ALBERT R. MANN LIBRARY AT CORNELL UNIVERSITY Cornell University Library QA 533.D85 Plane trigonometry, 3 1924 002 947 293 Cornell University Library The original of tiiis book is in tine Cornell University Library. There are no known copyright restrictions in the United States on the use of the text. http://www.archive.org/details/cu31924002947293 JOHN NAPIER OF MERCHISTON From the portrait painted from life in 1616, now in the gallery of the University of Edinburgh. Reproduced for Durell's Trigonometry by pennission of the University authorities. (See page 169.) PLANE TRIGONOMETRY BY FLETCHER DURELL, Ph.D. HEAD OF THE MATHEMATICAL DEPARTMEKT THE LAWKENCEVILLE SCHOOL NEW YORK CHARLES E. MERRILL CO. Durell's Mathematical Series ^^ School Algebra 607 pages, 12mo, cloth $1.10 Plane Geometry 341 pages, 12mo, cloth 75 cents Solid Geometry 213 pages, 12mo, cloth 75 cents Plane and Solid Geometry 514 pages, 12mo, cloth $1.25 Plane Trigonometry 184 pages, 8vo, cloth $1.00 Plane Trigonometry and Tables 298 pages, Svo, cloth $1.25 Plane and Spherical Trigonometry, with Tables 361 pages, Svo, cloth ... . $1.40 Plane Trigonometry with Surveying and Tables 390 pages, Svo, cloth $1.40 Plane and Spherical Trigonometry, with Surveying and Tables 448 pages, Svo, cloth $1.50 Logarithmic and Trigonometric Tables 114 pages, Svo, cloth 75 cents Copyright, 1910, by Charles E. Merrill Co. [4] PREFACE The principal object in writing this book has been the same as that which has governed the author in writing other mathematical textrbooks ; viz., to bring out the fundamental utilities whi'ch underlie and grow out of theT)rinciples pre- sented. 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 sexa- gesimally, 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 trigono- metric 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 trigo- nometry are reduced to a system. 3 4 PREFACE The subject-matter of the text-book is enlivened and made more vital and human by a chapter on the history of trigo- nometry. 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 ad- vantages 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 reduc- tion. (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 chang- ing 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 Exer- cise are mainly due. Professor William Betz of the East Rochester High School, Rochester, N.Y., Dr. Henry A. Con- verse 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 CHAPTEK I PAGE Logarithms 7 CHAPTER II Definitions. TrigoiJometric Functions 24 CHAPTEK III Eight Triangles 52 CHAPTER IV Goniometry 73 CHAPTER V Goniometry (Continued) 93 CHAPTER VI Oblique Triangles 107 CHAPTER VII Practical Applications 131 CHAPTER Vin Circular 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 PLAJS^E 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 = 10', hence log 1000 = 3, 10 being taken as the base ; again, if 8 be taken as the base, 4 = 8% hence log4 = |; 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, logjo 1000 = 3 ; logs 4 = 1; log^ 125 = 3 ; logs ^V = - 2 ; etc. In general, by the definition of a logarithm, number = (base)"'«''"""°, or N= B' ; hence log^= I. 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 TRIGONOMETRY 1000 = 10' • 100 = 10^ hence 1000 x 100 = 10° = 100,000, the multiplication being performed by the addition of exponents. Similarly, if 384 = 102-=8««+ and 25 = 10''9'9*+, 384 may be multiplied by 25 by adding the exponents of lO^-'^* and lQi.sa7M+^ thus obtaining 10'-s8227+^ ^nd then getting from a table of loga- rithms the value of 103-^27+^ ^j^. 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, tt, i^ir, — , 1/ 2, l/3, etc. 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 use- ful exercise to teach the class the use of the slide rule in con- nection 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 Briggsi^an System, in which the base is 10. This system is used almost exclusively when logarithms are employed to facilitate numerical compu- tations. LOGARITHMS 9 2. The system termed Natural or Napierian, in which the base is 2.7182818^. This system is generally used in alge- braic processes, as in demonstrating the properties of algebraic expressions, etc. EXERCISE I 1. Give the vahie of eacli of the following : logs 9) logs 27, logi 64, log4 tVj logs h logs TT) logio tV logio .01, logio .001. 2. Also of logj 32, logj ^Vj logs j^j, log4 8, logs 16. 3. Simplify loga 4 + logg 9 + logw .1 - logs i- 4. Write out the value of each power of 2 up to 2™ (thus 2' = 2, 2' = 4, 2' = 8, etc.) in the form of a table. 5. By means of this table multiply 32 by 8, converting the multi- plication 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 h- 1024. 8. Also convert each of the following involutions into a multiplica- tion : (32)=; (64)2; (32)^ 9. Also convert each of the following root extractions into a divi- sion: -v/64; a/1024; a/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 ifiake 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 = 10^ hence log 10,000 = 4 ; lOOO = 10^ hence log 1000 = 3 ; ^100 = 10^ hence log 100 = 2 ; ^10 = 10^ hence log 10 = 1. 10 TRIGONOMETRY Hence any number between 1000 and 10,000 has a loga- rithm 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 containiag/oMr figures has 3 for a characteristic. Similarly every number between 100 and 1000, and there- fore 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 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. y y '5. Characteristic of a : Decimal Fraction. 1 = = 10°. ■•• log 1 = 0; .1 = 1 10 10-1. • ■• log .1=-1; .01 1 100 1 lO'i =io-«. .•.log.01=- -2; .001 1 1000 _ 1 10= = 10-', .-. log .001 : = - 3, etc. Hence the logarithm of any number between .1 and 1 (as of .4 for instance) will lie between — 1 and 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 char- acteristic of a decimal fraction. Thus, (1) log .0384 =2.58433, the minus sign being placed over the char- acteristic 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. 462. 6. .08267. 11. T. 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? 1? 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 w;hich 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 103-82956 log 67.54 = log ^ = log ^^^ = log 10^-^ = 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). Using five-place tables find the logarithm of each of the following numbers : 1. 7627. 2. 6720. 3. 82. 4. 7862. 5. 75. 6. 157. 7. 36278. 8. 67.222. 9. 3.3427. 1 EXERCISE 3 ind the logaritl 10. .00672. 11. .000007. 12. 400000. 13. 14.6235. 14. .00226725. 15. 87. 16. .76. 17. .000125. 18. 100.25. 19. 17.6287. 20. 42. 21. .000001. 22. .0186789. 23. 32679. 24. 3267.9. 25. 326.79. 26. 32.679. 27. 3.2679. 28. Commit to' memory the mantissa for each of the following: 2, 3, 5, rr. Then write at sight the log of each of the following, 200, 3000, 50, 100 TT, 20, .002, 30, .0005, -^, .3, .2, 10 tt, 20,000. Using four-place tables, find the logarithm of each of the fol- lowing : 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 antilogarithtn). 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- ing: 1. 1.41863. 4. 7.68416. 7. 6.69068. 2. 2.19756. S. 9.22321-10. 8. 5.74706-10. 3. 0.98349. 6. 6.42857-10. 9. 8.00400. 10. Find log of 2.34578. 15. Find antilog of 3.21678. 11. Find antilog of 2.34578. 16. Find antilog of 6.00371. 12. Find log of 1.03678. 17. Find log of 6.00371. 13. Find antilog of 1.03678. 18. Find antilog of 4.98672. 14. 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. 26. 8.0283-10. 29. 2.6575. 21. 2.7332. 24. 2.0326. 27. 7.1170-10. 30. 4.3490-10. 22. 3.2335. 25. 1.0135. 28. 6.0019-10. 31. 2.8177. 32. Find antilog of 2.3041. 35. Find antilog of 0.4975. 33. Find log of 2.3041. 36. Find antilog of 1.6924. 34. Find log of 0.4975. 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^.ay = a'"*^; and also that {cty = a^. Using these properties of exponents, it can be shown that 1. log ('mn) = log'm + lQgn. 3. logm^ =p logm. /'m\ PI — logm 2. log I — j = logm-logw. 4. logV«t = — - — For m = 10'". .-. logm = a;. n = 10". .-. log n = y. .: mn = 10"^+" or log mn = x + y = log m -f- log n. (1) Also — = — - = lO'^"*, or log — = x — y = log m — log n, (2) 14 TRIGONOMETRY Also m'' = (10'')'' = 10^. .-. log wP =px =p • log m, (3) and V^ = 10K .•.log-e/m = 5=i5^^. (4) Hence : I. To multiply numbers : Add their logai'ithms mid find the antilogarithm of the sum. This will he 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 loga- rithms. log (561.75 X .03286)= log 561.75 + log .03286 log 661.75 = 2.74954 log .03286 = 8.51667 - 10 antilog,1.26621 =18.4691, Product. The following, however, is the arrangement of work used by many practical computers. It has the advantage of show- ing all the steps in a complex logarithmic computation. (See p. 12, etc.) 561.75 log 2.74964 ■03286 log 8.51667 - 10 Answer = 18.4591 log 1.26621 Observe that "561.76 log 2.74954" reads "561.75, its log is 2.74954," etc. Ex. 2. Compute the amount of $1 at 5 per cent com- pound interest for 20 years. LOGARITHMS 15 The amount of $ 1 at 5%, for 20 years = (1.05)™ 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)^" by con- tinued multiplication, and compare the labor in such a pro- cess 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. 632.768 log 2.72653 | log 0.90884. <-/ '' Boot = 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 cologarithm 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 cologa- rithm 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 = 1.67008. Hence, colog 37.16 = 8.42992 - 10. Ex. 2. Divide 52678 by 37.16 by the use of the cologa- rithm of the divisor. 62678 log 4.72163 37.16 log 1.57008 colog 8.42992 - 10. Quotient = 1417.58 log 3.16165. 11. In the extraction of the root of a decimal number it is best to add to anrJ ^-..r.-'xfict from th" t^- - •■'-''^--^ 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 Boot = .703667 log 9.84737 - 10 12. Computations involving Negative Numbers. In com- puting, 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. Ex. Compute — — ^. 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 the value of -— . 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. 26.27x52.67. 6. (2.67)«. 3. 8.25x25675. ^ 27.8675 ^ _1768 ■ 18.678' 211.6' 8. (.5278/. 9. V156.78. Also, if you can, extract the cube root of 156.78 with- out 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., -v'.86785. '■ ' - '—"t the onnore "'■ "* ^^^ imiavp vnnt of LOGARITHMS 17 .86785. About how much longer is this process than the logarithmic work ? 11. V- 76.526. 12. V- .00021. 13. V -. 00062367 x 7.867. Pind the compound interest on : 14. $16375 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 TT 300 X 600 compute each of the following : ^=^ ; „ ; OiO bo IT 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 600 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. \16.9 20. .001467 X 96.8 X 47.37. v 16.91 21. 556.86 X .00016277 x 4.6. .0016666 25. 22. (12.67)='. ■ .00042635- 23. (3.176)^. 26. ^42.67 x .10126 x 9.2. 27. ■v'.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 survej'or ? Find the compound interest on : 30. $ 369.67 for 8 years at 6%. 