0000 9 8 7 6 5 4 3 2 1 0 L. Cot. [110] .Tang. 0 1 2 3 4 5 6 7 8 9 0.0000 45~ 450 0.0000 0015 0030 0045 0061 0076 0091 0106 0121 0136 0152 44 46 0152 0167 0182 0197 0212 0228 0243 0258 0273 0288 0303 43 47 0303 0319 0334 0349 0364 0379 0395 0410 0425 0440 0456 42 48 0456 0471 0486 0501 0517 0532 0547 0562 0578 0593 0608 41 49 0608 0624 0639 0654 0670 0685 0700 0716 0731 0746 0762 40~ 500 0.0762 0777 0793 0808 0824 0839 0854 0870 0885 0901 0916 39 51 0916 0932 0947 0963 0978 0994 1010 1025 1041 1056 1072 38 52 1072 1088 1103 1119 1135 1150 1166 1182 1197 1213 1229 37 53 1229 1245 1260 1276 1292 1308 1324 1340 1356 1371 1387 36 54 1387 1403 1419 1435 1451 1467 1483 1499 1516 1532 1548 35 55 1548 1564 1580 1596 1612 1629 1645 1661 1677 1694 1710 34 56 1710 1726 1743 1759 1776 1792 1809 1825 1842 1858 1875 33 57 1875 1891 1908 1925 1941 1958 1975 1992 2008 2025 2042 32 58 2042 2059 2076 2093 2110 2127 2144 2161 2178 2195 2212 31 59 2212 2229 2247 2264 2281 2299 2316 2333 2351 2368 2386 30~ 60~ 0.2386 2403 2421 2438 2456 2474 2491 2509 2527 2545 2562 29 61 2562 2580 2598 2616 2634 2652 2670 2689 2707 2725 2743 28 62 2743 2762 2780 2798 2817 2835 2854 2872 2891 2910 2928 27 63 2928 2947 2966 2985 3004 3023 3042 3061 3080 3099 3118 26 64 3118 3137 3157 3176 3196 3215 3235 3254 3274 3294 3313 25 65 3313 3333 3353 3373 3393 3413 3433 3453 3473 3494 3514 24 66 3514 3535 3555 3576 3596 3617 3638 3659 3679 3700 3721 23 67 3721 3743 3764 3785 3806 3828 3849 3871 3892 3914 3936 22 68 3936 3958 3980 4002 4024 4046 4068 4091 4113 4136 4158 21 69 4158 4181 4204 4227 4250 4273 4296 4319 4342 4366 4389 20~ 700 0.4389 4413 4437 4461 4484 4509 4533 4557 4581 4606 4630 19 71 4630 4655 4680 4705 4730 4755 4780 4805 4831 4857 4882 18 72 4882 4908 4934 4960 4986 5013 5039 5066 5093 5120 5147 17 73 5147 5174 5201 5229 5256 5284 5312 5340 5368 5397 5425 16 74 5425 5454 5483 5512 5541 5570 5600 5629 5659 5689 5719 15 75 5719 5750 5780 5811 5842 5873 5905 5936 5968 6000 6032 14 76 6032 6065 6097 6130 6163 6196 6230 6264 6298 6332 6366 13 77 6366 6401 6436 6471 6507 6542 6578 6615 6651 6688 6725 12 78 6725 6763 6800 6838 6877 6915 6954 6994 7033 7073 7113 11 79 7113 7154 7195 7236 7278 7320 7363 7406 7449 7493 7537 10O 800 0.7537 7581 7626 7672 7718 7764 7811 7858 7906 7954 8003 9 81 8003 8052 8102 8152 8203 8255 8307 8360 8413 8467 8522 8 82 8522 8577 8633 8690 8748 8806 8865 8924 8985 9046 9109 7 83 9109 9172 9236 9301 9367 9433 9501 9570 9640 9711 9784 6 84 0.9784 9857 9932 *0008 *0085 *0164 *0244 *0326 *0409 *0494 *0580 5 85 1.0580 0669 0759 0850 0944 1040 1138 1238 1341 1446 1554 4 86 1554 1664 1777 1893 2012 2135 2261 2391 2525 2663 2806 3 87 2806 2954 3106 3264 3429 3599 3777 3962 4155 4357 4569 2 88 4569 4792 5027 5275 5539 5819 6119 6441 6789 7167 7581 1 89 1.7581 8038 8550 9130 9800 *0591 *1561 *2810 *4571 *7581 00 900 oo 9 8 7 6 5 4 3 2 1 0 L. Cot. [111] L. Tang. 0 1 2 3 4 5 6 7 8 9 80~.0 0.7537 7541 7546 7550 7555 7559 7563 7568 7572 7577 7581 9.9 80.1 7581 7586 7590 7595 7599 7604 7608 7613 7617 7622 7626 9.8 80.2 7626 7631 7635 7640 7644 7649 7654 7658 7663 7667 7672 9.7 80.3 7672 7676 7681 7685 7690 7695 7699 7704 7708 7713 7718 9.6 80.4 7718 7722 7727 7731 7736 7741 7745 7750 7755 7759 7764 9.5 80.5 7764 7769 7773 7778 7783 7787 7792 7797 7801 7806 7811 9.4 80.6 7811 7815 7820 7825 7830 7834 7839 7844 7849 7853 7858 9.3 80.7 7858 7863 7868 7872 7877 7882 7887 7891 7896 7901 7906 9.2 80.8 7906 7911 7915 7920 7925 7930 7935 7940 7944 7949 7954 9.1 80.9 7954 7959 7964 7969 7974 7978 7983 7988 7993 7998 8003 9~.0 81~.0 0.8003 8008 8013 8018 8023 8027 8032 8037 8042 8047 8052 8.9 81.1 8052 8057 8062 8067 8072 8077 8082 8087 8092 8097 8102 8.8 81.2 8102 8107 8112 8117 8122 8127 8132 8137 8142 8147 8152 8.7 81.3 8152 8158 8163 8168 8173 8178 8183 8188 8193 8198 8203 8.6 81.4 8203 8209 8214 8219 8224 8229 8234 8239 8245 8250 8255 8.5 81.5 8255 8260 8265 8271 8276 8281 8286 8291 8297 8302 8307 8.4 81.6 8307 8312 8318 8323 8328 8333 8339 8344 8349 8355 8360 8.3 81.7 8360 8365 8371 8376 8381 8387 8392 8397 8403 8408 8413 8.2 81.8 8413 8419 8424 8429 8435 8440 8446 8451 8456 8462 8467 8.1 81.9 8467 8473 8478 8484 8489 8495 8500 8506 8511 8516 8522 8~.0 82~.0 0.8522 8527 8533 8539 8544 8550 8555 8561 8566 8572 8577 7.9 82.1 8577 8583 8588 8594 8600 8605 8611 8616 8622 8628 8633 7.8 82.2 8633 8639 8645 8650 6856 8662 8667 8673 8679 8684 8690 7.7 82.3 8690 8696 8701 8707 8713 8719 8724 8730 8736 8742 8748 7.6 82.4 8748 8753 8759 8765 8771 8777 8782 8788 8794 8800 8806 7.5 82.5 8806 8812 8817 8823 8829 8835 8841 8847 8853 8859 8865 7.4 82.6 8865 8871 8877 8883 8888 8894 8900 8906 8912 8918 8924 7.3 82.7 8924 8930 8936 8942 8949 8955 8961 8967 8973 8979 8985 7.2 82.8 8985 8991 8997 9003 9009 9016 9022 9028 9034 9040 9046 7.1 82.9 9046 9053 9059 9065 9071 9077 9084 9090 9096 9102 9109 7~.0 83~.0 0.9109 9115 9121 9127 9134 9140 9146 9153 9159 9165 9172 6.9 83.1 9172 9178 9184 9191 9197 9204 9210 9216 9223 9229 9236 6.8 83.2 9236 9242 9249 9255 9262 9268 9275 9281 9288 9294 9301 6.7 83.3 9301 9307 9314 9320 9327 9333 9340 9347 9353 9360 9367 6.6 83.4 9367 9373 9380 9386 9393 9400 9407 9413 9420 9427 9433 6.5 83.5 9433 9440 9447 9454 9460 9467 9474 9481 9488 9494 9501 6.4 83.6 9501 9508 9515 9522 9529 9536 9543 9549 9556 9563 9570 6.3 83.7 9570 9577 9584 9591 9598 9605 9612 9619 9626 9633 9640 6.2 83.8 9640 9647 9654 9662 9669 9676 9683 9690 9697 9704 9711 6.1 83.9 9711 9719 9726. 