31. $ 100 for 37 years at 4%. 32. $4962.75 for 16 years at 6%. 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 215 AT Ko ^y ^^^ "®® °^ logarithms. '67 X 52 log Vi Before looking up the logarithm of any number in the table, it is im- portant 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 62 log 1.71600 colog 8.28400 - 10 2 )18.79037 - 20 Answer = .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. -p o n . V2i:8 . ^.03678 , ,, ., .^, liX. 2. Compute ^„„^^ by the use oi loerarithms. .28750 21.8 log 1.33846 ^ log 0.66923 .03678 log 8.56561 - 10 ^ log 9.52187 - 10 .28756 log 9.45873 - 10 colog 0.54127 Answer = 5.39976 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 .3* = 2. LOGARITHMS 19 Taking the logarithm of each member of the equation, (. X log .3 = log 2. H^nce* .= l2g2. = _0:30103_ = _0:3P103_^_5,5 ^^^^ log .3 9.47712-10 -0.52288 --^'^ > ^»»- 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. J VnM X V.0071725 g f .59 x 2209 .92678 " ■ \ 47 X. 3481' ^CoSSf ' *■ V(.19678)^-(.072567)». (V278:2 X 2.578)' V.00231 X V76l9 267.85 X 7 X .000925 x 468.765 6.^: ^ (21.67)2 X .00096725 x V667.266 7. Using the logarithms committed to memory in Ex. 28, Exer- cise 3, compute each of the following: / 300 X 500 ^ [mm^ / 200 X 30 \ '^ ' \3.1416' V 37x ■ 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. 6'^ = 67. U. 2.8' = .1967. 10. W+5 = 2167. 12. .SS"' = .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) . .-. loga; = l-log2-l. log .3. 2 log 0.30103 1 . 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, K= -\/s{s — a){s — 6)(s— c), whens = " + ^'''^ and a = 17.6, 6=21.675, c = 26.427. 16. Calculate the value of 6 in the equation, b = Vo^--?} when a = .17623 and c= .12673. (Use 6 = V(a + c)(a - c), etc.) 17. Find the volume of a sphere whose radius is 14.7, if F= | -n-W and TT = 3.1416. 18. Given J = 8, a = 32.17, find s, if s = i af. 19. Given s = "+^ + " and a = .1732, 6 = .14326, c = .2242, find n A, if A = - ■\Js{s — a){s — b)(s — c). c 20. Given M = 14.16 and ,r = -^S find S,ilS = i-n-IP. 21. Given t = -^ and i> = 23.8, find V, when V=i irD=. 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 ^ 7 529 . IMS 112:97 ^^' V67 X 51.8 ^=- 12.97>|i6:^ }.97. 1.78' 24. / .3756 X .265 . ^6. ^(125/ -(67/, V .227 X .1678 ^ / v / „ 47.326 / 55400x8 ^'- .10021 \i23456x. 007' 28. ■^.■2167x^72i;67 ni6765:. \ 32.77 Nf 1.76364 29. ' J yiOTS (26.72)^ 1 (36.27)^ X .01267 , Solve for x : 30. 2^ = 19. 32. 19.38^ = 81672. 31. 4^-^ = 11"^+^ 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 F=f 7ri2'andT=.?^. 36. Given i = 12 and g- = 32.17, find s, if s = ^f. 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 OP LOGARITHMS 15. The logarithm of unity in any system of logarithms is zero. For, if a be the base, l = a\ .-. log„l = 0. 16. ■ The logarithm of the base in any system of loga- rithms is unity. For a = a^. .-. log^ a = 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. For, since a>l, = — = ^=p a~". .-. log = — co. But in any system whose base is less than unity, the logarithm of zero is positive infinity. For, since a < 1, = a". .-. log„ = co. 18. Logarithm of a Product, Quotient, Power, and Root in any system. If a be taken as the base, and m and n be any two numbers, it can be shown in a paanner similar to that used in Art. 9 that 1. log„ nin = log„ m + log„ n. 2. log„ :^ =logani-logan. [Let the pupil supply the , proof. See Art. 9 : use 3. log„m. =plog„m. aiovlO.l 4. log„^^ = ^. 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 h, we use the following formula: j For, let logj r = x. Then l^ =r (1) by definition of a logarithm. Take the logarithm of each member of (1) to base a, then X log„ k = log„ r. Hence, x = ,"^" j, logaA or log*7- = -^SaZ. loga« It follows as a special case that if r = a, logi a = -, or logi a-log„ ^ = 1. Ex. Find the logarithm of .7 to the base 5, By the formula just proved, W 7 ^logiojT^ 9,84510-10 ^^ logio5 0.69897 - 0.1549 ^ _ Q 2216 +, Ans. 0.69897 EXERCISE 7 In working the first twelve examples in the following exercise use four-place tables in solving the odd-numbered examples, and five-place tables in solving the even-numbered examples. Find the value of : 1. Iog5 60. 5. log^^VB. 9. logs .7261. 2. log6 9.3. 6. loggolS. 10. log.„2i .08275. 3. logs., 26.2. 7- logi.8 -17362. 11. log,.2.9267. 4. log4.93. 8. log.8.2631. 12. log, V3.1416. LOGARITHMS 23 Find without the use of tables : 13. logs 27. 15. loga^V "• log2-125. 14. log2 32. 16. log, 8. 18. logs .0625. 19. Find the base of the system of logarithms in which the log of 16 = 4. 20. If the log of 27 = I, find the base. 21. If ^ = the log of 5, find the base. 22. Given the log of 5^ = — h 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 inter- est, 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 invented 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 ^ 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? 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 AB 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 A C, CB, and the Z C, the distance AB can be computed provided we 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 tri- angles which cannot be solved by ordinary 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 other sides {AO and CB) may be com- puted. By this method, for instance, the ^'°- *• 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 maybe determined. Later the word trigonometry comes to have a more ex- tended 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 fun- damental 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 BO to the other side, forming the right triangle ABC. Fig. 5. Then the ratio —r-= is termed the sine of the angle A. AB ° Similarly, cosine A = -— — , cotangent A = — — ,, cosecant A = -— — , AB •' BC BG tangent A = ——-,, secant A = -r-r, versed sine A=l — -> ^ AC AC AB coversed sine ^ = 1 — BC 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, esc, 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 hy h, and BChy a, we have c COS A = - sec ^ = - CSC ^ = - tan^: cotA = Similarly sin JS = - cosS = vers A = l — covers A = l- sec J5 = - CSC S = - tan B = - a ootB-- vers B = l — - c covers B = l — Or using abbreviations, ther acute Z = "^P" , cot of either acute Z = ^ ^n of ei' hyp- .opp. cos of either acute Z = , ^ ^' , sec of either acute Z = --2E^ hyp. ±adj. tan of either acute Z = "P? -, esc of either acute Z= -^P' - J-adj. -Lopp. 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 " (sinA)^" we write "sin^^." How then woiild "the cube of cos^" be written? "The nth power of tan J.?" In this book unless the contrary is stated, in the right triangle ABQ, 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 vir- tually 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 kiuds of algebraic processes. 24. The value of a trigonometric function of an angle depends only on the size of the angle, not on the length of the lines chosen to form the ratios. Thus, by similar triangles (in Fig. 7), B'C ^ BC_ B"C" AB' AB AB" sin4=:^=^ = etc. 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 6=4, find c and the trigonometric ratios of each acute angle. The hypotenuse c= VF+¥= V25 = 5 Hence sin ^ = f cos A = tan A = f etc. 4= B" sinB = | cos -B = f tan 5 = 1 etc. FiQ. 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 66. 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 capi- tal 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^T-^ee versa. Similarly the capital letter c should always be written in the form ^^ 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, 6 = 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, 6 = 8. 6. a = 39, 6 = 80. ^4. a = 8, 6 = 15. 7. a = .09, c = .41. 5. a = 12, c = 13. 8. 6 = 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 ^ tan A (4)l + tanM. (7) tan A ^''^ ^ ■ (2) sin^^ + cosM- (5) sea^ A-t&n'A. oosA (3) sin ^ CSC A (6) tan ^ cot A (8) cos ^ sec A By the use of squared paper construct the angle whose 11. Tangent = f. 16. sine 2 12. Tangent = |. 17. cosine =^. 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. Deter- mine 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, ZB = 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 geom- etry? 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. o = Vp" + g", 6 = V2pg, find sin A and cos A. — 28. a = 2 mn, c = m^ + n', determine sin A, sec A, and tan A. 29. b = 2pq, e=p' + q', find tan A, sin A, esc A. 30. o = Vm^ + mn, b = ^mn + v?, find all the functions of B. 31. If a = 2y/mn 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 6 = 2.64 and c = 2.65, find the functions of B. 34. If a = 2 6, find the functions of A. 35. If 6 = I c, find the functions of A. 36. If a + 6 = I c, find the functions of B. 37. If a — & = j7^ c, find the functions of A. 38. Find the functions of B, if a = 4 d and 6 = 3 cZ. By use of squared paper construct a rt. A, given : 39. c = 4 and tan^ = |. 40. & = 3 and sin A=\. 41. Find 6 if cos A = .36 and c = 4.5. 42. On Fig. 8, sin A = what ? cos B = what ? Does sin ^ = cos B ? In like manner, show that cos A = sin B, tan A = cot B, cot A = tan B sec A = CSC B, esc 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 sec A 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 = - . c c Hence, sin A = cos B, or sin ^ = cos (90° — A), since B = 90° - A. Let the pupil show in like manner that cos A= sinB = sin (90° - A), tan ^ = cot B = cot (90°,- A), and sec A = esc B- = esc (90° - A). Hence, in general. Any trigonometric function of an angle is equal to the co- function of the complement of the angle. By the use of this property, Any trigonometric function of an angle between 45° and 90° can he reduced to the function of an angle between 0° and 45°. Thus, sin 88° 10' = cos 1° 50'. 32 TRIGONOMETRY 5. CSC 21° 24' 30"- 6. sec 84° 16'. 7. sin 89° 59'. 8. cos 1° 18'. EXERCISE 9 Express each of the following trigonometric functions as a function of the complementary angle: —1. sin 60°. --2. cos 16°. 3. tan 65° 24'. 4. cot 55° 36'. '9. Given tan 60° = VS, find cot 30°. 10. Given sin 30° = ^, find cos 60°. 11. Given cos A = ~, find sin (90° — A). 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 trigono- metric functions of an acute angle, viz. : sin and esc cos and sec tan and cot For ^x^=l. c a .*. sin AxcscA = l ^x£ = l. c b .*. cos A x sec A = 1 ^x^=l. .'. tan A X cot A = 1 28. Four equations connect the trigonometric functions of an acute angle in important ways. For, from Fig. 9, a' + b' = c' (1) TRIGONOMETRIC FUNCTIONS 33 Dividing (1) by c^ that is, sin^ A + cos^ A = l. Dividing (1) by V, that is, tan^ A + l = sec^ A . Let the student prove in like manner that cot^ A + l = csc^ A. Also from Fig. 9. b c c' sin A that is, tan A = cos 4 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 ^ = Vl - cos' A. ^.^^^ ^ ^ cos^ . /T r^r-j * sinul cos A=vl — sin' A. cos^ ■, 1 CSC ^ = tan A = sin A "t-1 vers^= 1-cos^. covers .4 = 1 — sin .4. 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 siiiJ. = f, find the other trigonometric func- tions of A. cos A = V3r 'jL=vr^=vi=4V5, ^ . sin^ 2 . V5_ 2 _2 - cosjI 3 o ys o cot ^ = sec -4 = CSC J.= tan -4 _^ COS -4 1 ,V6 2 ■ V5. 3 '■32' Ex. 2. sin J. vers ^ = 1 — cos^ = l — ^ VB. covers ^ = 1 — sinJ. = l — 1 = ^. If tan a;= 2, find the other functions of x. sec'' x = l+ tan^ a;. (Art. 28.) .'. sec^ a; = 1 + 4 = 6. sec a; = VS. cos X = - 1 ^ /K . = __ = _V5. sin X =vr sec a; VS • cos^ a; = Vl-i = V| = I V5, etc. Geometeic 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^ = f, 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. %y/ Then AC = -V3'-2' = -V9-A=V5. ' Then from .the figure by the definitions of the trigonometric ratios a cos^-'^; tan^- ^ -^V5; cot ^ ^^. 3 -y/5 S 2 ' sec^ = 4- = iV5; csc^ = 5; vers^=l-:^. i =-V3 c FiQ. 10. ? V5 5 2' 3 ' covers A = l—% = 1. TRIGONOMETRIC FUNCTION'S 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: -1. sin^ = ^|^. 5. cotJ. = m. 9. tan^ = 0. -~- 2. tan A = \2-. 6. CSC A = VS. 10. sin A=l. 3. sec A — i^-. 7. sin ^ = 0. 11. sec J. = co . 4. cos^ = -|. 8. cos^ = 0. 12. sina; = 5j>. Find by geometric methods (squared paper may be used to advantage in constructing diagrams) the other functions of A (or x), given : 13. tan-4 = f. 16. cot^ = f. 19. tan^ = m. 14. cos^=j^^. 17. sin^ = |-. 20. sin^ = iV2. 15. CSC A = ^. 18. sec A = 4:. 21. cos x = l. Mnd by both methods the other functions of the angle named when : 22. csc^ = |^. 27. cosJ. = |. 23. tan^ = — ■• 28. sec^ = - ™_-» V6-V2 -24. cot^ = V2 + l. 29. cos A = K. 25. sm A=l. 26. tan 22^° = V2 - 1. 30. cotl6° = 2+V3. Express each of the other trigonometric functions of A in terms of : - 31. sin A. - 38. Given sin ^ = f, find cot A. 32. cos A. 39. Given cos >dL = |-|, find esc ^. 33. tan A. 40. Given tan A = VS, find sin A 34. cot A. 41. Given esc -4 = f , find sec A 35. sec A. 42. Given sec A = ^-f-, find cot A. 36. CSC A. 43. Given cot A = V2 — 1, find cos A. 37. vers A. 44. Given tan A = V6, find esc A. 36 TRIGONOMETRY - 45. Transform the expression sin' ^ + cos ^ so that the only trigo- nometric function contained in it shall be cos A. 46. Transform (1 + tan^ 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 cos^ x — sin^ x = sin x so that it shall contain only sin x. 49. Transform tan a; = 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)=a^ — 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 a^— a; = 2x — 2 is true only when a; = 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. Eelations 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 cot' ^ cos' ^ = cot' J. -cos' .4.. cotM cos2^ = 52!!4 cos^^ _ (1 — sin^ A) cos' A sin^A _ cos' J. _ sin' A cos' A ' sin' A sin' A = cot' A — cos' 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, cot' A cos' A = cot' A — cos' A and trans- form it as follows : 52?!^ cos' ^ = 55?!^- cos' ^, sin' A %var A cos* A cos' A — cos' A sin' A or ^ ■ • sin' A sin' A cos* A _ cos' ^ (1 — sin' A) _ sin' A sin' A cos* A _ cos* A sin' A sin' A Since the last is plainly an identity, we infer that cot' A cos' A = cot' A — cos' 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 — T/2 , =cotA. cos A tan-" A 38 TRIGONOMETRY Substitute - for sin ^ : - for cos ^ ; - for tan ^ ; - for cot A. Then c c b a a sin A _ « =6 = cotA cos A tan^ A b a? a c'b" EXERCISE II 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. 3. sin A sec A = tan A. COS AesoA= cot A. _ 1 + cos A sin A sin ^ 1 — cos A „ 1 + sin ^ _ cos A cos A 1 — sin ^ 4. cos A = sin A cot A. 8. sin^ A — 1 BOS- 'A = 2 sin^ A — 1. 9. (1 — sin^ A) tan" ^ = sin" A. 10. (tan A + cot A) sin A cos A = l. 11. (1 — sin" A) esc" A = cot" A. 12. (sin ^ + cos ^)" = 1 + 2 sin ^ cos A 13. (sin A + cos Ay + (sin A - cos A)^ = 2, 14. (csc"^ — l)sin" J. = eos" A , „ sin A , cos ^ ^ J 15. + - — - = sec A CSC A. cos A sin A ' 16. _^2t!4_ = cos"A 1 + cot"^ 17. tan A + cotA = sec ^ esc A. 18. tan^ + cot^=?5^^^^L5?^. sec AxcsoA 19. sin* u4 — cos* A = sin^ -4 — cos" A. „„ sin A , cos ^ • ^ , A 20. TT + q — I T = sin.4 + cosA 1 — cot A 1 — tan^ ^^* aIt ^ = esc .4 — cot ^. ■cos^ 22. TRIGONOMETRIC FUNCTIONS 39 1 4- tan A _ l — tan A 1 + cot ^ cot ^ — 1 23. cot A 4- tan A = sin A cos A 24. tan^ A — sin^ ^ = tan^ A sin^ ^. 25. csc^ A-2 csc^ ^ = cot* A — 1. 26. secM(l — sin*^) = 2tanM4-l. „_ esc A . 27. = cos A. tan A+cotA 28. -^ r-— • = sin^ ^ — cosM. 1 + cot- A 29 cot J. — COS A cot ^ cos A cot ^ cos ^ cot A + cos ^ 30. 1 — cotM = 2cscM-csc*A 31. Vl — sin^ A tan .4 = sin A. 32. siu''^4-cos^^ = l — 3sin^ J. cos^ A 33. cos' .4 — sin' j4. = (cos ^ — sin ^) (1 + sin ^ cos A). 34. Eeduce tan^ x sec* x to the form (tan" x + tan' a;) seo^ x. Transform : 35. tan' X into (tan* x — tan* x + tan^ a; — l)see^ a; + 1. 36. sec"?/ into seo^y (1 +4tan'?/ + 6 tan*y 4-4tan'!/4- tan'y). cos X 37. Vl + sin X into Vl — sin X 38. into sec' x — sec x tan a;, 1 + sin X 39. "^ ^'""^ into sec" x + sec a; tan x. cos^a; 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 ? TBIGONOMETEIC FUNCTIONS OF PARTICULAR ANGLES 32. Functions of 45°. The trigonometric functions of 30°, 45°, and 60° are nsed so frequently that it is of service to determine their values and commit these values to 40 TRIGONOMETRY l-VT xnemory. It is helpful to notice that we determine these values in each case by the use of a right angle, the hypote- nuse of which is taken as 1. Let ABC (Fig. 11) be an isos- celes right triangle, the hypotenuse of which, AB, is 1. Then, by geome- try, each leg is iV2_(for Z^ = 45°, .■.AC=BG; hut' AG' + BC^ = V, .-. 2 BO' =V, etc.). By the definitions of the trigonometric functions, sin 45° =(1V2)-^1=1V2. cos 45° = (iV2)4-l = iV2. tan 45° 1V2 ^ — = 1. 1V2 cot45° = ^=l. 1V2 sec45° =1-— ^ = A = V2. 2 V2 csc45° = l ^ — = A= V2 2 V2 33. Functions of 30° and 60°. Let ABD (Fig. 12) be an equilateral triangle in which the length of one side is 1. Let AC he ±BD. Then, by geometry ^BAI)=60°, and ZBAC=30°. Also AC bisects BD, hence BC = ^. AC=^AB'-BC'= Vi_i = iV3. Then in the right triangle ABC, 1 2- sin 30° cos 30° = 1V3. Pig. 12. TRIGONOMETRIC FUNCTIONS 41 tan 30° = ^ = ~ = 1.V3. JV3 V3 ' cot 30° 4^=V3. sec 30° = 1V3 CSC 30° = ^ = 2. 2 _2_ V3^ |V3. Let the pupil write out in like manner the functions of 60° (that is, oizABC 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 i i^M 4V3 cos •iVs iVs i tan iyfs 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 approaches zero and A C approaches 1. At the limit, sinO° ~T~ 0. cosO° 1 1 1. tanO° 1 0. cotO° i ~0~ 00 secO° 1. CSC 0° = T7 = 00. versO°= 1-1 = 0. covers 0° = 1 - == 1. 42 TRIGONOMETRY 35. Functions of 90°. Let ABC (Fig. 14) be a right triangle in which BAC is nearly a right angle and approaches 90° as a limit. AB remains fixed in length ; hence BC approaches 1 as a limit and AC approaches 0. At the limit. sin 90° = ^ = 1. cos 90° = ^ 0. tan 90° = ^ = CO , cot 90° = 0. sec 90° = - = CO . CSC 90° = ^ = 1. vers 90° = 1 - = 1. covers 90° = 1 - 1 = 0. 0° 90° sin 1 cos 1 tan 00 cot CD sec 1 CD CSC CD 1 The results obtained in Arts. 34 and 35 may be conveniently arranged in a table thus : 36. Representation of the Trigonometric Functions of an Acute Angle by Lines. If a quadrant of a circle GAB be drawn with center and radius OB equal to 1, the sine of any angle AGP' is EIL-^EZ-M'-p' GP' 1 -^'^^• Similarly the sine of Z AGP = MP, and sine of A AGP" = M"P". In other words the sine of any angle AGP in a quadrant whose radius is 1 is represented by the perpendicular let fall from P upon the radius OA. TRIGONOMETRIC FUNCTIONS 43 Hence it is easy to see that, since MP is the sine of ZAOP, if AOP becomes very small and = 0, MP = 0, and at the limit sin 0° = 0. Also if zAOP" increases and = 90°, sin z AOP" or M"P" = OB ov 1. Hence at the limit sin 90° = 1. Similarly cos ZAOP' = -^ = -^' = OM/ Hence also cos ZAOP = OM, cos ^AOr' = 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 and the foot of the line representing the sine. Hence cos 0° = OA or 1, and as ZA OP^ehanges from 0° to 90°, the cosine changes from 1 to 0. ':^ ^ Similarly, (Fig. 16), tanZ^OT= AT AT sec zAOT- OA OT 1 OT 1 = AT. OA cot ZA0T= tan Z BOB ^BB _BB OB 1 CSC Z AOT = sec Z BOB OB OB OT. V = BB. V FiQ. 16. = 0B. 'B The various lines which ■ represent V the trigonometric functions of an acute angle AOP may be combined in a single figure (Fig'. 17). Let the pupil find the lines on the figure which represent vers Z A OP and covers ZLAOP. 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 Fimctions. 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. 42° 30'. 6. 56° 37'. 7. 47.24°. 8. 72.37°. By use of the table of natural sines, construct : 9. 61° 23'. 10. 47° 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 sin^ 45° + 2 cos 45°. 16. tan 30° cos 90° - 4 sin 60° + eos^ 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 = 0°, Q = 30°, R = 45°, S = 60°, T= 90°, find the value of each of the following expressions : 20. sin Q + cos B — 1. 21. tan^P + tan^Q + tan^i?. 22. COS P cos Q cos P + sin B sin S sin T. 23. sec P + 2 sin Q + 2 cos^ R + \ tan^ 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 46.° TRIGONOMETRIC FUNCTIONS 45 30. Given that x is greater than 45° and less than 90°, show on a diagram similar to Fig. 17 that tan x is greater than cot x. 31. Given that x is less than 46°, show that sec x is less than esc 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 < osc 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 z whicli satisfies the equation sin x = ^. Since sin 30° = 1-, in the given equation x = 30°, Ans. Ex. 2. Solve sin x = cos a;. Dividing each member by cos x, tan x = l. .-. a; = 45°, Ans. Ex.3. Solve tan a; — 1 = 2 sin a; — 2 COS a;. Substituting for tan x, ^^^ — 1 = 2 sin a; — 2 cos k. cosx Hence, sin a; — cos a; = 2 sin a; cos a; — 2 cos^ x. Factoring, (sin a; — cos a;) (1 — 2 cos a;) = 0. Hence, sin a; — cos a; = 0. .-. tana; = l, a; = 45°. Also 1 — 2cosa; = 0. .-. cos a; = |, a; = 60°. Hence, a; = 45°, 60°, ^ns. 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 a;. .-. 90° — a; = 4 a;. 5 a; = 90°. X = 18°, Ans. EXERCISE 13 Solve each of the following equations : 1. tan2a; = 3. 12. 2 sin j/ + esc ?/ = 3. 2. sin^ x = \. 13. 2 sin x VB + 4 cos a; = 5. 3. cot a; = 3 tan a;. !*■ sec a; = 2 tan a;. 4. cot^a;=i. ^5- 4sin^a; — tan^x = cot^a;. 5. VI — sin^ a; = 1 + sin a;. 16. cot a; + 2 tan a; = — 6. sec^a; = 2. -^.j 3 cosa; + tana; = l +3 sina;, 7. tan a; + cot a; = 2. ig_ tan a; = 2 cot a; - 1. 8. seca! = V2tana;. 19. esc ?/ = 2 cot ?/. 9. cos^ a; — sin^ x = sin x. 20. 2 sin a; + cos a; = 2. 10. tan^a; + 2 sec^a; = 11. 21. 2 sec a; — cos a; = 1. 11. 3 cot^ x + cot a; = 4. 22. sin^ a; + sin a; = |. Solve : 23. sin a; = cos 5 x. 26. sec (46° + a;) = esc x. 24. tan ?/ = cot 8 2/. 27. sin ?/ — cos wj/. 25. cos ^ a; = sin a;. 28. sin 3 a; = cos 2 a;. 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 = 76, find B. 35. Given c = 120, & = 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 ex- plained in the Introduction to the Tables (Arts. 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 56° 16'. 13. log tan 86° 46' 6". 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' 0". 