9733 9740 9747 9755 9762 9769 9777 9784 60.0 84~.0 0.9784 9791 9798 9806 9813 9820 9828 9835 9843 9850 9857 5.9 84.1 9857 9865 9872 9880 9887 9895 9902 9910 9917 9925 9932 5.8 84.2 0.9932 9940 9947 9955 9962 9970 9978 9985 9993 *0000*0008 5.7 84.3 1.0008 0016 0023 0031 0039 0047 0054 0062 0070 0078 0085 5.6 84.4 0085 0093 0101 0109 0117 0125 0133 0140 0148 0156 0164 5.5 84.5 0164 0172 0180 0188 0196 0204 0212 0220 0228 0236 0244 5.4 84.6 0244 0253 0261 0269 0277 0285 0293 0301 0310 0318 0326 5.3 84.7 0326 0334 0343 0351 0359 0367 0376 0384 0392 0401 0409 5.2 84.8 0409 0418 0426 0435 0443 0451 0460 0468 0477 0485 0494 5.1 84.9 1.0494 0503 0511 0520 0528 0537 0546 0554 0563 0572 0580 50.0 9 8 7 6 5 4 3 2 1 0 L. Cot. [112] L. Tang. 0 1 2 3 4 5 6 7 8 9 850.0 1.0580 0589 0598 0607 0616 0624 0633 0642 0651 0660 0669 4.9 85.1 0669 0678 0687 0695 0704 0713 0722 0731 0740 0750 0759 4.8 85.2 0759 0768 0777 0786 0795 0804 0814 0823 0832 0841 0850 4.7 85.3 0850 0860 0869 0878 0888 0897 0907 0916 0925 0935 0944 4.6 85.4 0944 0954 0963 0973 0982 0992 1002 1011 1021 1030 1040 4.5 85.5.1040 1050 1060 1069 1079 1089 1099 1109 1118 1128 1138 4.4 85.6 1138 1148 1158 1168 1178 1188 1198 1208 1218 1228 1238 4.3 85.7 1238 1249 1259 1269 1279 1289 1300 1310 1320 1331 1341 4.2 85.8 1341 1351 1362 1372 1383 1393 1404 1414 1425 1435 1446 4.1 85.9 1446 1457 1467 1478 1489 1499 1510 1521 1532 1543 1554 4~.0 ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~. 860.0 86.1 86.2 86.3 86.4 86.5 86.6 86.7 86.8 86.9 87~.0 87.1 87.2 87.3 87.4 87.5 87.6 87.7 87.8 87.9 -i 1.1554 1664 1777 1893 2012 2135 2261 2391 2525 2663 1.2806 2954 3106 3264 3429 1564 1675 1788 1905 2025 2148 2274 2404 2539 2677 2821 2969 3122 3281 3445 1575 1686 1800 1917 2037 2160 2280 2418 2552 2692 2835 2984 3137 3297 3462. 1586 1698 1812 1929 2049 2173 2300 2431 2566 2706 2850 2999 3153 3313 3479 1597 1709 1823 1941 2061 2185 2313 2444 2580 2720 2864 3014 3169 3329 3496 I I I II L 1608 1720 1835 1952 2073 2198 2326 2458 2594 2734 2879 3029 3185 3346 3513 1619 1731 1846 1964 2086 2210 2339 2471 2608 2748 2894 3044 3200 3362 3530 1630 1743 1858 1976 2098 2223 2352 2485 2621 2763 2909 3060 3216 3379 3547 1642 1754 1870 1988 2110 2236 2365 2498 2635 2777 2924 3075 3232 3395 3564 1653 1766 1881 2000 2123 2249 2378 2512 2649 2792 2939 3091 3248 3412 3582 1664 1777 1893 2012 2135 2261 2391 2525 2663 2806 2954 3106 3264 3429 3599 3.9 3.8 3.7 3.6 3.5 3.4 3.3 3.2 3.1 30.0 2.9 2.8 2.7 2.6 2.5 -1 3599 3777 3962 4155 4357 3616 3795 3981 4175 4378 3634 3813 4000 4195 4399 - 3652 3831 4019 4215 4420 3669 3850 4038 4235 4441 3687 3868 4057 4255 4462 3705 3887 4077 4275 4483 3723 3905 4096 4295 4504 3740 3924 4116 4316 4526 3758 3943 4135 4336 4547 3777 3962 4155 4357 4569 2.4 2.3 2.2 2.1 20.0 880.0 1.4569 4591 4613 4635 4657 4679 4702 4724 4747 4769 4792 1.9 88.1 4792 4815 4838 4861 4885 4908 4932 4955 4979 5003 5027 1.8 88.2 5027 5051 5076 5100 5125 5149 5174 5199 5225 5250 5275 1.7 88.3 5275 5301 5327 5353 5379 5405 5432 5458 5485 5512 5539 1.6 88.4 5539 5566 5594 5621 5649 5677 5705 5733 5762 5790 5819 1.5 88.5 5819 5848 5878 5907 5937 5967 5997 6027 6057 6088 6119 1.4 88.6 6119 6150 6182 6213 6245 6277 6309 6342 6375 6408 6441 1.3 88.7 6441 6475 6508 6542 6577 6611 6646 6682 6717 6753 6789 1.2 88.8 6789 6825 6862 6899 6936 6974 7012 7050 7088 7127 7167 1.1 88.9 7167 7206 7246 7287 7328 7369 7410 7452 7495 7538 7581 1~.0 89~.0 1.7581 7624 7669 7713 7758 7804 7850 7896 7943 7990 8038 0.9 89.1 8038 8087 8136 8186 8236 8287 8338 8390 8443 8496 8550 0.8 89.2 8550 8605 8660 8716 8773 8830 8889 8948 9008 9068 9130 0.7 89.3 9130 9193 9256 9320 9386 9452 9519 9588 9657 9728 9800 0.6 89.4 1.9800 9873 9947 *0022 *0099 *0177 *0257 *0338 *0421 *5005 *0591 0.5 89.5 2.0591 0679 0769 0860 0954 1049 1147 1246 1349 1453 1561 0.4 89.6 1561 1671 1783 1899 2018 2140 2266 2396 2530 2668 2810 0.3 89.7 2810 2957 3110 3268 3431 3602 3779 3964 4157 4359 4571 0.2 89.8 4571 4794 5028 5277 5540 5820 6120 6442 6789 7167 7581 0.1 89.9 2.7581 8039 8550 9130 9800 *0592 *1561 *2810 *4571 *7581 -o 0.0 9 8 _. 7 6 _1 5 4 3 2 1 0 L. Cot. [ 113] TABLE VIII CONVERSION OF f " INTO DECIMAL PARTS OF A DEGREE 1' 0.0160 11' 0.1830 21' 0.3500 31' 0.5160 41' 0.6830 51' 0.8500 2'.033 12'.200 22'.366 32'.533 42'.700 52'.866 3'.050 13'.216 23'.383 33'.550 43'.716 53'.883 4'.066 14'.233 24'.400 34'.566 44'.733 54'.900 5'.083 15'.250 25'.416 35'.583 45'.750 55'.916 6'.100 16'.266 26'.433 36'.600 46'.766 56'.933 7'.116 17'.283 27'.450 37'.616 47'.783 57'.950 8'.133 18'.300 28'.466 38'.633 48'.800 58'.966 9'.150 19'.316 29'.483 39'.650 49'.816 59'.983 10'.166 20'.333 30'.500 40'.666 50'.833 60' 1.000 1" 0.000280 6" 0.001660 10" 0.002770 2".00056 7".00194 20".00555 3".00083 8".00222 30".00833 4".00111 9".00250 40".01111 5".00138 50".01388 TABLE IX CONVERSION OF DECIMAL PARTS OF A DEGREE INTO f " I 0.01~ 0' 36".02 1' 12".03 1' 48".04 2' 24".05 3'.06 3' 36".07 4' 12".08 4' 48".09 5' 24".10 6' 0.110.12.13.14.15.16.17.18.19.20 6' 36" 7' 12" 7' 48" 8' 24" 9' 9' 36" 10' 12" 10' 48" 11' 24" 12' -1 0.41~.42.43.44.45.46.47.48.49.50 24' 36" 25' 12" 25' 48" 26' 24" 27' 27' 36" 28' 12" 28' 48" 29' 24" 30' 0.51~.52.53.54.55.56.57.58.59.60 30' 36" 31' 12" 31' 48" 32' 24" 33' 33' 36" 34' 12" 34' 48" 35' 24" 36' 1 0.610.62.63.64.65.66.67.68.69.70 36' 36" 37' 12" 37' 48" 38' 24" 39' 39' 36" 40' 12" 40' 48" 41' 24" 42' 0.710.72.73.74.75.76.77.78.79.80 42' 36" 43' 12" 43' 48" 44' 24" 45' 45' 36" 46' 12" 46' 48" 47' 24" 48' 0.210.22.23.24.25.26.27.28.29.30 12' 36" 13' 12" 13' 48" 14' 24" 15' 15' 36" 16' 12" 16' 48" 17' 24" 18' 0.310.32.33.34.35.36.37.38.39.40 18' 36" 19' 12" 19' 48" 20' 24" 21' 21' 36" 22' 12" 22' 48" 23' 24" 24' 0.81~.82.83.84.85.86.87.88.89.90 48' 36" 49' 12" 49' 48" 50' 24" 51' 51' 36" 52' 12" 52' 48" 53' 24" 54' 0.910.92.93.94.95.96.97.98.99 1.00 54' 36" 55' 12" 55' 48" 56' 24" 57' 57' 36" 58' 12" 58' 48" 59' 24" 60' I! 0.0010.002.003.004.005.006.007.008.009 3.6" 7.2" 10.8" 14.4" 18 " 21.6" 25.2" 28.8" 32.4" I r I1 I/I 1 L -Il- J ANSWERS Exercise 1 I. logs 9 = 2. logs 27 = 3. log64 = 4. log4 =-2. logs =-2. log3 81=-4. logo - =- 1.logog.01 =-2. loglo.001 - 3. 2. log232 = 5. log312 = -5. log48 =. log2 18 =-7. logs 16 = 4. 3. 1. 9. s4 = 4. 1024= 4. /4096=8. Exercise 2 1.2. 3. 2. 5. ( 2.4. 4. 1. 6.16. 3=4. 2=3. 5 9- 10 = 0. 1. 3.88235. 8. 2. 3.82737. 9. 3. 1.91381. 10. 4. 3.89553. 11. 5. 1.87506. 12. 6. 2.19590. 13. 7. 4.55965. 14. 28. log 200 = 2.30103. 2.49715. log 20 =1.3 ). 7. 0. 9. - 3. 11. 0. 13. -4. 15. 1. -2. 8. 0. 10. -5. 12. 3. 14. 2. =6. 1=2. 0=1. 4=5. 8-10 = 1. 7 - 10=2. Exercise 3 1.82751. 15. 1.93952. 22. 8.27135- 10. 0.52410. 16. 9.88081 - 10. 23. 4.51427. 7.82737 -10. 17. 6.09691 - 10. 24. 3.51427. 4.84510- 10. 18. 2.00109. 25. 2.51427. 5.60206. 19. 1.24622. 26. 1.51427. 1.16505-10. 20. 1.62325. 27. 0.51427. 7.35550. 21. 4.0000 - 10. log 3000 = 3.47712. log 50 = 1.69897. log 100 o = 0103. log.002 = 7.30103 - 10. log 30 = 1.47712. log -r- = 8.49715 - 10. log.3 = 9.47712 - 10. 100 3( log.0005 = 6.69897 - 10. log.2 = 9.30103 - 10. lo 29. 1.1028. 35 30. 2.8824. 36 31. 1.6302. 37 32..0887. 38 33. 8.4200 - 10. 39 34. 7.1030-10. og 10 r = 1.49715. log 20000 = 4.30103...0011. i. 1.3923.. 9.0459-10. 3. 1.0676.. 7.1030 -10. 40. 2.9847. 41. 0.1666. 42. 0.2462. 43. 5.5655 - 10. 44. 7.4213 - 10. 45. 4.4619. 46. 1.2916. 47. 9.9358 - 10. 48. 8.0012 - 10. 49. 0.3474. Exercise 4 1. 26.22. 2. 157.6. 3. 9.627. 4. 48323333.3. 5..16719. 6..00026827. 7. 3896545.45. 8..000055855. 9. 100925581.4. 10..37029. 11. 221.705. 12..01569. 13. 10.88375. 14..50742. 15. 1647.3. 16. 1008581.4. 17..78488. 18. 96988. 19..69781. 20. 25.6. 21. 541. 22. 1712. 23..14277. 24. 107.8. 25. 10.315. 26..0106725. 27..001309. 28..000010044. 29. 454.44. 30..0000022337. 31. 657.166. 32. 201.409. 33..3625. 34. 9.6968- 10. 35. 3.1443. 36. 49.25. 37..2285. 3 4 ANSWERS Exercise 5 1. 53295. 4. 8.3552. 7. 1.492. 10..96518. 13. -.34526. 2. 1383.62. 5. 514.055. 8..01141. 11. -1.8583. 14. $33945. 3. 211820. 6. 19.033913. 9. 5.3921. 12. -.059439. 15. $491.04. 17. 2=.53191. 100= 5.4165. 300 x 500 47746.67. 376 58 wr 18. 1.3774 A., 3.4435 A., 45.9134 A. 19. 33.38. 21..4171. 23. 3261. 25. 3.908. 27..0939. 31. $325.60. 20. 6.727. 22. 2034.3. 24. 1.16467. 26. 3.413. 30. $213.47. 32. $5874.75. Exercise 6 1..972. 2. 99.266. 3. 8.9254. 4..182916. 5. 1602.4 6. 2.37242. 7. 218.51. 6.6943. 7.1845. 8. 500 m. = 1640.5 ft. 7294 m. = 23931.11 ft. 300 m. = 984.26 ft. 9. 2.34667. 10. -.0447. 11. - 1.5793. 12. 24.1394. 13. 19.85. 14. 24.035 15. 189.66. 16..12246. 17. 13306.06. 18. 1029.4. 19..11069. 20. 2519.6. 21. 7061.67. 22. 65.97 66 yr. 23..5342. 24. 1.6167. 25. 1.1377. 26. 22.33. 27. 10695. 28..1705. 29. 6080000. 30. 4.245. 31. 17.49. 32. 1.272. 33..4163. 34. 12.07. 35. 5.77. 36. 2316.8. Exercise 7 1. 2.544. 2. 1.2445. 3. 2.495. 4. -.053474. 5. 1.465. 6..65959. 7. -29.78. 8. 5.9837. 9. -.46187. 10..64509. 11. -.4167. 12..29414. 13. 3. 14. 5. 15. - 2. 16. -.4 17. -3. 18. - 4. 19. 2. 20. 81. 21. 25. 22. -8. 23. 32. 24. 17.677. 11.894. 25. 5%. Exercise 8 1. sinB b tan B=, sec B c' a a' 2. sil A 4, tan A 4 secA- 3. sin A = tan A= -, sec A =, 4. sin A= T1, tanA=A85, secA= 7, 5. sinA A=2, tanA= 1-, sec A=-1, 6. sin A= 39 tan A = - secA= —, 7. sin A =4-, tan A=9 — secA= 4, 8. sinA=)-9, tanA=-0-, secA=-4, 9. III. sin B=, tan B=43, sec B=, I.. sin B = 1 tan A = 1-, sec = 11-2, V. sin B — -5 tan B= '2, sec B= -, VI. sinB-, tan B=, secB=0, VII. sin B=-, tan B=-4, sec B=-, VIII. sin B = 1, tan B = 2, secB= 16 0. (1) 5- (2) 1. (3) 1. (4) B-, 1(2). (3) 1. (4) 9. a a c cosB= -, cot B- a csc B= c. e b b cosA=-, cotA=',cscA='. cos A=, cot A =, cse A =. cosA=-, cot A= 42 Csc A=. cosA=-1, cot A=1-, csc A= —7. cos A = -, cot AA = - csc A = cos, cot= csc cos A=-, cotA =, csc A = 1 cos B= 3, cot c= 4 CSC B= _. cos A=-4 co, cot =, csc B=A- eos B = -1- 3, eot B = 1-52, CSC 9 B. = 153 cos B=|-, cotB= A8, csc AB= -. cos -B=-, cot B=-0, csc B=-6. cos _ _319 cot -3 9cs. 8 -cos B-= 9, cot B = - csc B = 1 (5) 1. (6) 1. (7) 0. (8) 1. ANSWERS 5 AB = 283.86. 22. AD = 218.4. CD = 358.7. 23..854. DB = 181.3. 24. 56.75. /Vp2 + q2 27. sin A= --,+ p+q 28. sinA 2 _mn p,2 _ -q2 29. sin A = 30. sin B =-Vmn + 2 SLn + n cos B = f + ra, mn + i 31. sin B = — M mT + n1 2 m cos B ---- mq + n 32. sec A=-61, tan B=, 33. sin B - 2j, tan B- 264 2~5, 2 34. sin A=- 2 5, tan A=2, 35. sinA=-, tan Ao=3 2 36. sin B= 4 — 2 63 cos B-44+ '2 6 37. sin A =-, tan A= 52, 38. sin B-=, tanl B-=, 41. 1.62. 42. 5, I. xv2 pq cos A = Ptq sec A =2 + tanA= 22 mn m-n2' n2 n2_ 2 ta 2 - q2 pcse 2 + q2 2 pq p2- - q2 tajn B=\ ~ 92+ sec B = q +tan + - ' V ---- + m29?' cot B = /N/mn+ 2 csc.B= - + n V/' —15 ~ —' ' /99515+ 4 a95?7ppt + na2 Vm + 22 tan B (n - 95) vn\/i sec B = (+ n ) Vn9)/n 2 /r 2 n + ^1 cot B = -- cscB =Yo - 9 m - 9 cot B= S, osin A = 6o sec Be=26-, cos B= 2-5, cotB= 2 —3, csc B-26 sec A- = 5, cos A= /5, cot A= 2, cscA=. 2 sec A=, cos A=, cot A= —2, csc A = x/5. tan B 9 — 4/2 sec B= 3(4 -2) 7 7 cot B 9 + 4 /2 csc = 3(4 +~ /2) 7 ' 7 sec A= -1-, cos A= 3, cot A= —, csc A=. sec B= -, cos B= 4, cot B= 4, csc B=. e-5 1. cos 30~. 2. sin 75~. 3. cot24036'. Exercise 9 4. tan 34~ 24'. 7. cos 1t. 5. sec68~ 35 30". 8. sin 88~42'. 6. csc5~ 44'. 9. -/3. 10. 1 x 11. -. y 12. p. Exercise 10 1. tan A =-5, sec A: -1, cos A=- 7, cot A=-, csc A= 7. 2. sin A= -, sec A= 15, cos A= -5 cot A- 5, csc A= - 3. sinA= 4 tan A= 4-, cos A= -, cot A=, csc A=-4. 4. sin A=-, tan A= -, secA= cotA —5, csc A =-5. 3 ' 2 5 5. sinA= t +V tanA co1 A = n+1 se Am2 + 1 n m2 +1 m cse A = \/m2 + 1. 6 ANSWERS 6. sinA=- tan A =, se A cos A =, cot A =2. 5 2 5 7. tanA=0, sec A =, osA= 1, ct A =, csc A =oo. 8. sinA=l, tan A =, sec A = oo, cot A =, csc A =l. 9. sinA=0, secA=l, cosA=1, cot A =, csc A=oo. 10. tan A =oo, sec A = oo, cos A = 0, cot A=, cscA=l. 11. sinA=l, tan A = o, cos A = 0, cot A=, cscA= 1. 12. tan x = 5se COS X I l- 25 p2' 12. tan xa= =_-_5P, sec x -- cos x -= /1 25p2, l - 25 p2 1x/I- 25p2 cot x = -CSC = 1- 5p 5p 13. sin A = -, sec A =, cos A =, cot A = 4, cs A = -, 14. sin A = 12 tan A = -1, sec A = -, cot A = 5, csc A - 13. 15. sin A= 15, tan A = s-A5, sec A = 1-, cos A = 1y, cot A = 85. 16. sin A=, tanA=, secA = -3 cos A =33, csc A= /3 13 3 13 2 17. tan A = -V3, sec A = /3 cosA A = \/33 csc A = 2. 18. sin A =, tan A = x/i5, cos A =, cot A = - V15, csc A = 4 x/15. 