17. We have proved (see Art. 3.S) 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 = 6 tan A. (Why ?) If ^ = 18° 16' and h = 18.63, find a. 20. In the rt. A ABC, & = c cos A. (Why ?) Find 6 if c = 18.675 and ^ = 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°. 29. log cos 40.16°. 28. log cot 12.65°. 30. log cot 29.23°. 31. In the rt. ABAC, b=acotA. (Why?) If ^ = 18.67° and a = .2167 feet, find b. 32. In the rt. A ABC, a = c sin A. (Why?) If c = 17.65 and A = 59.72°, find a. Also find b, if b = c cos A. EXERCISE 15 Using five-place tables, find A, given : 1. log sin ^=9.59632 -10. 7. log cos ^ = 9.53390 - 10. 2. log tan ^ = 9.73777 - 10. 8. log tan ^ = 1.06575. 3. log cos A = 9.90951 - 10. 9. log sin A = 9.95788 - 10. 4. log cot A = 10.07029 - 10. 10. log cot A = 1.02921. 5. log sin ^ = 9.96159 - 10. 11. log sin ^ = 8.84501 - 10. 6. log tan ^=0.44540. 12. log cos ^ = 8.84501 - 10. By use of four-place tables, find A, given : 13. log sin A = 9.6496 - 10. 20. log cos A = 9.8409 — 10. 14. log cos ^ = 9.8063 - 10. 21. log tan J. = 0.2575. 15. log tan ^ = 9.7384 -10. 22. log cot ^ = 2.0248. 16. log cot J. = 0.4755. 23. logtan J. = 1.5718' 17. log cot A = 9.8248 - 10. 24. log sin A = 9.9596 - 10. 18. log tan ui = 0.4422. 25. log cos J. = 9.3129 - 10. 19. log cos ^ =9.6351 - 10. 26. log cot ^=0.5881. EXERCISE 16 By use of five-place tables find : 1. log sin 0° 66' 18". 5. log cot 1° 18' 36". 2. log tan 1° 16' 37". 6. log cos 89° 7' 19". 3. log cos 88° 13' 26". 7. log sin 1° 6' 12". 4. log tan 88° 54' 60". 8. log cot 88° 16' 32". Find the angle A if : 9. log tan ^=7.88154 -10. 13. log tan ^ = 3.06992. 10. log cos ^ = 8.28910 - 10. 14. log cot ^ = 2.88206. 11. log sin ^ = 8.09600 -10. 15. log sin ^ = 6.88800 - 10. 12. log cot ^ = 7.90390 - 10. 16. log cos ^ = 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. Z log tan 8.09699-10. 18. Z log tan 10.90261 -10. 21. Z log sin 8.09599 -10. 19. Z log cos 9.06000 -10. 22. Z log tan = 2.77651. 23. In the A ABC, a = G sin A. Find a if c = 18.6 and A = 26° 18' 48." Find the value of the following : 528.7 X cos 83° 16' 24" x tan^ 75° 18' 24" 24 25. 672 cot^ 18° 32' 64" x sin 69° -h cos^ 16° 16' 34" 265 X tan 66° 18' x cos^ 14° 28' 12" 19 cot^ 11° 16' 24" X sin 75° 16' 45" x .l' By use of four-place tables, find : 26. log cos 88.76°. 30. log tan 88.763°. 27. log sin 0.762°. 31. log cot 0.765°. 28. log cot 89.267°. 32. log sin 1.267°. 29. log tan 1.067°. 33. log cos 89.467°. Find angle A if : 34. log cot ^ = 8.1067 - 10. Find: 35. log tan ^ = 8.2574 -10. 42. log cot 88.676°. 36. log cos ^ = 8.1360 -10. 43. log tan 88.676°. 37. log sin A = 8.0440 - 10. 44. Z log cot 8.1078 - 10. 38. log tan ^ = 2.1080. 45. Z log tan 8.0295 - 10. 39. log cot ^ = 2.0532. 46. Z log cos 8.0959 -10. 40. log sin A = 7.9100 - 10. 47. Z log sin 8.0371 - 10. 41. log cos ^ = 7.9932 -10. 48. log tan 88.68°. 49. In the rt. A ABC, a = c sin A. (Why ?) Find a if c = 126.27, and A = 1.267°. 50. In the rt. A ABC, b.= a cot A. (Why?) Find 6 if a = 0.4267, and ^ = 2.166°. =, -n-- .q *!, 1 f 632.7 X cos 78.16° X tan^ 71.62° 51. Find the value 01 -r^r—;: -. — . - _„. ^,, ^„ .-o o' 426.8 X sm 13.25° X cot'' 12.47° x .8 ^ T,- J i, 1 c 326 X tan 38.26 x cos^ 88.627 52. Find the value of ^g-^_^^^g2^^3—,^;^g^- 50 TRIGONOMETRY EXERCISE 17. REVIEW 1. In the right A ABC, given tan A=^ and a = 16, find 6, c, and the other functions of A. 2. If cos A = —, find the value of sin ^ + tan ^ ^ 17 cos A — cot A 3. Show that cos 60° cos 30° + sin 60° sin 30° = cos 30°. 4. Show that eot45° + cot90° ^^^ 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 ^ = f , is ^ greater or less than 45° ? 7. If tan ^ = 2, is ^ greater or less than 60° ? 8. Which is the greater, tan 37° or cot 37° ? 9. If J. = 60°, show that sin | ^ = yj~ ■ cos A 10. If ^ = 60°, show that cot i ^ = J^±^^A. ' ^ ^1 — cos^ 11. Which is greater, sin 45° or \ sin 90° ? sin 60° or 2 sin 30° ? tan 30° or J tan 60°? 12. If a; = 30° and y = 60°, show that sin x cos y + cos x sin y = sin (x + y). 13 Prove ^ + ""t J. ^ sec A + esc A 1 — cot A sec A — esc A , . -D 1 + tan^ A sin^ A 14. Prove — ■ — = — 1 + cot^ A cos^ A 15. Prove ^ + 0°^-^ = (esc A + cot AY- 1 — cos ^ • 16. If a; = 30°, show that tan 2 a; = 2 tan a; 1 — tan^ X 17. If a; = 30°, show that sin 3 a; = 3 sin a; — 4 sin' x. 18. If a; = 30°, show that cos 3 a; = 4 cos' a; — 3 cos x. Solve the following trigonometric equations : — 19. tan a; + 3 cot a; = 4. 20. 2 sec^ X — tan^ a; = 5. 21. 3 esc" a; — 2 cot a; = 4. TRIGONOMETRIC FUNCTIONS 51 If P = 0°, Q = 30°, B = 45°, ;S'= 60°, T = 90°, find the value of : 22. cos^ Q + cos" S + cos^ r + 2 cos Q cos ^S" cos T. 23. sec Q(l + tan R) - sin^ r(cos 22 + sin aS cos Q). 24. I + ^^°' f + 3(cos P sin" R - sin ^S). 2 — tan^i? 25. If 26 sin A = 7, find cot ^ and csc A. 26. If j9 cot ^ = vr— y, find sin ^. 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 = f 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. 166) ? 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 sin" A + cos" A = l (see p. 172) ? tan a; = ?HL? ? cos a; 35. Give an account of the computation of trigonometric tables (see pp. 168-170). CHAPTER III RIGHT TRUNGLES 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 loill give the un- known angle. The unknown sides may then be found by means of the following : 1. Either leg = {sine of Z. opposite) x hypotenuse. 2. Either leg = {cosine of Z. adjacent) x hypotenuse. 3. Either leg = {tangent of Z. opposite) x other leg. 4. Hypotenuse = {secant of either acute Z) x {leg adjacent to thatZ). Also (either leg) = (cot of Z adjacent) x (other leg) ; B hyp. = (esc of either acute Z) X (leg opposite that Z). / Proof / By def., sin A = -. c .\a = c sin A. V a Also cos 5 = ^. c .■.a = c cos B. / 1 J Also tan^ = «. h sec £ = - • a .■.a = b tan A. h ( Fio. 18. .•.c = a sec B. 62 EIGHT TRIANGLES 53 Similarly it may be proved that : b = c sin B, b = c cos A, b = a tan B, and c = 6 sec A. Ex. 1. Given A = 55° 43' 29", c = 415.18, find the remain- ing 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 65° 43' 29" for A). We also indicate the parts to be computed (a, b, B) by annex- ing the = mark to each of these. During the numerical computation, aa soon as the result for any part is ascertained, this result should be entered on the diagram after the proper = mark. ZS = 90° - 55° 43' 29" = 34° 16' 31"- a = 415.18 sin 55° 43' 29". (Art. 41, 1) .-. log a = log 415.18 + log sin 65° 43' 29". 415.18 log 2.61824 55° 43' 29" log sin 9.91716 - 10 a = 343.085 l^g 2.53640 Also 6 = 415.18 cos 55° 43' 29". (Art. 41, 2) .-. log 6 = log 415.18 + log cos 55° 43' 29". 415.18 log 2.61824 55° 43' 29" log cos 9.75064 - 10 6 = 233.821 log 2.36888 (As a check use a = b tan A.) Ex. 2. Given a= .0723, 5 = 31° 47' 7", find the remain- ing parts ol the right triangle. Z ^ = 90° - 31° 47' 7" = 58° 12' 53". 6= .0723 tan 31° 47' 7" .0723 log 8.85914-10 31° 47' 7" log tan 9.79216 - 10 & = . 448022 log 8.65130-10 c=.0723sec31°47'7" ^ .0723 cos 31° 47" 7' .0723 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 &= c cos A.) 54 TRIGONOMETRY Ex. 3. By use of four-place tables solve the right triangle in which 6 = 21.635, ^ = 47.23°. Z B = 90° - 47.23° = 42.77°. Also a = 21.635 tan 47.23°. (Art. 41, 3) B .-.* log a = log 21.636 + log tan 47.23°. 21.635 log 1.3362 47.23° log tan 0.0339 a =23.394 log 1.3691 •^ By Art. 41, 4, c = 21.636 sec 47.23° = 21.635 cos 47.23° log c = log 21.635 -l-colog cos 47.23° 21.635 log 1.3352 47.23° colog cos 0.1681 c = 31.864 log 1.5033 (As a clieck use a = c cos B.) 42. First Estimates. Graphical Solutions. In the solutions of triangles fully one half the mistaltes 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 416.18, each of the legs will be less than 416.18 ; 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 calcula- tion, the pupil finds a leg greater than 415.18, or a less than 6, 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 numeri- cal 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 obtaiaed 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 frac- 6 tion §^M85 g^ ^ o^t i^ f^ouj tije t^ble the value of tan 65° 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. (3) (1) (2) 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. k = 28°, 6 = 12. 6. ^ = 46° 18', & = 48.527. --5. ^ = 78°, c = 26.736. 7. ^ = 28° 17', = 24.16- 56 TRIGONOMETRY a 5 = 54° 43' c = 1123. ^10. ^ = 38° 16' 24", c = 3.6289. 9. 5 = 37° 19', 6 = 293.8. 11. 5 = 72° 16' 42", a = 22.684. 12. Given c = . 62684, J5 = 63° 18' 48" ; find a. 13. Given ^ = 37° 25' 20", c = .356; find 6. rind the remaining paxts in eacli of the following right triangles, given : 14. ^ = 63° 28' 40", a = 256.43. 15. c = 13.867, ^ = 87° 16' 30". 16. ^ = 51° 9' 6", c=. 19678. 17. a = 126.78, ^ = 26° 18' 36". 18. Given ^ = 5° 16' 32", 6 = .96156; find c. 19. Given ^ = 37° 14' 16", 6 = 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 Colo- rado is known to be a horizontal distance of 3 miles from the Colorado Eiver and the angle of depression of the river is 17°, how deep is the canon 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, hav- ing given : X 24. A = 32.6°, 6 = 18. 28. A = 37.67°, c = 126.7. 25. ^ = 56°, c = 2.678. 29. 5 = 76.25°, a = .926. 26. 5 = 38.2°, c = .7685. 30. A = 21.32°, a = 16.256. 27. 5 = 82.5°, a = 12.56. 31. 5 = 66.27°, 6 = .0087. 32. Given c=.6243, 5 = 51.25°; find a. 33. Given ^ = 77.26°, c = .5163; find 6. 34. Given 5 = 39.29°, 6 = 41.67; find a. RIGHT TRIANGLES 67 Find the remaining parts in each of the following right triangles, given : 35. c = 13.13, A = 88.17°. 36. 5 = 42.16°, a = .5252. 37. Given ^ = 5.26°, 6 = 128.6; find c. 38. Given B = 87.267°, c = 22.67 ; find a. 39. Given A = 4.276°, a = 26.32 ; find 6. 40. "Work Exs. 20-23 by four-place tables. Solve without the use of tables, having given : 41. ^ = 30°, 6 = 7. 45. ^ = 60°, a = 2000. 42. A = 4.5°, c = 12. 46. B = 30°, c = 1200. 43. 5 = 60°, 6 = 25. 47. ^ = 45°, 6 = 200. 44. 5 = 30°, a = 1000. 48. ^ = 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 Siues, etc., pp. 91-96). 50. How many of Exs. 41^8 can you solve at sight vrithout draw- ing a figure ? 51. On the figure if A ADB and DCB are right A, find BD, BG, and DC at sight. ^^ 52. On Fig. 52, p. 93, if 0P= 1, what is the value of 0©? ofPQ? of ^iV? oiON'i 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 : A.OVV- 1. sine of either acute ^ = - 2. cosine of either acute ^ hyp. _^adj. hyp. j_ opp. 3. tangent of either acute ^ = -j — -rr-. 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 fo und by one of the formulas, a = V(c + b)(c — b), b = V(c + a)(c- a)). Ex. 1. Given a = 317, c = 438, find the remaining parts of the right triangle ABC. (Art. 44, 1) • A 317 r^=438- Hence log sin A = log 317 + colog 438 317 log 2.50106 438 log 2.64147 colog 7.35853 - 10 A = 46° 21' 55" log sin 9.85969 - 10 B = 90° - 46° 21' 55" = 43° 38' 5". 6 = 438 cos 46° 21' 55". (Art. 41, 2) 438 log 2.64147 46° 21' 55" log cos 9.83888 - 10 b = 302.24 log 2.48035 ( As a check use tan A = --] Ex. 2. By use of four-place tables, solve the right triangle in which a = 3. 104, & = 2.965. , b tan A = 3.104 2.965 JB = 90°-A 3.104 c = -■ cos B 3.104 log 2.965 colog 0.4920 9.5279 - 10 .4 = 46.31° log tan 0.0199 5 = 90° -46.31° = 43.69°. 