19. sinA =m/m2+1 cosA= v'm2 +1 cotA= 1 seA =Vm2T1, m2+ 1 m2+ 1 ' cscA = V/2 + 1 m 20. sinA= -, tanA=l, secA=v/2, cosA=-,2 cotA=l, cscA-=/2. 2sn 2 21. sinx =0, tanx=0, ' secx=l, cotx=00, cscx= oo. 22. sin A = tan A = 4, se A =-, cos A=, cot A =, 23. sin A = 2m2 cosA= - cotA= 2-2 secA- - 2- 2 m2+ m2 + n +2 csc A= m2 + -- 2 mn 24. sin A = 1 V\/2- /2, tanA= x/2-1, cosA A ~/2 + ~/2, sec A = V/4-2 /2, csc A = A/4 + 2 V/2. 25. tan A =oo, secA= c, cos A - 0, cotA=0, cscA =1. 26. sin 221~ = ~V2 - x/2, cos 22-~ =/2 + '/2 cot 22~~ = /2 + 1, 2 see 22~ = -/4 - 2 v/2, csc 22~ = V/4 + 2 v/2. 27. sin A = ' tan A /-, seeA=g, cot A = -x/39, cscA - — x/39. 8 5 9 28. sin A = — 2 tanA=2+Vx/, cos A=6- 2, cot A = 2-V3 esc A= 2 2 - -/3. ANSWERS 7 29. sin A =/ 2 —, tanA=, cotA =v secA=-, K 1-K2 K csc A = 1 - K2 30. sin15~ \/2, tan15~=2 -~, cos15~ 0= 2+ /3 2 2 sec 15~ 2V2 - \/3, csc 15~ = 2/2 + V3. 31 31. cos A =l - sin2A, tanA — sinA, cscA = /1 - sin2 A cot A =,/ —sin2A secA = 1 sin A /1 - sin2 A 32. sinA= / - cos2 A, tan A= /1 -cos2A cos A cosA ctAs ___ ___ cos A x/1- cos2 A secA= 1, cscA= cos A -/1 - cos2 A tanA 1 1 33. sinA= ta cosA = cotA = -/1 +tta tan A tan A tan A' sec A = V1 + tan2A, csc A= /1 + tan2 A. tan A 34. tan A= c1A, csc A = 1+ cot2A, sinA 1 cot A' / 1 + cot2 A cosA=z- cotA, sec + cot A cos A. see A,/1 + A c ot A 1 1 35. cos A = tan A = /sec2A -1, cotA= = sec A' sec2 A - 1 cscA sec A in A sec2A - 1 v/sec2 A- 1 sec A 36. sinA= 1 cosA = /csc2A-1 tanA = csc A csc A x/csc2 A -1 secA = csc A cot A = /csc2A - 1. Vcsc2 A - 1 37. cos A= 1-versA, secA = I - vers A 2 ves A - 2 ver 1vers A tan A = cot A = - 1 - vers A V 2 vers A - vers2 A sin A = /2 vers A- vers2 A, csc A = /2 vers A - vers2 A 38 x/7 46. 1 38.. 42.. c46. 3 A cosA 1 39. 4/879. 43. -V2- V2. 47 sincos 40. ~/3. 44. }\/4-2. 40. 23 446. V42. 48. 2 sin2 x + sin x = 1. 41. -8A3-9. 45. 1 -cos2 A + cos A. 49 tanx-2tan 1. 49. tan x — 2 tan x= l. 8 ANSWERS Exercise 12 13. 22. 14. 1 3(b + c). 15. 2 + /2. 16. 1 —2/3. 17. -1-/V2. 18. -63. 20. (/2-1 ). 21. A 3. 22. -V6. 23. 5. 35. 86.6. 36. 150; 259.8. 38. 961.3+. 39. 165. Exercise 13' 1. 60~. 4. 60~. 2. 60~. 5. 00. 3. 30. 6. 45~. 22. 270 13/ 12". 23. 15~. 24. 10~. 25. 60~. 7. 45~. 8. 45~. 9. 30~. 26. 22}~. 27 90~ n + 1 10. 60~. 11. 45~. 12. 300, 900. 28. 29. 30. 13. 600. 14. 30~. 15. 45~. 18~. 45~. 38~ 50'. 16. 30~. 17. 45~. 18. 45~. 33. 34. 35. 19. 600 20. 90~, 21. 0~. 30~. 60~. 300. Exercise 14 1. 9.64647 - 10. 2. 9.98997 - 10. 3. 9.86603 - 10. 4. 9.38699 - 0. 5. 0.15908. 6. 9.43707-10. 7. 8.73767 - 10. 8. 9.86126 - 10. 9. 8.95017 - 10. 10. 9.97991 - 10. 11, 0.11532. 12. 9.99194-10. 13. 1.24820. 14. 8.91931 - 10. 15. 9.84324-10. 16. 9.74610 - 10. 19. 6.1493. 20. 14.991. 21. 9.4214- 10. 22. 9.8297-10. 23. 0.1759. 24. 0.7033. 25. 9.6622-10. 26. 9.9523 — 10. 27. 0.3076. 28. 0.6489. 29. 9.8832 - 10. 30. 0.2522. 31. 0.6413. 32. 15.24. Exercise 15 1. 23~15. 2. 280 40/. 3. 35~ 43. 4. 40~ 23 5. 660 15' 24". 6. 70~ 16' 21/. 7. 70~0' 26". 8. 850 5' 15'1. 9. 65~ 10 20. 10. 5~20129". 11. 4~ 047u. 12. 85~ 591 13. 13. 26.5~. 14. 50.2~. 15. 28.7~. 16. 18.50. 17. 56.26~. 18. 70.14~. 19. 64.430. 20. 46.11~. 21. 61.07~. 22. 0.541~. 23. 88.465~. 24. 65.67~. 25. 78.14~. 26. 14.470. Exercise 16 1. 8.21421 - 10. 2. 8.34812 - 10. 3. 8.49128 - 10. 4. 1.72220. 5. 1.64078. 6. 8.18538-10. 7. 8.28456 - 10. 8. 8.47866 - 10. 9. 0~ 26/ 10". 10. 88~53' 6". 11. 0~ 42' 531. 12. 89~ 32' 271. 13. 89~ 57'. 14. 0~ 4t 3111. 15. 0~ 2' 39t. 16. 890 451 6. 17. 42~ 5 26". 18. 82~ 521 1. 19. 83~ 24t 25". 20. 0~ 171 7.3". 21. 0~ 17'7.11. 22. 89~ 54 15". 23. 8.245. 24..1504. 25. 1.6687. 26. 8.3353 - 10. 27. 8.1238-10. 28 8.1070 - 10. 29. 8.2701 - 10. 30. 1.6657. 31. 1.8744. 32. 8.3446 - 10. 33. 7.9686 - 10. 34. 89.266~. 35, 1.036~. 36. 89.2160. 37..6340. 38. 89.553~. 39,.507~. 40. 4.6620. 41. 84.35~. 42. 8.3638 - 10. 43. 1.6362. 44. 89.2660. 45..613~. 46. 89.2850. 47..6240. 48. 1.6375. 49. 2.792. 50. 112.82. 51..7348. 52..026694. ANSWERS 9 1. SineA= -T. Cosecant A = -1-. b 2. - 2-. 5. sin 49~ > cos 49~. 6. A<45~. 7. A>60~. 25. cotA= 3v, csc. Exercise 17 Cosine A = 1.. Cotangent A = 18o = 30. c = 34. 8. cot 37~ > tan 37~. 22. 19. x = 45~. 23. 20. x = 60~. 2 21. x = 45~. A =. 26..P 27..3056. 28. r Secant A = -,. -f~, 1. 4V3-_I2-1. 11- 3V3 2 300. 29. 270.12 Exercise 18 4. B =62~. 7. B= 61~43'. 10. B = 510 43' 36". a = 6.3804. a = 11.448. a = 2.2478. c = 13.591. b = 21.276. b = 2.849. 5. B= 12. 8. A = 35~ 17'. 11. A = 170 43'18". a = 26.15. a = 648.67. b =70.985. b = 5.5585. b = 916.7. c = 74.5217. 6. B =43~42'. 9. A =52~ 41. 12..23661. a = 50.78. a = 385.436. 13..282726. c = 70.24. c = 484.644. 14. B = 26~ 31' 20". 15. A = 2~ 43' 30". 16. B = 38~ 50' 54". b = 127.976. a = 13.85129. a =.153254. c = 286.5875. b =.674616. b =.12343. 17. B = 63~ 41' 24,. 18..96565. b = 256.406. 19. 164.93. c = 286.033. 20. 1416.13. 21. 1614.26 yd. = depth of cafion. 5521.125 yd. = distance of river. 24. B =57.4. 30. B = 68.68~. 39. 352.1. a = 11.5125. b = 41.65. 41. B 60. c = 21.37. c = 44.71. a= 7 =4.0425. 25. B = 34. 31. A = 23.73~. c = 14 /3 = 8.083. a = 2.22. a =.003824. 42. a = 62 = 8.484. b = 1.4976. c =.009504. 43. a = -5-V/3= 14.43. 26. A 51.8~. 32..3907. = 3 2886. c = b-o \'3 28.86. a.604. 33..11388. 44 b = 030 = 5774. b =.4753. ob=.4753. 34. 50.933. c = 2900Q 3 = 1154.7. 27. A = 7.5 0 27. A= 75 35. B = 1.830. 45. b = -2-_o_ 3 = 1154.8. b = 95.42. b = 95.42. a = 13.125. c = A4o 0 /3 = 2309.5. c = 96.225. c = 96.225. -=.4194. 46. a = 600V/ = 1039.25. 28 B = 62.33. 36. A = 47.840. b = 600. a = 77.43. b =.4757. b = 527.3. =.475. 47. ao 200. b=52.33. c =.7086. c = 200/2 = 282.8. 29 A-13.75~. 37. 129.15. a b = 3.7845.48 = 10 d c = 3.89583. 38. 1.081. b = 10 d/3-= 17.32 d. 10 ANSWERS 49. Same as the respective answers for numbers 6 and 7 in this exercise. 51. DB-=50. BC=-25. DC = -- V3 = 21.65x. 1. A = 35~ 33' 27". b =14.969. 2. A = 33~ 18' 3". b = 31.147. 3. A = 42~ 24' 43". b = 29.2557. 4. A = 39 48' 201. c = 7.81016. 5. A = 49 44' 5". b =.579587. 6. A=49~. a = 16.3608. 7. A = 52~ 12' 25". c =.079471. 8. A =43~ 52'. b =.184875. 9. 53~ 7! 48". 10. 21~ 531 58". 11. 42~ 24' 39". 12. c = 8.48. 13. 25~ 48' 40". 14. B = 16~ 11' 7". b = 32.702. 15. A = 83 31' 31". a = 53.666. Exercise 19 16. B = 17~ 56' 5. b = 8.6188. 17. 13~ 71 18". 18. Z = 670 22' 481,.-. 7' 12" too small. 21. A = 41.49~. b =17.755. 22. A =45.17~. a =.39855. 23. A =50.66~. c = 43.04. 24. A = 32.02~. c = 9.432. 25. A = 46.31~. a = 7.015. 26. A = 48.43~. c =.19107. 27. A = 40.67~. a = 86.64. 28. A =40.95~. b =.0839. 29. A = 52.33~. c = 2987.33. 30. A = 43.44~. 31. 50.43. 32. A = 18.96~. a = 50.91. 33. B = 7.812~. b = 117.166. 34. 57.260. 35. 26.770. 37. A=B — 45~. c= 13/2 = 18.384. 38. A =300. b= 9/3 = 15.888. 39. B =30~. a = 100V = 173.2, 40. B =30~. c=2. 41. A =60~. b =3. 42. A =45~. b = 1. 43. A = 60~. b =50. 44. A =30~. a =6. c=12. 1. Leg = 120. Vertex Z = 60~. 2. Base = 353.87. 3. Base = 9.6837. Vertex Z = 67~ 24'. 4. Leg = 50.699. Base = 79.578. Vertex Z = 103Q 24' 20"t. 5. Vertex Z = 69~ 231 12". Leg = 927.84. Base = 1056.225. 6. Leg = 8.8204. Base Z = 62~ 10t. Vertex Z = 55~ 401". 7. Base Z = 33~ 21' 30"'. Leg =.075978. Exercise 20 8. Base Z = 46~ 161 41". Vertex / = 87~ 26' 38". Leg = 6690.16. 9. r = 8.2583. R = 10.208. Perimeter = 60. Area = 247.75. 10. r = 1.5388. R = 1.618. Perimeter = 10. Area = 7.694. 11. Side = 8.282. r = 15.455. Area - 768. 12. Side = 9.112. r=17. Area = 929.24. ANSWERS 11 13. Side = 8.6524. r = 5.9546. Perimeter = 43.262. Area = 128.8. 14. Perimeter = 4.70498. Area = 1.6417. 15. h = IsinD. m =2 1 cos D. C = 180~ -2 D. 16. tan D = 2h. I = h2 ( 2 tan 1 = 2t. 21. 12.7001. 34. 22. 95.94. 23. 15.1848. 35. 26. 8.1183. 27. 48.2055. 36. 28. Base = 61.86. VertexZ = 114.8~. 37. 29. Leg = 2081.5. Vertex Z = 45.2~. 38. ] 30. Leg = 34.47. 39. Base = 59.026. Base L = 31.14~. 40. 31. Base Z = 52.86~. Leg =.61014. 32. Base / = 61.1~. Base = 124.4. 33. Base = 114.2. Vertex Z = 114.54~. 17. sin D = h 1 h cos- C = 2 1 m = 2 212 - h2. 18. C = 1800 - 2 D. h = - m tan D. I = msec D. 2 19. D = 90 - C. h = cotl C. 2 2 = m sc 1 C. 2 2 20. Base = 3.889. Base Z = 42~ 15' 34". Vertex Z = 95~ 281 52". Base =.0588. 41. Side = 9.318. Leg =.12027. Base Z = 54.275~. Leg = 26.77. Base =.8462. Base = 14.15~. r = 16.9. Area = 946.5. Perimeter = 143.166. Side = 1.0878. r = 1.6737. Side = 20.22. r =21. R =23.3. Area = 1486.34. Perimeter = 141.54. r =17.387. Area = 972. 42. 22.025. 43. 111.4. 44. Altitude = 25 /3 = 14.435. Base Z = 30~. 45. Base Z = 30~. Base = 50 /3 = 86.6. 46. - /V3= 11.547 = leg = base. 47. Base Z = 450. Vertex Z = 90~. Altitude = 6. 48. 120~. 49. 7.07. 1. 12560.57. 2. 5911.7. 9. b = 3.416. c = 4.2331. A = 36~ 111' 53t. 10. a = 2.67815. b = 5.41875. c = 6.0445. 11. a = 13.1945. b = 8.4405. A = 57~ 23' 36". 12. 42.847. Exercise 21 3. 172.756. 5 4. 545.44. 6 13..088996. 14..0287326. 15. 244.79. 16. 300.61. 17. h = 5.2496. I = 6.1403. A = 58~ 451 17". 18. 1 = 1.5086. c = 2.6811. h =.69175. i. 3122. i. 21511 7..19936. 9.5. 8. 202281.818. 19. 1 = 7.1773. c = 12.299. h = 3.7011. 20..7723. 21. 9.58675. 22. 1.5458. 23..8874. 24. R = 3.22046. c = 2.2029. r = 3.0263. 12 ANSWERS 25. Perimeter = 21.265. 26. p = 23.187. R = 3.9448. 28. 938. 29. 47577. 30. 882. 31..01618. 32. 31.47. 33. 137.33. 34. 6000000. 35..00003529. 36. a = 8.283. A = 52.44~. c = 10.45. 37. c = 77.22. a = 68.9. b = 34.84. 38. Impossible. 39..13833. 40. 149.07. 41. 4813.3. 42. 151.4. 43. 80.8. 44..2084. 45. h = 8.828. A = 22.03~. I = 23.54. 46. 1 = 1.2351. h =.7478. c = 1.9656. 47. 1 = 54.51. c = 91.06. h = 30.04. 48. c =.8598. h =.2384. A = 29~. 49. 58.75. 50..8308. 51. 36950. 52. 15.172. 53. R = 2.262. c = 1.9624. r = 2.038. 54. R = 18.34. c = 10.3332. r= 17.6. 55. R = 4.031. c = 2.7575. r 3.788. 56. 101.36. 57. 2886.8 = 10oaaI V3. 58. 180000 /V3 = 301760. 59. 298.78. 60. 4050 v3 =7014.6. 61. 3200/3 = 5542.4. 62. 800. 63. 2000000 /3=3464000. 64. 7200. 65. 2500A 3= 4330. 66. -aoo0/3 = 5773.3. 67. 4003= 692.8. 68. 80,000. Exercise 22 In this exercise, where two answers are given to an example, the first is the result obtained by use of five-place log tables, and the second answer is the result obtained by use of four-place tables. 1. 389.7 = Ht. 2. 474.788. 474.8. 3. 114.1. 4. 10~ 33 25". 10.560. 5. 491.511. 491.44. 6. Base = 76.79. Base = 76.8. Alt. = 49.6955. Alt. = 49.7. Area = 1908.5. Area = 1908.08. 7. 37~ 58'46". 37.975~. 8. Distance of ladder from house = 12.588. 12.58. Z ladder makes with house = 30~ 14' 8" = 30.22~. 9. 695.414. 695.35. 10. 17~ 31' 7". 17.52~. 11. 82.056. 82.06. 12. 287.25. 287.47. 13. 231.7. 231.68. 14. 1534.96. 1535. 16. Ht. of hill 1673.038. Ht. of hill 1673.67. Dis. of ship 6215.143. Dis. of ship 6215.7. 17. KR = 12-3 = 20.784. KA =24. KT = 6/3 = 10.392. RT = 18. FT = 18/3 = 31.176. RF = 36. 19. 23.013. 23,012. 20. 5246.25. 5246.6. 21. 43.3 = ht. of tree. 25 = width of river. 22. KR = 12. RP = 6 x/3= 10.392. RS = 6 /6 = 14.694. ST = 12/3 = 20.784. SF = 24. TF= 12. 23. 13.071. 13.053. 24. 71.264. 71.28. 25. 616.771. 616.5. 26. 45~ 0 37". 45~. 50.6375. 60.62. ANSWERS 18 27. AB= sin y. OB = cos y. BC = sin x cos y. OC = cos x cos y. 29de = 26.0.9 A = 108~ 14' 40'. 108.26~. 71~ 45' 20". 71.74~. Exercise 23 1. 2. 3. 3. 5. 4. 7. 4. 9. 3. 11. 1. 13. 4. 15. 4. 2. 2. 4. 4. 6. 1. 8. 3. 10. 3. 12. 2. 14. 2. 16. (1) Same as the signs of the functions in the second quadrant. (3) Same as the signs of the functions in the third quadrant. (5) Same as the signs of the functions in the fourth quadrant. 17. 385~. 18. 330~. 19. 460~. 20. 260~. 745~. 6900. 820~. 620~. - 335. - 390~. -260~. - 460. - 695~. - 750~. -620~. - 820~. 21. 65~. 22. 600. 23. 60~. 24. 155~. 25. 40~. 26. 53c 27. Second. 29. Second. 31. Fourth. 28. Third. 30. Third. 32. Second. 33. 8.052 (by use of five-place tables). 8.06 (by use of four-place tables). 