3.104 log 0.4920 43.69° colog cos 0.1408 c = 4.293 log 0.6328 45. Sources of Power in Trigonometrical Solution of Tri- angles. There is danger that the pupil form mechanical habits of solving triangles without realizing the nature or RIGHT TRIANGLES 69 meaning of what he is doing. He should constantly realize that he is able to do what he is doing because some one be- fore him has computed the legs of every possible right tri- angle 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 men- tioned in terms of the sine. Ex. Given c = 412, & = 410, solve the triangle. By the formula, a = V(412 +410)(412-410) = V822x2. ,-. loga = ^(log822 + log2). 822 log 2.91487 2 log 0.30103 ^^^^^ Al • . 40.546 a = 40.546 log 1.60795 ^^^° sin A = -^^ 40.546 log 1.60795 412 colog 7.38510 - 10 ^ = 5° 38' 52" log sin 8.99305 - 10 S = 90° - 5° 38' 52" = 84° 21' 8". 60 TRIGONOMETRY EXERCISE 19 Using five-place tables, solve in full the following right triangles, given : (In working each example outline aU 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. & = 14.222, c = 21.678, 3. a = 26.725, c = 39.626. 7. a = .0628, 6 = .0487. 4. a = 5, 6 = 6. a a = .1777, c = . 25643. 9. Given a = 4 yd., 6 = 9 ft., find A. 10. Given a = 8.701 yd., b = 21.645 yd., find Z A. 11. Given 6 = .26725, c = .39626, find Z B. 12. Solve in full if a = 6, 6 = 6. \ 13. rind ^ if a = .02678, 6 = .05537. ^14. Solve in full if c = 117.32, a = 112.67. SuGGESTioif. First use 6 = Vc^ — a?= V(c + a){c — a). 15. Solve in full if 6 = 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 | of a right angle is to con- struct 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= 2.3.7, a = 15.7. 25. 6 = 6.7, c = 9.7. ^22. c = .562, 6 = .3962. 26. 6 =.12675, a =.14296. 23. a = 33.29, 6 = 27.28. 27. c = 132.96, 6 = 100.82. 24. a = 5, 6 = 8. 28. a =.07282, c = . 11111. 29. a = 2367, 6 = 1827.6. RIGHT TRIANGLES 61 30. Given a = 11, c = 16, find A. 31. Given a = 27.82, 6 = 33.67, find B. 32. Given c = 156.7, 6 = 148.2, solve in full. First use a = Vc^ — b^ = Vc + 6)(c— 6). 33. Given c = 862, a = 854, solve in full. 34. Given a = 98.6, 6 = 63.4, find A. 35. Given c = .4367, 6 = .1967, find B. 36. Work Exs. 17-20 by the four-place tables. Witbout the use of tables solve in full each of the following right triangles, given: 37. a = 13, 6 = 13. 41. c = 6, a=3V3. 38. 'c = 18, a = 9. 42. c=V2, 6 = 1. 39. c = 200, 6 = 100. 43. c = 100, a = 50 Vs. 40. a=V3, 6 = 1. 44. a-|-c=18, 6 = 6V3. 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 deter- mined 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. Draw the altitude OD. Then Z^OD=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 ^i) = 47.6 sin 21° 15'. 47.6 log 1.67761 21° 15' log sin 9.55923 - 10 JZ) = 17.252 log 1.23684 AB = 2AD = 34.504. Fio. 25. 62 TRIGONOMETRY Ex. 2. By use of four-place tables, solve the isosceles triangle whose base is 12.25 and vertex angle 28.22°. Draw the altitude AD. Then Z B^Z) = 1(28.22°) = 14.11°. ZB = 90°- 14.11° = 75.89°. AB = 6.125 sec 75.89° = ^'l^^,,, cos 75.89° 6.125 log 0.7872 75.89° colog cos 0.6130 AB = 25.129 log 1.4002 48. A regular polygon may be divided into equal right tri- angles by lines drawn from the center to the vertices and by the apothems to the sides. Hence if certain parts of a reg- ular polygon are given, the remaining parts may often be determined by divid- ing the polygon into right triangles and solving one of these triangles. It is to be observed that one of the right triangles, as A CD of Fig. 27, has 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 ACA' ■= , where n denotes the n number of sides of the polygon, and ZACD of the right , . 1 180° triangle = . n EXERCISE 20 Using five-place tables, solve each of the folio-wing isosceles triangles, given : 1. Base = 120, base Z = 60°. 2. Leg = 216, vertex Z= 110°. 3. Base Z = 56° 18', leg = 8.7265. 4. Base Z = 38° 17' 50", altitude = 31.42. RIGHT TRIANGLES 63 5. Base Z = 55° 18' 24", altitude = 762.89. 6. Base = 8.2364, altitude = 7-8. 7. Vertex Z = 113° 17', base = .12692. 8. Altitude = 4835, base =9248. 9. One side of a regular pentagon is 12. Find the apotbem, 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 m 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 I. 16. D, I, and C, in terms of m and h. 17. D, C, and m, in terms of h and I. 18. C, h, and I, in terms of D and m. 19. D, h, and I, in terms of C and m. 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 arc of 64° 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 in- scribed in the same circle. 24. The radius of a circle is E; show that each side of a regular /'180°\ Inscribed polygon of n sides is 2i?sin i J, and that each side of a /180°\ regular circumscribed polygon is 2 iJtan I )• 64 TRIGONOMETRY 25. Eacli side of a regular polygon of n sides is m ; show that the radius of the circumscribed circle is equal to ^ esc i -J, and the radius of the inscribed circle is equal to — cot f )• 2 \ 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 Z = 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 2-v/3. 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 & denote the base, a the altitude, and K the area of a right triangle, by geometry K = \ab. .•. log -K = log a + log 6 + colog 2. Ex. 1. Given JL = 37°19', & = 308, find the area of the right triangle. To find log a and then the area we proceed as follows : a = 308 tan 37° 19'. (Art. 41) 308 log 2.48855 37° 19' log tan 9.88210 - 10 o log 2.37065 308 log 2.48855 2 colog 9.69897 -10 jr = 36155 log 4.65817 Ex. 2. Find the area of a right triangle in which the hypotenuse is 417 and the base 356. a = V?^^^ = V(417)2 - (356)^ = V(417 + 356)(417-356) = V773 x 61. .-. log a = I (log 773 + log 61). 66 TRIGONOMETRY K= \ ab. Fig. 29. , log K= log a + log 6 + colog 2. 773 log 2.88818 ^ log 1.44409 61 log 1.78533 i log 0.89267 356 log 2.56145 2 colog 9.69897 - 10 ^=38652.7 log 4.58718 Ex. 3. By -use of four-place tables find the area of the right triangle in which J. = 37.32° and 6= 308 (see Fig. 28). log ^ = 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 ^=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, S= ^ ab. When an acute angle and the hypotenuse are given, Sl = ^c^sinA cos A (or = ^ c^ sin B cos B). When the hypotenuse and a leg are given, K = ^a^{c + a){c-a} (or =i &V(c + &)(c-fe)). When an acute angle and a leg are given, K^^aHanB (or = 1 &Han J.) , or K = ^a^ cot A {pic = ^¥' coiB). 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 com- puting 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. .4 = 28° 18', 6 = 216. 5. S = 63°18', c = 124.72. 2. 5 = 72°, a = 196. 6. a = 192.7, 6 = 212.97. 3. ^ = 21° 16' 30", c = 31.967. 7. a = 0.73216, c = .9125. 4. c = 46.72, 6 = 32.54. 8. c = 927.8 ft, 6 = 759.8 ft. 9. Given a = 2.5 and K= 4.27, find 6, c, and A. 10. Given K= 7.256 and A = 26° 18', find a, 6, and c. 11. Given K= 55.686 and c = 16.67, find a, 6, 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 differ- ence between its area and that of the inscribed circle. 24. Given m = 9 and /f = 30, find r, c, and B. 25. Given n = 11 and K=35, find the perimeter. 26. Given n = 5 and K = 37, find ^ and ij. 27. If n denotes the number of sides, li the radius, and C the cen- tral angle of any regular polygon, prove that ir= nR^ sin ^ C cos ^ 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. 6 = 8.42, c = 11.26. 29. B = 84°, a = 100. 33. B = 39.24°, c = 23.68. 30. A = 18.62°, 6 = 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. jS'= 1200, ^=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 B. 54. Given n = 11 and ^= 1000, find r, c, and B. 55. Given n = 9 and £"= 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 ^ = 60°. 61. a = 80 and c = 160. 58. 6 = 600 and c = 1200. 62. 6 = 40 and c = 40 V2. 59. a = 26.3 and 6 = 21.2. 63. c = 4000 and ^ = 30°. GO. B = 60° and . T' \ T, ^ — r' Fig .38. Fig .39. Similarly (in Fig. 39), if E1E4 is tangent to the circle at the point B, cot Z AOPi = tan Z BOP^ = ^ = ^ = BRi; OB i cot AAOP2 = BR, ; cot AAOPb'= BR^ ; cot ZAOP4 = BR^ ; 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 OTi, OT^, OT3, OTf-, 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 repre- sented by ORi, OR2, OB3, ORi ; 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 B B The lines which represent the various trigonometric func- tions 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 hnes, than by the use of the ratios which the hnes 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. M^P^, M^P^, M^P^, MiP^ (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 iViPi, JV2P2, N^P^, N^Pi are +, — , — , +, respectively; and in Fig. 38 the tangent lines AT^, AT^, ATg, AT^ are +, -, +, —, respectively. Since the sign 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 BR^ BRi, BRs, BBi are +, -, +, -. The secant is considered as plus when it is drawn in the same direction from the center as the terminal radius (thus OTj,, Fig. 38, is opposite in direction from OP2 and is therefore negative). Hence the secant lines OT^, OT^, OT^ OTi have the signs +, — , — , +, respec- 80 TRIGONOMETRY tively. Similarly the cosecant lines (Fig. 39) ORi, OB2, OR3, OBi have the signs +, +, —, — . The results thus obtained may be arranged in a table as follows : I II Ill IV sine and cosecant + + - - cosine and secant + -■ - + tangent and cotangent + - + - EXERCISE 23 In which quadrant is each of the following angles ? '1. 123°. 6. 415°. 11. 1111°. 2. 155°. 7. -18°. 12. -222°. 3. 215°. 8. -125°. 13. -1826°. 4. 285°. 9. 612°. 14. 2625°. . S. 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 co- terminal with : 17. 25°. 18. -30°. 19. 100°. 20. -100°. Find the smallest possible angle coterminal with : 21. 425°. 23. -300°. - 25. -1760°. 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 see 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 horizon- tal, 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' [23.6°] and the other a slope of 32° 45' [32.75°], how wide is the base ? 35. Make up a similar example for yourself. •^„ FiQ. M. 58. Functions of 0°, 90°, 180'', 270^ 360°. In Arts. 34 and 35 it is shown that sin 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 to 1 ; in the second quadrant it decreases from 1 to ; in the third it decreases from 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, — 1, 0. Similarly in the first quadrant (Fig. 45) the cosine decreases from 1 to ; in the second quadrant it decreases from to — 1 ; in the third quadrant it increases from — 1 to ; in the fourth quadrant it increases from to 1. Hence the cosines of 0°, 90°, 180°, 270°, 360°, in order, are Ij 0, — 1, 0, 1. Fig. 45. Similarly from Fig. 38, or from the formula tan x = ?lE£j it is clear cosx that the tangent in the different quadrants changes from to oo ; from — 00 to ; from to oo ; from — oo to 0. Hence the tangents of 0°, 90°, 180°, 270°, 360°, in order, are 0, ± oo, 0, ± oo, 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 cot = — ; sec = — ; tan cos csc = - Thus, sec 180° = cos 180° 82 TRIGONOMETRY Obtaining the values of the required functions thus and arranging all the results obtained in a table, we have 0° 90° 180° 370° 360° sin 1 -1 cos 1 ■ -1 1 tan 00 00 cot CO CO 00 sec 1 03 -1 CO 1 CSC CD 1 00 -1 CD In thq 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. 