34. 55.73. Exercise 24 1. 2. 2. oo. 3. 0. 5. 4. 4. c2 -a2 + 4ac. 6. -2 a. 7. 0. 8. 3m. 1. sin 390~ =. cos 390~ = 1 3. tan 390~ = 1- /3. sec 390 = 2 /3. 2. cos 780~ =. tan 780~ = \/3. sin 780~ = /3. cot 780~ = 1\/3. 4. sin = ~ /3. cos= 1. tan = /3. cot = 1/3. 5. sin = ~. cos = 1/3 tan = l\/3. cot = /3. 6. sin = 2. cos = 1 /2. tan = 1. cot = 1. Exercise 25 7. sin =. cos = 1 /3 tan = 1 /3. cot = x3. 8. sin = ' 3. COs = 1 tan = /3. cot = 1 /3. 9. sin = /2. cos = x/2. tan = 1. cot = 1. 10. sin x = ~ *. tan x= f 4. cot x = T:. sec x - - csc x = i. 11. sin x = ~ 1. cos x = T. cot x = - sec x =- T 1-7 csc x = ~ }1. 12. cos x = T- -1. tan x = ~ 52. sec x +- 1. cot x = + 12 CSC X = - 13. 13. sin x =-. 5 2V cos x - - tan x = 1 cot x = 2. sec x -- 2 csc x = - 5. 14. sin x = 2 1 COS X = - - m 14 ANSWERS tan x = - Vm2 -1. cot x - I n2- sec x = - m. m Vm2 -1 16. CSC x =V/2- 1 15. sin x = - 3 /~ 10 cos X= N I 10 tan x = - 3. cot x =- 1. sec x = V10. csc x = - sin x =- COS X = -6 tan x = V35. sec x = - 6. 35 35 Exercise 26 V3. 5. - V2. - tan 45~. - sin 20~. - sin 27~. - cot 25~. sec 30~. - sin 27~. cot 22~. -cos 10~ 16'. 18. sin y = - 1\5. 5 csc y = - V6. 19. sin x = - -. a cosx-. 2 cot x = -. tan x =- - sec x=2. 3 csc x = -2. 20. -3. 21. - 2t3 6. -1. 7. V/3. 8 —~. 9. -. 25. - cot 30~ 17'. 26. - sec 25~. 27. sin 8~. 28. -tan 20~. 29. - cot 30~. 32. 91. 33. 11 cos x. 35. p sin x cos x. 1. -. 2. 2. 3. -V. 4. - 17 10. -- _+5 17. 10 2 18 11. - J/3-4. 19 20 12. sin 38. 13. - tan 17~. 2 14. sin 40~. 22 23 15. -sec 5~. 2 16. tan 5~. 34. a cos x + b sin x- ctan x. p I, I, 1.I 1. L. 36 - (a + b) cos x - (a - b) sin x. 1. ~/2. 2.,/3. 3. 10. sin =- cos 29~. cos = - sin 29~. tan = cot 29~. cot = tan 29~. sec = - csc 29~. csc =- sec 29~. 11. sin =- cos 9~. cos sin 9~. tan - cot 9~. cot - tan 9~. csc - sec 9~. sec csc 9~. 12. sin sin 15~. cos - cos 15~. tan tan15~. cot - ct 15~. sec - sec 15~. csc = csc 15~. Exercise 27 -4. 4 —/3. 5. — 3. 6. 13. sin = - sin 15~. cos = cos 15~. tan =- tan 15~. cot =- cot 15~. sec = sec 15~. csc = - csc 15~. 14. sin = cos 17~. cos = - sin 17~. tan = - cot 17~. cot = - tan 17~. sec = - csc 17g. csc = sec 17~. 15. sin = cos 10~. cos = sin 10~. tan = cot 10~. cot = tan 10~. sec = csc 10~. csc = sec 10~. 0. 7. -2. 8. ~V3. 9. -~v2. 16. sin = sin 0~. cos =- cos 0~. tan = tan 0~. cot = cot 0~. sec = - sec 0~. csc = csc 0~. 17. sin = sin 36~ 43'. cos = - cos 36~ 43'. tan = - tan 36~ 43g. cot =- cot 36~ 43'. sec = - sec 36~ 43'. csc = csc 36~ 43?. 18. sin = cos 37.24~. cos = sin 37.24~. tan = cot 37.24~. cot = tan 37.24~. sec = csc 37.24~. csc = sec 37.24~. ANSWERS 15 21. -cos x. 23. — sinx. 22. - cos x. 24. tan x. 28. -acos x bsinx-ctanx. 29. -m cosA-pcot A-qcotA. 25. -secx. 26. -sec x. 30. sin2 x cos x. 31. - cos x. 1. 30~, 2. 30~, 3. 45~, 4. 30~, 13. 30~, 14. 60~, 45~, 150~. 1500, 135~, 150~, 150~, 120~, 135~, 210~, 225~, 210~, 45~, 240~, 225~, 330~. 315~. 3303. 2250, 300~, 315~. Exercise 2 5. 30~, 1500. 6. 60~, 300~, 7. 30~, 150~. 8. 45~, 225~. 315~. 15. 3' 6' 16. 3' 17. 31 9. 45~, 180~. 10. 60~, 11. 45~, 12. 45~, 0~, 150~, 210~, 0~, 120~, 240~, 0~, 150~, 210~, 0~, 150~. 27. -3 cosx. 225~. 240~. 225~. 135~, 225~, 315~. 330~, 300~. 330~. 18. 30~, 150~. Where two answers are given, the first answer is found by the five-place tables, the second answer is found by the four-place tables. 19. 66.35 mi. east. 66.34 mi. east. 27.14 mi. north. 20. 39~ 10'25". 39.18~. 21. 760.316. 760.33. 22. Distance of the spring from the house = 217.39. 217.4. Distance of the spring from the barn = 229.12. 229.16. Exercise 29 1. sin (x + y) = —. 2. sin (x - y) = 3. cos (x + y) = 1 cos (- y) = o5 3. sin (x + 450) = /2 (sin x + cos x). cos (30~-) = /3 cos x + sin x 2 sin (x- 60~) = sinx - cos xx/ 2 8. 9. /3 —2. 10. - 4 1 - tan y tan (450 - y)= 1 - tan y 1 + tan y 4. (x + y)=oo. 5. cot(x- y)= 0. 6. 2+ V. 7. 2+V'3. 6- /2 11. /2-/- V 12. sin 90~=1. 4 4 cos 90~=0.. cot (60 + y)= /3cot2y-4cot y + 3 cot2y - 1 c 0 3 cot2 y + 2 cot y. cot (30~ + y) = 3cot2y - 3 cot2 y - 3 1. sin 60~= 1 3. cos 60~ = 1 2. tan 60~ = /3. 3. sin 1200 = /3. tan 120~ =-./3 9. 3 sin x - 4 sin3 x. Exercise 30 10. 4 cos3 - 3 cos x. 1 3 tan x - tan3 x. 1 - 3 tan2 x 13. -15. 8' 14. —. 21. cos4 x + cos2 + a. 8v~rm 2 lcY 8 16 ANSWERS Exercise 31 2. sin 15~ -= /2- /3 =.258 tan 15~ = 2- /3 =.2679. cos 15~ = V2/ + /3 =.965 3. cot22~ = 2 1 = 2.4142. cos 221 = V2 + /2 =.923 sin 221 = -V2 - x/2 =.382 4. sin 45~ = cos 45~ = -V2 = tan 45~ = cot 45~= 1. sec 45~ = csc 45~ = x/2 = 1. 5. cos A=V18+6\x/5. cot!A= 3 +V 2 2 tan1 A 3 2 2 16. A = 79~ 36' 40". A = 79.726~. b =22. 13. sin (A + B)= - __ 3 15 - 1 cosin (A- B) 8 3 /V5- 1 cos (A + B) =.. cos (A-B) = 3/5 + sin 2 A = 1 /3. sin 2 B = 1 /15. cos 2 A = 2 cos 2 B = 7. 38. 6. cos =1/2 +2 a. 2 2 9. cot0= 1 -/1- a2. 2 1-a. 1 2 tan -= I v/1-a2. 9. 2 l+a 701 12. coS A =41 + cos 2 A 7071. 12. cosA = 2 4142. sinm =l -o A cotA / ~ + cos2 A 1 - cos 2 A 13. -15 14. 4 14. 3 \/5- + 25. 21 17. 20 44 40/. 2.744~. Exercise 32 14. sin (60~ + 30~) = 1. sin 60~ + sin 300 -/3 +2. sin 29.5~ cos 7.5~ sin 27~ sin 11~ 16. 2 cos 3 A sin A cos 6 A 17. sin (A + B) sin (A- B). 18. 3.44..2136. Exercise 34 1. csc 0 = - -. cot 0 =. sin I 0 = \/5. tan (180~ - 0) = - 4. sin (- ) =. 2.,. 3. 2 +/3. 4. sin 2x = -32/7 the sign depending on whether x is taken in the first or fourth quadrants. In like manner: tan 2 x =:F -- V/. 5. 6. (a) (b) (c) cos 15~ = '/2 + V3. csc 15~ = 2V/2 + 3. tan 15~ = 2 - V. 3- 4V3 10 4 -33 10 4 + 3\/3. 10 ANSWERS 17 (d) (e) (f) (g) (h) (i) (j) 7. (a) (b) (c) (d) 8. (a) __ 1 - 24 25 -/3- 48 11 25 V3 - 48 39 -1 2 - 2, - 2. sin( - C X. cos X - sin x. \ 2/ tan (x- - =- cotx. 2/ cot - — tan x. 35. -. =.3 41. 17 -4 ~ ~ ~~g.. sin (r - ) = sin 0. cos (7r - ) = - cos 0. - ) tan (7r - )= - tan 9. cot (r - 6) =- cot 0. sin (xI-)=ocos. \ 2J cos (x- = - sinx. (c) tanX - - _=- Cot x. ( 2 ) cot X - - tan x. sin (r + x) = -sin. cos (r + x) =- cos x. tan (7r + x)= tan x. cot (7r + x) = cot x. 34. -. 