6 sin 90° + 7 cos 180° + 8 sin 30°. v^. msin0°+i)cos90° + ccot360°. 3. 6 cos 90° - c tan 180° + b cot 270°. 4. {a' - (f) cos 180° + 4 ac sin 90°. 5. 2 tan 0° sin 90° - 4 sec 0° sin 270° + 5 esc 90° cos 0° cot 270°. .€. a cos 180° sec 360°- 6 tan 180° sin 270°- a sin 90° sec 0° + & sin 90° cos 270°. 7. m sin 270° esc 90° + n cos 180° esc 270° cot 270° - m sec 180°. ja. 6 m CSC 90° cos^ 0° - 17 w sec^ 0° cot^ 270° + 3 m sin 270°'sec 360°. 9. Show that 4 cos^ 45° sec 0° + 6 tan^ 30° sin 270° + 12 cot^ 45° cos 180° - 4 tan'' 45° esc 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 trigono- metric 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 concern- ing the function of an acute angle are true for the functions of any angle. Thus, on each of the Figs. 40-43, MP' + OM^ = 0P\ ' That is, sin^ x + cos^ a; = 1. Also in each quadrant the A OMP, OAT, OBR are simi- lar. :. AT: 0A = MP : OM, or tan a; : 1 = sin a; : cos a;, sin a; or ■ tan x = cos a; Let the pupil prove in like manner, 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. P 4- ord. P = dist. P , . /abs.PV / ord. P y^^ ■ ■ l^dist. p) Vdist. P) Hence, sin^ x + cos' x = 1. Let the pupil prove in a similar manner that ^'' tan' x + 1 = sec' x, and cof x + 1 = esc' X. : ord.P ,1 . ord. P dist. P sin x . sin x Also tan.'e = = = ,ovtanx= . abs. P abs. P cos x cosx dist. P Also ord. P j^ dist. P ^^ abs. P dist. P ^ j^ I ord. P abs. P ^ ^ . ' dist. P ' ord. P ' dist. P abs. P ' abs. P ord. P ' ' or sin x x esc x = 1, cos x x secx = 1, tan x x cotx= 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 a; = — f, find the other functions of x. By the table of signs (Art. 57) a negative cosine occurs in both the second and third quadrants. 2c? quadrant. sin x = Vl — (f )^ = Vl — if = V^ = f , tana; = sin X = — f , etc. 3c? quadrant. sin x = Vl — (-f)" = V^ = . 3 tan X = - cos X — f 2 = 1, etc. Ex. 2. Given tan x = 2, find the remaining functions of x. The positive tangent occurs (see Art. 67) in both the first and third quadrants. 1st quadrant, sec^ x = l + tan^ a; = 1 + 4 = 5, sec x = VS, cos x = 3d quadrant, sec^ x = = — — = - V6, etc. sec X -y'S 5 : 1 + 4, sec a; = — VS, : — = — - VB, etc. -V5 5 In case solutions are sought by the geometrical method, the follow- ing figures may be used in Exs. 1 and 2 respectively. Fig. 46. GONIOMETRY 85 EXERCISE 25 yl. 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°. Find the values of sin, cos, tan, and cot of /3. 765°. 5. -330°. 7. 750°. 9. 2205°. 4. 1860°. 6. -675°. 8. -1740°. JIO. Given cos aj = — |, find the other functions of x. 11. Given tan x= — J/-, find the other functions of x.' ,12. Given sin x= —^, find the other functions of x. 13. Given cot x = 2 and sin x negative, find the other functions of x. 14. Given sec 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 posi- tive. 17. Verify geometrically the results obtained in Exs. 10-16. 18. Given cot y = f VS and cos y negative, find sin y and esc y. 19. Given tan a; = — |- V3 and cos x positive, find the other func- tions of x. 20. If 6 is in the second quadrant and if cosec 6 = -i^, find the value [c cot 6 + sea 6 tan 6 + cos 6 21. Find the value of °^ -, if 6 is in the fourth quadrant and tan 6=— ^-. esc 6 -H sec d 62. Trigonometric Functions of 90° 4- a; in terms of func- tions of a?. The trigonometric functions of 90° -fee may be reduced to functions of x by use of the following formulas : sin (90° +x') = cos x. cot (90° -f a;) = - tan x. cos (90° -I- X) = — sin as. sec (90° +x')=— esc x. tan (90° -f ac) = - cot x. esc (90° + x) = secx. 86 TRIGONOMETRY Fig. 48 a. For, let Z AOP (Fig 48 a) be any angle x in the first quadrant. Let POQ P be a right angle. Let OP = OQ=i- A Then Z RQO = /.MOP. {sides ±) .-. ARQO = AMOP- {hyp. and acute Z = ) :. sin {90° + x) = BQ = OM= cos x. cos (90° + x) = OR= -MP= - sin x. tan (90° + x) = ^^^-g^ = -^^^^ = - cot :.. _ ' cos (90° + a;) -sinx Let the pupil supply the proofs for cot (90° + x), sec (90° + x), and CSC (90° + 4 The same results may readily be obtained for angles end- ing in the second, third, and fourth quadrants by use of the following diagrams. Fig. 48 6. Fig. 48 e. Fig. 48 d. Ex. 1. Find the value of sin 300°. sin 300° = sin (90° + 210°) = cos 210° = - sin 120° = - cos 30° = -iV3. Ex. 2. Eeduce 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. cot840°-3tan420°+2sec480°. j ,'' 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°. 2.1. 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°. 70° sin 610' - sin 330° tan 225° cos 390° = 0. tan^ 135° sin 1890° + 8 cot 45° cos 1140° + esc 630° tan 225° cos 720° sin 1830°. Sl.'^rove tan 45° sec 1080° cos 570° sin 610° 32.'^ind the value of 6 sec^ 1080° tan^ 135° sin 1890° Simplify the following expressions : 33. 5 sin (90° + a;) - 6 cos (180° + x). 34. a sin (90° + a;) + 6 cos (270° + a;) - c tan (180° + a;). 35. p sin (180° + X) cos (180° + a;). 36. (a + 6) sin (270° + a;) - (a - 6) cos (270° + a;). 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 ( — «;) = — sin oc. cot ( — £c) = — cot x,. cos ( — x) = COS 3C. sec (—!«) = sec 3C. tan ( — x) = — tan as. esc ( — ac) = — csc a?. 88 TRIGONOMETRY For let ZAOP (Fig. 49) be a positive angle, x, and Z AOQ an equal negative angle. Let OF = OQ = 1- Then the right triangles OMF and OMQ are equal. Hence, sm ( - x) = MQ = - MP = - sin X cos ( — x) = 0M= cos X sin {— x) — sin £C tan {— x) = cosx Fig. 49. COS {— x) = — tan X. Let the pupil supply the proofs for cot ( — x), sec ( — x), and esc { — 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)' = -cos45° = -^V2, Ans. Ex. 2. Simplify cot (180° -A). cot (180° - ^) = - 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 a;. cos (180° -X) = — cos X. tan (90° -X) = cot X. tan (180° -X) — — tan X cot (90° -X) = tan X. cot (180° -X) = — cotx sec (90° -X) = CSC X. sec (180° -X) = —sec a; CSC (90° ~x) = seccc. CSC (180° -X) = CSC a; 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° ± a; or 360° ± a; 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° + a; is an angle in the third quadrant and therefore sin (180° + a;) is negative. But x mean- time would be an angle in the first quadrant, hence sin x would be positive. Hence, in general, sin (180° + a;) = - sin x. Let the pupil show in like manner that, by the above rule, sin (360° — a;) = — sin aj ; also that sin (270°— a;) = — 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 deter- mining 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° -f a; on the diagram, it is an advantage to construct the 180°, or 270° first, beginning with the initial line, and then to annex the angle x to the 180°, or 270°, after it has been constructed. Thus, to prove that tan (270° + a;) = —cot a; when X is an angle in the second quadrant (i.e. an obtuse angle) we first take (Fig. 60) the positive angle AOB^ (270°) and annex to it ZB'OP' (=x or Z^OP). Then * 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) = BE. But ZB'OT = ZAOB (construction) Subtracting 90° from each of these angles we have Z AOT = Z BOP. .: A AOT = ABOP. (leg and acute Z =) .-. AT = BB, in absolute magnitude. (liom. sides of= A) .•. tan (270° + x) and cot x are equal in absolute magnitude. But ^Tand BR are opposite in sign. .-. tan (270° + x)= — cot x. Similarly, to prove sin(270°— «)=— cosk when X is an angle in the second quadrant (Fig. 61) we take Z AOB' (270°) and from it deduct ZB'OP'(=ZAOP or a;). Hence, sin (270°-a;)=il!fP', while cosa; = JVP. Since A OMP' = A ONP, MP' and JVPare equal in absolute magnitude. They are also opposite in sign. .-. sin (270° — a;) = — cos x. F^ ^~^v p/\ \ ^ ^ 1 \ r Fig. 51. EXERCISE 27 Find the numerical value of : l.^n(-225°). 4. cot (-210°). 2.M;an (- 300°). 5. tan (-600°). 3. cos (-120°). 6. sin (-900°). 7. sec (-240°). a tan (-150°). 9. sin (-135°). Eeduce the functions of the following negative angles to the functions of positive angles not greater than 46° : 10. -119°. 13. -16°. 16. -900°. ll.V-81°. 12. -195° 14. -253°. 17. -216° 43'. 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°) = |. By the general rule stated in Art. 64 reduce each of the following to a function of x : 21. cos (180° + a;) . 22. sin (270° + a;). 23. cos (270° - a;). 24. tan (180° + a;). 25. sec (180° — x). 26. CSC (270° + a;). 60NI0METRY 91 Simplify the following expressions : 27. 5 sin (90° - a;) + 8 cos (180° -x). 28. a sin (270° -x)-b cos (270° -x)+c tan (180° - x). 29. m cos (180° + ^) + p cot (180° - ^) + g tan (270° + A). 30. sin (270° + x) cos (270° - x) sin (180° - x). 31. sin (a; - 90°) + cot (a; - 90°) + tan (a; - 180°). 65. General Solutions of Trigonometric Equations. If there be no limit to the size of an angle, an indefinite num- ber 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 w = or any positive integer. Ex. 2. Solve tana; =±V3- ,^ (60° ±«(360°),120°±«(360°), , ' \ 240° ± n (360°), 300° ± n (360°). Ex.3. Solve sin^ a; = cos^ a;. 1 — cos^ X = cos^ X. 2cos^a; = 1. cos CB = ± 1 V2. r46°± n(360°), 316° ± n(360°), |l35°± n(360°), 225° ± w(360°). Or more briefly, a; = ±w(180°) ± 45°. Ans. The pupil should observe that the values of a; 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 6. 10. 2V3cot«-f csc^e=l. 11. tan ^ + sec^ e = 3. 12. cos''fl + cot2^ = 3sin2e. 13. 1^ cot fl — cos ^ + sin tf = ^. 14. sec" 6 csc^ e + 2 csc^ 6 = 8. 15. 2 V3 tan = 3 sec^ 6-6. 16. 4sec20-7tan''«=3. 17. cot e + 2 tan e = I sec 6. 18. sin 6+-V3cos6 = 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., ^C= 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 B is a barn. If QR = 217 ft., Z PQB = 63° 40' [63.67°], Z PBQ = 58° 15' [58.25°], find the distance of the spring from the house and also from the barn, by solving right triangles only. ±. S^U X = f. 2. •'e0s^a; = f. 3. tan^a; = 1. 4. tana;=^ cot a;. 5. sin X + CSC X = ^. 6. tan^ a; — sec a; = 1. 7. 2 cos^ a; — 3 sin a; = 0. 8. tan a; + cot a; = 2. 9. cot a; + csc''a; = 3. CHAPTER V GONIOMETRY (Continued) 66. Formulas for sin (as + ^) and cos(!W + ^). In Fig. 52 let AOQ be an angle x, and QOF an angle y, the sum of X and y being less than a right angle. p Let OF = 1. Draw FM± OA, FQA.OQ, QE±FM. Then ZliFQ=Zx {sides ±), FQ = sin y, 0Q= cos y. sin {x + y) = FM= QN+FR. In rt. A OQN, QN= sinxOQ (Art. 41) = sin x cos y. In rt. A RFQ, FR = cos x FQ = cos x sin ?/. Hence, sin (pD + y)=- sin as cos y + cos a; sin y. Also on Fig. 52, cos(a; + y)= Oif = ON-RQ. In rt. A QiV, OiV= cos a; ^ = cos x cos ?/. In rt. ^RFQ, RQ=smxFQ=smxsiny. Hence, cos (oc + y) = cosx cos y — sin as sin y. If a; and y be acute angles whose sum is an obtuse angle, the above proofs will hold good without any change except that it is necessary to notice that in the statement cos {x + y) = 0M= ON- RQ, OM is a neg- ative line and is obtained by subtracting the positive line RQ from the smaller positive line ON. See Fig. 53. If either x ox y is obtuse, the above formulas may be proved as follows : 94 TRIGONOMETRY Taking x and y as still acute, sin (90° + x + y) = Gos(x + y) (Art. 62) = cos X COS y — smx sin y. But cosa; = sin (90° + x), - sin a;=cos (90° + a;). (Art. 62) .