39. tan - 36.. 37. - 2 /3. 3 3-4 cos4x + cos8 x 38. -2b. sin =-. 128 54. -x4(35 -64 cos 2 x + 32 sin2 2 x cos 2 x + 28 cos 4 x + cos 8 x). 3. 7. Exercise 35 a = ccos B. (I) a-b = tan (A - 45") and a right triangle. I) + b (a ) (2 + 3) an isosceles triangle with the angles 30, 30 (II) a + b = (a - b)(2 + '/3) an isosceles triangle with the angles 300, 300, 120~. 9. sin B = b. a sin A = ab 1. c = 9.1226. C = 41~ 7'. b = 13.288. 2. A = 134~ 20'. b = 74.9916 c = 242.755 3. A = 57 52'. a = 1116.98. c = 1260.26. Exercise 36 4. A = 109~ 19'. a = 4899.56. b = 4106. 5. C= 69~ 57' 36"t. a =.85442. b =.81196. 6. A = 29~ 1' 2'. a = 56.541. b = 90.164. 7. A = 99~ 29 12. b = 1.0943. c =.488667. 8. B = 43~ 18' 36f. b = 1.3487. c = 1.8286. 9. C = 68~ 15/ 30'. a =.182095. b =.188745. 18 ANSWERS 10. b = 5.267 V. 16. = 7.4486. c =2.6335(/6+ /2). = 11.175. 17. C = 105~. 11. C = 75~. a = 500(3x/2 - v6). 18. = 896.55. b = 500(2 /3 - 2). = 732.1. 19. 12. 4.0954. 11.697. 13. b = 17.08. c = 15.097. 20. C = 56.73~. 14. a = 634.3. b = 632.89. 21. A = 81.32~. 15. c = 1.022. a=1.4815. 22. B = 25.57~. c = 38.52. b = 57.412. A = 79.9~. a = 13283.34. c = 13346.67. A = 80~ 46'. a = 600.4. b = 602. C=.75~. c = 7.295. b = 14.83. A = 117.67~. b =.2592. a =.2181. C = 55.87~. a = 186.25. c = 32.5. A = 101.96~. c = 4377. b = 5641.43. A = 55.69~. 23. a = 20.343. c = 28.66. B = 27.77~. 24. a = 838.83. b = 595.1. C = 56.6~. 25. b=c= a = 100. B = C = A = 60~. 26. C = 303. a = 200/3 = 346.42. b = c = 200. 27. C= 45~. b = 250(3 2 — x/6) =448.3. c = 250(2/3 - 2) = 365.7. 28. B=30~. c = 200 /2 = 282.8. a = 100( V/6 + /2) = 386.4. 29. 925.8. 30. Distance of balloon from first point = 2033 yd. Distance of balloon from second point = 2363 yd. Height of balloon = 1739 yd. Exercise 37 1. c = 26.8675. B = 39~ 45' 17". A = 72~ 14' 43". 2. a = 385.43. B = 74~ 381 19t". C = 37~ 3' 41". 3. C = 110~ 22' 10". B = 39~ 25' 30". a =.1912. 4. A = 48~ 42' 12". C = 67~ 42' 18". b =.0748566. 5. C = 34~ 6' 36". B = 22~ 36' 54". a = 4.70177. 6. a = 336.446. B = 99~ 55' 36". C = 27~ 58' 24". 7. 8.185 = c. 8. C = 109~ 36' 5". B = 38~ 5' 25". a = 14.962. 9. C = 6 49'41". b = 317.8. A = 4~ 51' 41". 10. A = 49.06~. c = 208.1. B = 79.117~. 11. a =.9418. B = 117.99~. C = 33.85~. 12. A = 32.24~. = 35.58~. b =.6566. 13. B = 141.99~. A = 25.89~. c = 3.972. 14. A = 79.82~. C = 21.56~. b = 1712.3. 15. a = 7.93. 16. B = 6.23~. C =4.97~. a = 5.906. 17. c= 102.425. A = 65.83~. B = 45.93~. 18. A =33.84~. B= 102.98~. c = 1474.67. 19. b =10.7. Where two answers are given, the first answer is obtained by using the fiveplace tables, and the second answer is obtained by the use of the four-place tables. ANSWERS 19 20. Distance = 234.34 ft. Distance = 234.32 ft. 21. 4.36 mi. 22. Resultant = 14.989. Resultant = 15.08. Z with OA = 77~ 11t 20". Z with OA = 77.23~. Z with OB = 43~ 31' 40". Z with OB = 43.49~. 23. 3.59. 24. 152.268. 152.22. 238.31. 238.22. Exercise 38 1. A = 78~ 5' 36". 78.1~. B = 58~ 23' 28"t. 58.38~. C = 43~ 30' 58". 43.52~. 2. A = 44~ 32' 4". 44.53~. B = 86~ 25'. 86.41~. C = 49~ 2' 58". 49.05~. 3. A = 26~ 19/ 54". 26.33~. B = 98~ 18t 54/'. 98.32~. C = 55~ 21' 14". 55.36~. 4. A = 45~ 11' 50". 45.19~. B = 101~ 22' 18". 101.38~. C = 33~ 25' 58". 33.43~. 5. A = 43~ 53t 14". 43.88~. B = 60~ 3' 36". 60.06~. C =76 3' 18". 76.06~. 6. A =61~ 53' 38". 61.88~. B = 72~ 46' 4". 72.78~. C = 45~ 20' 20". 45.34~. 7. A =91~ 48'. 91.80~. B = 470 36' 56". 47.61~. C = 40~ 35' 10". 40.59~. 16..53224..5323. 17..1188. 8. A =37~ 50' 40". 37.84~. B = 127~ 3'. 127.05~. C = 15~ 6' 22". 15.11~. 9. A = 40~ 38' 22". 40.64~. B = 49~ 21' 56". 49.36~. C = 89~ 59' 46". 90~. 10. A = 520 20' 30". 52.34~. B = 107~ 19' 121". 107.32~. C = 20~ 20' 26t". 20.34~. 11. A = 13~12' 8". 13.2~. B =30~ 2' 46'. 30.04~. C = 136~ 45' 6"t. 136.76~. 12. A = 46~ 19' 52"t. 46.33~. B = 31~ 17' 50". 31.3~. C = 102~ 22' 18". 102.37~. 13. A = 107~ 55' 12. 107.92~. B = 35~ 15' 34". 35.26~. C = 36~ 49' 18". 36.82~. 14. 104~ 28' 42"'. 104.48~. 15. 160 44' 6". 16.736~. 18. 14.8586. 14.86. 20. Q is 53~ 7' 48" (53.14~) north of west from P. Q is 38~ 52' 48" (38.88~) north of west from R. P is due west of R. P is 36~ 52' 12" (36.86~) east of south from Q. R is due east of P. R is 38~ 52' 48" (38.88~) south of east from Q. When R is northeast from P: Q is 8~ 7' 48" (8.14~) north of west from P. Q is 6~ 7' 12" (6.12~) south of west from R. R is 6~ 71 12" (6.12~) north of east from Q. P is southwest from R. P is 8~ 7' 48" (8.14~) south of east from Q. 21. 28~ 57' 17". 28.96~. 20 ANSWERS Exercise 39 1. One solution. 15. A = 32~ 551 57t. A' = 147~ 4 3'. 2. Two solutions. C = 131~ 33' 51" 3. One solution. C' = 17~ 25' 45". c = 1643.96. 4. No solution. ct = 661.15. 5. No solution. 16. A = 43~ 38'. B = 58~ 3'42". 6. One solution. B = 5 342". b =.32868. 7. One solution, a right A. 17. A = 900. 8. No solution. c = 25.64. 9. Two solutions. 18. B = 28~ 16' 25". C = 20~ 25' 11. 10. B = 32~ 36'33". b =.56045. C = 109~ 5' 27"t. c = 211.48. 19. A = 103~ 501 22". A' = 13 7' 8" = A. 11. B = 40 40. 15.354. a = 15.354. BI = 160 44t' B' = 16~ 44'. a' = 3.589. C = 780 2'. B = 44~ 38' 23". Co = 101~ 58t~ C' _ 1010~ 58' B' = 135~ 21/ 37t. b = 15.787. b' = 6.9753 20. A = 35.91~. A' = 144.09~. 12. B =42~ 44'23".. 11172 C -111.720. A = 33~ 1'49l. =354 C' = 3.54~. a = 92.942. c-= 219.7. 13. A 18~ 19' 43". c' = 14.6. C=139~17'59".. 21. B = 55~. c= 1.3952. B =10.26~. 14. B =70~ 47'. C 67.63~. B' = 14 35'. C'= 112.37~. C=61~54'. b 20.118. C' = 118~ 6'. b =4.372~. b = 128.455. b' = 34.2515. 28. Other side = 129 25. 129.125. 41.66. Other diagonal = 41.62. 173~ 15' 8". Larger angle of parallelogram = 173 15 82 173260. f 6~ 44'52". Smaller angle of parallelogram = 74~ {6~74415" 22. A = 25.22~. C = 49.51~. a = 135.46. 23. A =20.79~. B =132.99~. b = 136.733. 24. A =16.25~. A' = 163.75~. C= 149.45~. C' = 1.95~. c = 36.63. c' = 2.4518. 