-. sin (90° + a; + y) = sin (90° + x) cos y + cos (90° + x) sin y. Replacing 90° + a; by x', sin {x' + y) = sin x' cos ?/ + 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. 6y 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° sin 30° = iV2.|V3 + iV2.i = iV2(V3 + l), Ans. 67. Formulas for sin (pc — y) and cos (x — y). In Fig. 54 let J. 0$ be a positive acute angle x, and POQ a, smaller angle y, subtracted from x. ThenZ^OP = a;-?/. Let 0P=1; draw PMl-OA, PQJLOQ, QNl. OA, PE± QN. Then Z RQP = Z a;. {sides ±). Also PQ = sm.y, OQ = coay. : , B\n{x-y) = PM=QN-RQ, In rt. A OQN, QN= sin x OQ = sin x cos y. GONIOMETRY 95 In rt. A RQP, 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) = 0M= 0N+ RP. In rt. A OQN, 0N= cos x 0Q = cos x cos y. In rt. A RQP, RP = sin x PQ = sin x sin y. Hence, cos {x — y) = cos 05 cos 1/ + sin a; sin 2/- By t. -^ same method as that used in Art. 66 these formulas can be pre .red true when x and y are any angles. Ex. Obtain the numerical value of cos 15°. cos 15° = cos (45° - 30°), = cos 45° cos 30° + sin 45° sin 30° = iV2.iV3 + iV2.^ = iV6 + iV2, Ans. 68. Formulas for tan {x + y) and tan {oc — y). By Art. 66, , , sin (x + v) sin x cos y + cos x sin y tan(x + w) = ) — -^ = ^ -. r-^- ^' cos (x + 2/) cos X cos J/ — sm x sm j/ Divide both numerator and denominator of the last fraction by cos X cos y. sin X cos y cos a; sin ?/ m, , , V cos a; cos w cos x cos ■?/ Then, tan (x + w) = ^ ■ -■ — - ^ ^' cos a; cos 2/ sm a; sm 2/ cos X COS 2/ cos a; cos y , ^ tan X + tan 1/ or, tan (x-\-y) = - — 7 — , . ^ ^ 1 - tan X tan y Similarly, let the pupil show that , , tan X — tan y tan {x-y) = — — , ^ l + tana5 tant/ J ^ , , cot a; cot 2/ T 1 and cot {x±y) = — :^ : — zQK.y± cot X 96 TRIGONOMETRY Ex. Find the numerical value of tan 105°. tan 105° = tan (60° + 45°) _ tan 60° + tan 45° — 2 - V3, Result. 1 — tan 60° tan 45° ^ V3 + 1 ^ 1+V3 1-V3.1 1-V3 / ' EXERCISE 29 1. If sin a! = ^, cos a; = f , sin 2/ = ^, cos y = ^, find the value of sin (a; + y). 2. Also of sin (x — y), cos (a; + y), and cos (a; — y). 3. Find sin (a; + 45°), cos (30° — a;), and sin (a; — &/) in terms of sin xalaA cos x. r 4. H tan x = \, and tan 2/ = 2, find the value of tan {oa-\-y\ 5. If cot X = —2, and cot y = ^, find the value of cot (a; — y). Findrthe numerical value of : * 6.'^30s 75°. a.'sin 105°. 10. sin 15°. 7. V 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°. Prowthe following identities : 17. cot (45? + A) = 99LA=zA. ^ ' 1 + cot^ 18. cot (45° -A\ = cot ^ + 1. ^ '^ cot ^ - 1 19.ym (60° + ^) - sin (60° -A) = sin A. 20. cos a; — sin a; = \/2 cos (a; + 45°). 21. cos a; + sin a; = V2 cos (a; — 45°). 22. Find the smallest value of x which will satisfy the equation tan (a; + 45°) + cot (a; - 45°) = 0. GOTSriOMETRY 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 a; cos a; + cos x sin x or, sin 2 ae = 2 sin x cos x. Similarly from the formulas for cos {x + y), tan {x + y), and cot {x + y), let the pupil obtain cos 2x = cos^ X — sin^ x. 2 tan a? tan 2x = cot 2 a? = l-tan^a5 cot* 05-1 2 cot 0? Substituting 1 — sin* x for cos* x in the formula for cos 2 a;, cos 2 as = 1 — 2 sin* as. Substituting 1 — cos* x for sin* x in the same formula, cos 2 a; = 2 cos* a? — 1. Ex. Find cos 120° from the functions of 60°. cos 120° = cos 2 X 60° = 008^60° -sin" 60° =(iy-av3)» EXERCISE 30 1. Given sin 30° = ^, and cos 30° = iV3, find sin 60°. Also cos 60°. 2. Given tan 30° = \ VS, 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^ = -^i^Ii^. 6. ?i5l^_22i2^ = seca;. 1 + tan^ A sin x cos x g ._ l-tan'^ ■ l + sin2g ^ (tane + iy . cos ^ i+tan"^' ■ l-sin26» (tan^-l)"' 98 TRIGONOMETRY 8. State the formulas for sin 2 x and cos 2 a; in general language. 9. Find sin 3 a; in terms of sin x. 10. Find cos 3 a; in terms of cos x. 11. Find tan 3 a; in terms of tan x. 12. Prove sin 4 5 = 4 sin d cos 6 — 8 sin' 6 cos 6. 13. Given tan fl = |, find tan 2 6. 14. Given cos 6 = f , find cot 2 6. In a right triangle, C being the right angle, prove : 15. tan B = cot A. 16. tan 2^= ^"^ . 17. sinM-.B) + cos2^ = 0. b^ — aF -_ cjt. i.1, i. • 2 1 — cos 2 a; j ■ m 1 — cos 4 a; 18. Show that sin'^ x = , and sin'' 2 a; = . . _ oT, J.1, J. 9 1 + cos 2 a; j ,0 1 + cos 4 x 19. Show that cos'' x = — ■ — , and cos^ 2 a; = — ■ — . 20. Using the results of Exs. 18 and 19, transform sin* x into \ cos 4 a; — ^ cos 2 a; + f . 21. Also transform cos* a; into an expression in terms of cos 2 x and cos 4 X. 22. Also show that eos^ x may be changed to the form tV (5 + 8 cos 2 a; — 2 sin^ 2 x cos 2 a; + 3 cos 4 a;). 70. Functions of the Half Angle. From Art. 69, cos 2 ^ = 1 - 2 sin^ A. Hence, 2 sin^ A=\- cos 2 A. Let A = \x; then 2 J. = x. Hence, 2 sin^ \ x— 1 — cos x. . • 1 . .» /l — cos 3C Similarly, from cos 2^ = 2 cos^ A — 1, we obtain, . cos ^ a; = ■± yjl±^21^. GONIOMETRY 99 AT i. 1 sin 1 a; ^ 1 — cos Also tan4x = 2_=-j-\^^ cos^a; 1 + cosa; =^4 i. 1 ™ , -./■'-"COS X tan ix = ± \ 1 + cos X This formula may be reduced to another convenient form, thus : tan ^x = J (1-cosxr ^^ (l-cosx)- ^j_-cosx_ (1 + cosa;)(l — cosa;) 1 — cos a; smx 1 1 — cos X tan ix = sin X Similarly, cot i x = i±50S^. sin 05 Ex. Find tan 221° from the functions of 45°. , „ „„,„ 1 - cos 45° 1-J-V2 2-V2 ^ ^ ^ EXERCISE 31 1. State the formulas for sin | ^, cos ^ A, and tan ^A in general language. 2. Given cos 30° = i-VS, find sin 15°, tan 15°, cos 15°. 3. Given sin 46° = iV2, find cot 22^°, cos 22^°, sin 22^°. 4. Given cos 90° =0, find the functions of 45°. 5. Given sin A= }, and A acute, find cos ^ A, cot ^ A, tan ^ A. BBS 6. Given cos 6 = a, find cos -, cot -, tan -■ Z Ji A Prove the following identities : 7. tan|^ = ^-5HLA_. 9. sec2^ = . 1 + cos ^ 2 sec e + 1 8. cot \A = ^-^i^- 10. csc^ ? = -2^?^ . ^ 1 — cos ^ 2 sec ^ — 1 11. sin 1^^ + cos ^ ^ = VI + sin A. 12. Express cos A, sin A, and cot A, in terms of cos 2 A 13. Find the value of 2 — ± if a; is in the second quadrant cot ^ 0! + cos a; and sin a;=f. 100 TRIGONOMETRY 14. If X is in the fourth quadrant and esc a; = — |, find the numerical value of sini^±sec^. cot ^ a; + cos x 15. In a right triangle show that tan ^A = -v— 5 + 6 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 (Log- arithmic Formulas). Adding and subtracting the formulas of Art. 66, and also those of Art 67, sin {x + y) + sin {x — y) = 2smxcosy . . . (a) sin. (x + y) — sm.(x — y) = 2cosxsmy ...(b) cos {x + y) + cos (x — 2/) = 2 cos x cos y ...(c) cos (x + y) — cos {x — y)= —2 sinxsiny . . . (d) If we let x + y = A, and x — y = B, then a; = 1(A + B), and y = ^(A- B). Hence, by substitution in (a), (&), (c), (d), sin^ + sin^ = 2sin^(^ + -B)cos|-(^--B) ... (1) sin.4-sin^ = 2cos^(^ + ^)sinl(^-i) ... (2) cos^ + cos-B = 2cosi(^-f ^) cos^(^-^) . . . (3). cos4-cos^= -2sin|(^ + ^)sin|(^-J5) . . (4) These formulas enable us to convert the sum or difEerence 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 60° + sin 30° = 2 sin ^(60° + 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° , ^ qo ,. sin 5x + sin x , r, *• ST^s-^ ^ = tan 18 . 6. — 21 = tan 3 x. cos 33 + cos 3 cos 5 a; + cos x g cos 27° + cos 3° ^^go cos 80° + cos 20° ^ - ■ sin 27°+ sin 3° ' " sin 80° - sin 20° ^^• 3 sin^ + sinB ^_^^^^^_ cos A — cos B - cos 4 a; + cos 2 a! , „ 9. — — - — ^^—. = cot 3 X. sin 2 a; + sin 4 x , „ sin J. — sin 5 .A + B 10. = — cot — ^ — cos A — cos B 2 11. cos 20° + cos 100° + cos 140° = 0. ,_ • , • o , • t; sin^3a; 12. sin a; + sin 3 a; + sin 5 a; = — -. • sin a; 13. Given sin J. = ^ 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°. Eeduce each of the following to a form adapted to logarithmic com- putation (that is, to products or quotients) : sin 37° + sin "22° ^^ sin 4 ^ - sin 2 A cos 38° - cos 16°' ■ cos 6^ 17. sinM-sin^S. 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 re- lations 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 ^~.^"^^^ = tan A. smZ A 1-0--2 sin^ A) ^ sin A 2 sin A cos A cos A 2 sin^ A sin A 2 sin A cos A cos A sin A __ sin A cos A cos A Or if the teacher prefers, the proof may be put in the following form : l-cos2^ ^ 1-(l-2sin'^) ^ 2sin^^ ^ sin^ ^^ ^ . sin 2 A 2 sin A cos A 2 sin A cos A cos A Ex. 2. Prove sin (A + B) sin {A-B) = sin' A - sin^ B. (sin ^ cos B + cos A sin B)(sm AcosB — cos A siuB) = sin^ A — sin* B. sin^ A cos^ B — cos^ A sin* B = sin* A - sin* B. sin* ^ (1 - sin* B)-(l- sin* A) sin* B = sin* A - sin* B. sin' A — sin* A sin* £ - sin* B + sin* ^ sin* B = sin* ^ - sin* B. sin*^— sin*S=sin*^-sin*5. . 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 ^ + 5 + C = 1 80°, prove that sin A + sin B + sin C = 4 cos 1 A cos ^ B cos ^ C. A + B = 180''-C a,ndi{A + B) = 90°-ia GONIOMETRY 103 Hence sin i(A + B) = sin (90° - 1 0) = cos | G. sin J. + sin J5 + sin = sin A+smB + sin [180° -(A + 5)] = sin J. + sin B + sin {A + B) = 2 sin^ (^ + B) cos ^ (J.-5) + 2 sin i (^ + B) cos i (-4+ B) (Arts. 69, 71) = 2 sin \ (^+B)[cos \ (A-B)+coa | (A+B)^ = 4 cos I Gcos ^ J. cos 1^ -B. EXERCISE 33 Prove the following identities : ^ cos g + sin g _ sin 2 g + 1 cos 6— sin 6 cos 2 6 2. 2 cos (45° + ^- A) cos (46° - } ^) = cos A 3. cos {A + B) cos {A — B) = cos^ B - sin^ A. 4. tan (45° + a;) — tan (45° — a;) = 2 tan 2 x. 5. ( Vl + sin a; — VI — sin xf = 4 sin^ ^ x. g cos (x + y) + cos (x — y) _ cos (x — y) — cos (a; + y) _ cos a; cos y sin a; sin y 7 tan (45° + j ^) + tan (45° -iA) _ ^„^ ^^ tan (45° + i -4) - tan (45° - ^ ^) _ cos 3 A , sin 3 A „ t. n a 8. — : 1 = 2 cot 2 A. sm A cos A - cos A — sin J. o ^ J. o ^ 9. : — 7 = sec 2 A — tan 2 A. cos ^ 4- sin A 10. tane- sine + sin2e 11. l+cose + cos2e cot e - 1 1 - sin 2 e cot e + 1 cos 2 e ,„ 1 — tan^Aa; 12. — ^4- = cos X. 1 + tan'' ^ X li A + B + = 180°, prove that 13. cos J. + cos B + cos C'=l + 4sin-|-^sin|-Bsin^C 14. tan ^4- tan B + tan 0= tan ^ tan B tan C. 15. cos (^ + B - (7) = — cos 2 C 104 TRIGONOMETRY EXERCISE 34. REVIEW 1. Given cosfl = — f and is in the third quadrant, find esc 5, cot e, sin i e, tan (180° - 6), sin (- ff). 2. Given tan ^x = ^ (and x acute), find sin x. 3. Given sin 2 a; = -^ V3, find cot ^ x. 4. Given cos ^ a; = |, find sin 2 x and tan 2 a;. 5. Given cot 30°= VS, find cos 15°, esc 15°, and tan 15° 6. Given sin -4 = | and A acute, cos B = \ and B acute, find (a)Hm{A-B); (h) cos {A+B); (c) cos(^-B); {d)sva.2B; (e)cos25; (/)tan 2jB; (y) cot 2 A; (h) tan (^--B); (i) cot (A + B); (J) cos^B. 7. Given cot $ = —2 and is the second quadrant, find (a) sec ^; (b) tan (180° - 5) ; (c) cot (180° + 0) ; (d) cos (- 0). 8. Pind sin, cos, tan, cot, of : (a) («= - f ) ; (6) ('^ - ^) ; (c) (*« - t) ' ^'^ ^'^ "^ "'^ ' ""^^'^ " "" ^^^''' Prove the following : „ , 1 — cos2a; ,. sina; + sin2a; +„„ „ 9. tan a; = : — 12. ■ ^— = tan x. sin 2 a; • l + cosaj + cosia; 10. tani^ = i^^. 13. !iB_yi§=tan^ + tanS. '• smA cos -4 cos -B 2 sin ^ — sin 2 J. _ 1 — cos ^ ^^ sin 21° + sin 5° _ ^^^ ^j, 2sin^ + sin2J. 1 + cos^ ' cos2r + cos5° 15. cos9g + cos5g + cosg^^.^^gg sin 9 e + sin 5 e + sin 16. cos" X tan' x + sin' x cot' a; = 1. 