25. B = 122.81~. B' = 12.45~. C = 34.81~. C' = 145.19~. b = 441.7. b' = 113.2. 26. A 70.78~. C = 45.91~. a =10.08. 27. A = 72.16~. A' = 9.22~. B = 58.53~. B' = 121.470. a =.19685. a' =.03313. 29. 1010.58. 1010.2. ANSWERS 21 1. 106.79. 106.8. 2..30733..30726. Exercise 40 4. 14290.6. 14290. 5. 38983.64. 38983.33. 8. 1056.66. 1056.25. 9. 1283.5. 10. 42150. 42130.77. 3. 125.229. 6. 113.55. 125.225. 7..054776..0547875. 11. Area of parallelogram = cdsin A. 13. 600/3 = 1039.2. 14. 106.798. 106.8. Exercise 41 In this exercise when two answers are given to an example, the first answer is found by the use of five-place tables, and the second answer is found by four-place tables. 4. 69.372. 5. 72.268. 69.37. 72.27. 6. 8968.5 ft. above the Colorado plain. 8958 ft. above the Colorado plain. 14144.5 ft. above sea level. 14134 ft. above sea level. 7. 373.3. 11. Height = 97.083. 8. 69.98. Height = 97.08. 9. 136.9. Distance = 71.787. Distance = 71.78. 10. 1016.6. 1016.8. 12. 10.274. 6.61. 13. 16.83. 14. Other side =4343.4 Other diagonal = 58342. 58.346. 146~ 52' 47". 146.88~. 33~ 71 13". 33. 12~. 15. Height = 42.93. Height = 42.92 ft. Distance = 104.63. Distance = 104.675 ft. 16. 11.36. 18. 4.2818. 5.573. 4.283. 17..1189. 19. 1496.517. 1496.66. 20. First answer = 4.4867 mi., 4.488 mi. Second answer = 9.16 mi. 21. 996.94. 25. 220.7. 997.6. 22. 401.52. 26. 16.58. 401.54. 23. 443.54. 443.5. 24. 974.145. 973.9. 27. 6739 m. 6740 n. 28. 9~6'. 29. Difference of latitude = difference of departure = 247.5 mi. New latitude = 34~ 231 North. New longitude = 48~ 9' W. 80. 152.69 ft. 31. 114.5 ft. 152.7 ft. 32. 85.854 ft. } = distance between observers. 85.89 ft. 38.566 ft. = distance of first observer from the rock. 33. 2008 = resultant. 720 16' = angle the resultant makes with OX. 72.270 22 ANSWERS 35. 298. 39 367.89 ft. =side opposite tower. 36. 1813 ft. 367.9 ft. 37. 161.8 ft 90.032ft. 90.04 ft. and _ the othertwo sides 38. 97~ 2' 32 379.125 ft. respectively. 97.06~ and 379.1ft. 14~ 57' 28" 14.94~ respectively. 40. 48 ft. and 108 ft. respectively. 41. 40~ 0' 16" } =angle the slope makes with the embankment. 400 29.45 ft. = width of base. 42. 161.3. 43. 22~49t 46". 44. 85.27 mi. 22.83~. Exercise 42 I 1. 300 2. _ 30. 3. 1 =.01745 i 6 6 16' =.000077" 135 o~0 3. _~ ~ 21' 15i" =.000654' 135~ = = 45. 5~ 14' =.0913374 4 4 60~ - = 60. 4. 2 radians = 11 3 3 3.2 radians = 1 o7~2w ~.003 radian = 0 902 =r 7r 120~ 2 3 5. Arc 21 in. long= 210~ =77r 4 = 144. Arc 7 in. long= 6 5 6. R=28. 270~ =-3. 3 = 1080. 7. Radians = 1.118 2 5 Angle = 64~ 3' 2250 =5. 7 2520. 8. Angles = 85~; 4 5 = 1,473' 72~ = 2. 8= 960. rac 5 15 315~ 77r. 4 9. Complement of r =7r, 60~; supplement = 150~. 6 3' 6 Complement of ~r 7rr Complement of 3- =7, 30~; supplement = 2-, 120~. 36 3 Complement of 7r, 450 = 45~; supplement =, 135~. 4 4 4 Complement of 7r =, 70~; supplement =7r 160~. 9 18 9 Complement of 5r =2r, 40~; supplement = 137r 1300. 18 9 18 radian. 57 radian. 5 radian. 4 radian. 140 351 30". 83~ 20' 48". 0 10' 18.8". = 3 radian. - radian. 22.5". 25~. 25 radians;.43625 lian. ANSWERS 23 10. sin=7r=. cos=l-. s 62 2 tan = 1/3. cot = /3. sec = /3. csc = 2. sin =21. cos = 3 2 2 tan = V. cot = ~ 3. sec =2. csc = V/3. sin =cosr=l. 4 42 tan r = cot = 1. 4 4 sec = csc=2. 11. 4 4 sin=l. cot-=0. 13. 2 2 cos = 0. sec = o. 14. 2 2 tan =0o. csc-= 1. 15. 2 2 16. p =.26175. 17. p=.64565. 18. a = 10.9935. R = 154.89. 19. 22. 437320 mi. 23. 35374500 mi. sin 37 1 /2. cos =- 1/. 4 2 2 tan = cot - 1. sec = - v2. csc = V2. sin7: ~. cos = - 1 V/. sin761r =-2. COS =- -\a 6 - 2 tan = o/3. cot = 3. sec = - 3. csc =-2. sin7r =- 1 /2. cos = - 2. 4 2 2 tan = cot = - 1. sec = V2. csc = - 2. 11 radians = 68~ 45' 18". 5 (ULIU-V ~ L R =4. A = 143~ 14' 22.5". a = 12.5. A = 14~ 19' 264'. p = 8. A = 458~ 22'. 4' 35". 20. 4 69.102 ft. 21. 24. 3 2 - 4t 20". 1117 mi. - 6. 7r 2 7r 4 7r 5 7 3 2 3 ' 3 r 3 r 5 7r 7 7r 4' -', ' _ 3 2 8 I 4 7 ' 5 7 3' 3' 3 Exercise 44 5. 57r. 6' 6 7r 5 7r 6. -I -. 3' 3 7r 5 7 7 11 r 67 6' 6' 6 7r 3 r 5 r 7 7r 7r 2wr 4 r 57r 8. 4' 4 4 ' 3 3 ' 3 ' 3 4' 4' 4' 4 3' 3 ' 3 ' -T 7r 3 r 7r 5 r 9 2 ' 6' 6 10. 7r 5r 6 6 11. 0~, 7. 3 7r 7 7r 7r 37r 12. 7 4' 4 ' ' 2 7r 5 7r ~r 3 r 5 7r 7 7r 13. 6' 6'4' 4' 4' T ir 2 7 14. -,, r. 15. 2 7r 4 7r 5 7 3' 3' 3 3 16. -6' 6' 6 7r 5 r 7 7r 11 7r 0 ir 6' 66' -6 ' 3 7r 3 7r 7r 57r 18. - -w 4 ' 4 ' 3' 3 19. 0,,, 5. 3' 6' 6 20. 0, 6- r, 2 6 6 2 wr 2 7r 4 r 5 r 21. 0, 3 r, 22 r r 7r 2 7r 4 7r 5 7r 22. 2 3' w, -, 6' 2' 3' 2' 47 ' 24 1. 0 = 30~, 210~. x= 100, - 100. 2. 0 =36.5~, 216.5~. x = 200, -200. 3. 0 = 58.51~, 301.49~. x = 500, -500. 1. cos-1 1/2 = 450, ' 4 tan-' /3 = 60~, 7 3 sin-1' = 30~,! sec-1 2 = 45~, 7 4 csc- /3 = 60~, - 3 cot-l/ = 30~, - 6 cos-i = 600, % sec-1 2 = 60~, K 3 sin-l 1 x/3= 60~, 0 3 cot- 1 \/ = 60~,. 3 tan-' = 300, 7r 3 -30, 6 32. q ~2 nTr, 4 7. 2 n7r. 4 33. - 2 n-, 3 4 2 nr. 3 34. - 2 nr, 2 7r n 3 ANSWERS Exercise 45 4. x= 50~ 7. x=36.87~. y = 40~. y = 22.62~. 5. x=1000. 8. x=1000. y =2000. 0 = 72.5~. 6. x =60~. 9. x = a cos A + b sin A. y = 45~. y = b cos A -a sin A. Exercise 46 * 2. cos (cot-l) =.) 3. tan (sin-1) I= 1. 4. sec (tan-1' )== 1 1 + a2 5. sin (cot-' a) = /l + a2 6. cot (coS-l a ab2- a2 b i b2 - a2 7. tan (2 sin-1 ) = v/3. 8. sin (2 tan-1 5)= - 2. 9. cos (2 sec-l 17) = - i6 10. sin (1 cos-1 )= - /3. 11. cot (I tan-1 -5)=. 12. sin (3 sin-1 ) = 1. 13. sin (sin- - cos-1 ) = 2 -14. tan (tan-12 + cot- 3) = 7. ~ ~r q-2 n Tr, 30. 6-+2nr, 31. r 2 nr, 6 6 5 r 2 nTr. 7r 2 nr. 6 6 35. 7r ~2, 38. ~2 nw, 6' 4 r ll 2 n7r. 77r 2nr 6 4 36. + 2 2n, 39. 72 nr, 2 6 2 6 _i 2nr. 11~+ 2nw. 37. ~2 n7r, 42. x-4w7 37.!q~2 nw, 42. x=-. 6 3 ~r2 n. 6..2n. ANSWERS 25 43. 30~ = sin-1 ~ = cos-'1 /3 = tan-1 /3 = cot-1 3 60~ = sin -1 V/3 = cos-1 I = tan- /3 = cot-1 V3. 90~ = sin-l 1 = cos-1 0 = tan-l oo = cot- 0. 45~ = sin-l1 /2 = cos-1 V/2 = tan- 1 = cot-11. 0~ = sin-1 0 = cos-l 1 = tan-i 0 = cot-l o. n 180~ = sin-10 = tan-l O. n 900 cos-l 0 = cot- 0.