19. tan X + cot a; 4- 1 _ 2 + sin 2 a; tan a; + cot a; — 1 2 — sin 2 x cos 75° + cos 15°^^ 20 cos2a; + l^_^^^,^_ sin 75° - sin 15° cos 2 a; -1 ^g sin ^ + sin .B ^ ^^^ ^ ,^_ ^. ^i. sin (a! + y) ^ cotx + coty cos £ — cos J. ^ ' sin (a; — y) cot y— cot a; 22. cos A = 2 tang + D + tang-l) 23_ sin^^±^lsin^izll = tan'a:-tan'2,. cos' X cos' 2/ 24. cos 5 a; + cos 3 a; = 2 cos 4 a; cos a;. sin 2 a; + 1 _ 2 tan x + tan' x + 1 25 sin 2 a; — 1 2 tan a; — tan' a; — 1 GONIOMETRY lO.*; 26. sin (46° + x)+ sin (46° — x) = V2 cos x. l + cot^/'E + a;') l-cot^f^-x\ 27. )p. Z = csc2a;. 28. _ )p Z=_sin2aj. l-cot^fE+x\ l + cot^Q-a;] „ 1 + cos X + COS 2 X _ sin a; + sin 2 a; _ cos a; sin x 30. cos 12 a; + cos Ga; + cos 4 a; + cos 2 a; = 4 cos 6 a; cos 4 a; cos 3 a;. ■sin a; -sin a; 31. tanf45°+|W!-^ \ 2 J ' 1 — sir 32. (sin X cos y — cos a; sin y)" + (cos a; cos 2/ + sin x sin y)'' = 1. 33. cos^ ^ a; (tan ^ a; — 1)^ = 1 — sin x. OSG -^ OOS fl 1 • • 34. Find tlie value of ^^——, — when cot = — -, and is in quad- sec e+sm.0 2 rant II. 35. Find the value of -^ when sin = — - and is in the oj J , cot ^ + sec ^ 6 3d quadrant. 36. Simplify cos 300° - cot f~ + 60°"\ + cot 150° - tan [" - -\ 37. Simplify sin 660° + tan f?j- - 60°"^ + cot 330° + cos (- 30°). 38. Simplify: (a - b) sin £ - (a + 6) tan 226° + (a^ + 6^) cot ^ - a cos (^=|^^ • 39. If tan 2d = ^, find tan B and sin &, 9 being in the 3d quadrant. p sin (A + B) _ tan A + tan B _ cot B + cot A sin (.4 — B) tan A — tan B cot B — cot A 41. If ^ is an angle in the second quadrant and sin A = ^, find the value of sin 2 ^ 4- cos 2 A. If ^ + 5 + C = 180?, prove : 42. sin A + sin B — sin (7= 4 sin ^ ^ sin ^^ S cos \ C. 43. cot \A + cot \ B +cot ^C=cot^Aoot^B cot ^ C 44. sin2^ + sin2S+sin2 C=4sin J. sinS sin C. 45. cos 2 A + cos 2B+ cos 2 (7= — (4 cos A cos 5 cos C+ 1). 46. tan .4 — cot B = sec A esc B cos C. 106 TRIGONOMETRY In a right triangle, C being the right angle, prove -A: « 47. sin2i£=£^. 49. tani , 2 2c 2 h+c 6 + c 2c ■ 48. (cos\A + sm.\A}\=^^^. 50. 008^-^ = V 2 2 y c 2 TT ■ • 1 ■ r> • !> 1 — COS 2 a! 1 1 + COS 2 a; Using sm X cos a; = - sin 2 a;, sm^ a; = , cos'' x = — ■ — , . f 2 ' 2 ' 2 ' transform : 51. sin^ X cos^ X into -1^(1 — cos 4 a;). 52. sin* X cos^ X into Jj(l — cos ix) — ^ sin^ 2 a; cos 2 aj. 53. sin'' X cos'* a; into an expression in terms of the cosines of even multiples of x. 54. sin* a; into an expression of the same general kind as in Ex. 63. 55. What nation first used the formula for sin^A? 56. What man discovered the formula for sin 2 ^ ? 57. Who first published the formulas for sin (A — B) and cos (A — B), and at what date ? CHAPTER VI OBLIQUE TRIANGLES TRIGONOMETRIC PROPERTIES OP 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, a Fig. 56. In Fig. 55 the angles A and B are both acute. In Fig. 56 the angle A is acute, and angle ABO obtuse. Let CD, denoted hj 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 = asmB; (Art. 41) .■.hsmA = asmB. (Ax. 1) In Fig. 56, in the rt. A ACD, p = b sin A; in the rt. A BCD, p = a sin (180° - Z ABC) = a sin Z ABC. (Art. 64) Hence in A ABC in both figures, hsinA = a sin B, or a:b = sin A : sin B. In like manner, b -.0 = sin B -.sin C, and a : c = sin J. : sin C. Or, collecting results, a _ & _ c sin J- sin 5 sin C 107 108 TRIGONOMETRY 75. Law of Tangents in a triangle. In any triangle the sum of any tioo sides is to their difference as the tangent of half the sum of the, angles opposite the given sides is to the tangent of half the difference of these angles. In a triangle ABC (Figs. 55 and 56), a : & = sin J. : sin B. (Art. 74) By composition and division, a + b _ sin A + sin B a — b sin A — sin B _ 2 sin^ (J. + B) cos ^{A - B) . 2 cos 1 ( J. + -S) sin ¥ (^ - -^) ' (Art. 71) Or, In like manner, a + & _ tan|(J. + ^) a-b tani(J.-^)* & + C t.an|(^ + C) tanl(5-C)' and b-c c + a tan^(C+A) c — a ta.n^{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), produce ^Oto D, making CD= CB oi a. On CB mark ofE CE = AC or 6. Draw the straight line DB. Then AD=CD+CA=:a + b. Also EB = CB — CE=a — b. Z.DCB, being an exterior angle of AA0E, = x + x — 2x. Also Z.DCB, being an exterior angle of A ACB, =A + B{oiA ACB). .■.2x = A+B (Ax. 1), or a; = ^ (A+B). Also, ZFAB = A-x = A-^(A + E) = i(A-B). „ Fig. 57. Also A ADF and EFB are similar (two A equal). OBLIQUE TRIANGLES 109 .-. Z. AFT) = Z EFB. .: AF± DB. In AAFD&uAEFB, DF:FB = a + b :a — b. In A AFD and AFB, ta.n x:ta,nZ FAB = — : — = DF:FB. AF AF By Ax. 1, a + h:a — b = tan x : tan Z i^'J.B . =tan^(^ + S):tan|(^-£). 76. Law of Cosines in a triangle. In the triangle ABC, Fig. 55, by geometry, a .2 _ &^ + c^ - 2 c X AD. But in the rt. /\AGD, AD = 6 cos A. .". a^ = &^ + c^ — 2 &c cos A. If J. is an obtuse angle, Fig. 58, by geometry, a^ = 6'' + c' + 2 c X AD. But inthert. A J.Ci?, AD = h cos Z CJ.Z) = &COS (180° - A) = .-. a^ = V + c^-2lccosA. Hence in either case, 2 he cos A = V + & - a\ h cos A. or cos -4. = h'+c? a 2 he In like manner it may be proved that Fig. 58. COS -B = COS C ■■ 2ab 77. Formulas derived from the Cosine Formula. The for- mula 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 compute)! 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: 2cosnA = l + cosA = l+^±^r=^ 2 be ^ 2 bc + b^ + c' - a' ^ (b + cY -a" 2 be 2 be _ (b + c + a)(b +c — a) 2 be Let 2s = a + b + c; then, subtracting 2 a from each memher, 2s — 2a = b + c — a. Hence, 2 cos^ 4 ^ = 2 s (2s -2a) ^ 2 6c -. . j(s or cos ^ ^ he In like manner, Also from Art. 70, 2 sin' i J. = 1 - cos ^ = 1 - ^' + c'-«' ^ 2&C ^ 2bc-b^ - side adj one A. 2. side opp. A < side adj no A. II. If given Z yl 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.) x {sin given /.)... two A. (2) side opp. = {side adj-) x {sin given Z) . . one right A- (3) side opp. < {side adj.) x {sin given ^) . . . . no A. In practice, the cases of no solution and of one right tri- angle or one isosceles triangle as the solution do not often occur. Hence we usually need merely a method of discrimi- nating between the cases where one oblique triangle or two 124 TRIGONOMETKY oblique triangles form tlie solution. We may state this test in the form of question and answer thus : Q. In geneial, 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, hoiv may the two triangles he con- structed ? 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 that the vertex of the given angle comes at the left end of the unknown base. Thus given ZC=38°,&=152,c=103,wehave 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 /.: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, & = 48.5, 4 = 21° 31', solve the tri- angle. OBLIQUE TRIANGLES 125 Since the side opposite the given angle, 84, is greater than the other given side, 48.6, there is but one solution. sin B . 48.5 84' •.sinS sin 21° 31' 48.5 sin 21° 31' 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 9.32686 - 10. C=180°-(^ + B) = 146° 15' 27". By Case I we find c = 127.211. Ex. 2. a = 22,b = Bi,A = 30° 20', solve the triangle. Since the side o opposite the given angle A is less than the other given side (A being acute, and 22 > 34 sin 30° 20') there are two solu- tions to the given triangle. In this case it is well to draw the smaller triangle separately as well as the general figure. C= C'= Fig. 67. By the law of sines (Art. 74), sin B ^34 sin 30° 20' ~ 22' 34 log 1.53148 30° 20' log sin 9.70332 -10 22 log 1.34242 colog 8.65758 - 10 B = 51° 18' 27" log sin 9.89238 - 10 .-. on Fig. 67a, B'=180°-51° 18' 27" = 128° 41' 33". F16. 67a. sin B = 34 sin 30° 20' 22 To complete the solution of AACB, ZACB = 180° -{ZA + ZABG) = 180° - 81° 38' 27" = 98° 21' 33". Hence by Case I we find c = 43.098. To complete the solution of A AC'B' (Fig. 67a). C" = 180°-(J.-|-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 o = 22, 6 = 34, J. = 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. 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 Fig. 68a. sin B = 34 sin 30.33° 22 To complete the solution of AACB, ZACB = 180° - (30.33° + 51.32°) = 98.35°. Hence by Case I, obtain c = 43.1. .-. ZB' = 180° - 51.32° = 128.68°. To complete the solution of AACB' (Fig. 68tt), 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 con- struct a figure for each example, lettering it according to the method specified in Art. 82 : 1. ^ = 30°, 6 = 50, a = 60. 2. J3 = 30°, a = 100, & = 70. 3. C = 45°, a=60, c = 60. 4. ^ = 60°, 6 = 12, a = 10. 5. C=80°, 6 = 16, c = 15.5. 6. 5=54°, a =23, 6 = 36. 7. C = 30°, a=18, c = 9. 8. 5 = 50°, a = 50, 6 = .37. 9. ^ = 75.16°, c^l8, a = 17.6. Using five-place tables, solve the following triangles, having given: ,10. .4 = 38° 18', 6 = 120.6, a = 138.7. { U. ^ = 61° 18', c=23.7, a = 21.25. OBLIQUE TRIANGLES 127 12. G = 104° 13' 48", 6 = 115.72, c = 165.28. 13. .8 = 22° 22', a = .6728, 6 = .81434. 14. ^ = 47° 19', a = 100, c = 120. 15. B = 15° 30' 12", a = 1200, 6 = 590. 16. C = 78° 18' 18", tt = . 26725, c = . 37926. 17. jB = 26°18'36", a = 28.604, 6 = 12.678. 18. A = 131° 18' 24", a = .8888, c = .4128. 19. = 31° 31' 15", 6 = 11.111, c = 8.267. Using four-place tables, solve the following triangles, having given; 20. B = 32.37°, 6 = 126.6, a = 138.7. 21. ^ = 67.366°, c = 22.7, a = 20.672. 22. 5 = 105.273°, 6 = 306.72, c = 241.8. 23. C = 26.223°, a = 66.35, c = 82.59. 24. 5=14.3°, a = 20.17, 6 = 17.8. 25. ^ = 22.37°, c = 300, a = 200. 26. 5 = 63.31°, c = 7.67, 6 = 9.54. 27. C = 49.31°, 6 = .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. ABBA 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 ZB be acute ; in Fig. Q%, let ZABC be obtuse. c c 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. 69o, in the rt. a CBD, p = asm B. (Art. 41) In Fig. 696 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=^ac sinB. In case the given parts are a, b, C, or 6, c, A, let the pupil state what the formula becomes. Let the pupil also state these formulas in general language. Ex. 1. J. = 66° 4' 19", b= 21.66, c= 36.94, find the area of the triangle ABC. By the formula K = ^bc sin A, K= \{21M X 36.94 x sin 66° 4' 19"). .-. log K= log 21.66 + log 36.94 + log sin 66° 4' 19" + colog 2. 21.66 log 1.33566 36.94 log 1.56750 66° 4' 19" log sin 9.96097-10 2 colog 9.69897 - 10 Area= 365.682 log 2.66310 Ex. 2. Given A = 66.07°, h = 21.66, c = 36.94, find the area of the triangle ABC. OBLIQUE TRIANGLES 129 By the above rule, K=^ (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 1.3367 36.94 log 1.5675 66.07° log sin 9.9610 - 10 2 colog 9.6990 - 10 ^»-ea = 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. ^ g sin C a sin C a sin (7 _ ■'■ ^ ~ sin A ~ sin [180° - (5 + C)] ~ sin {B + C) Substituting this result in the formula for K in Art. 83, j-_ a^ sin B sin C ~ 2 sin {B+€) ' 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 = -Vsis — 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 ol 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, 6 = 21.6, C'=,36°18'24" [36.31°]. 2. a = .86, B = 52° 18' [62.3°], C = 66° 42' [66.7°]. 130 TRIGONOMETRY 3. a = 18, 6 = 14, c = 24. 4. 6 = 200, c = 150, A = 72° 18' 30" [72.31°]. 5. 6 = 600, A = 18° 26' [18.43°], = 31° 44' [31.73°]. ^6. 6 = 14.7, a = 18.6, A = 74° 18' [74.3°]. 7. a = .8167, b = .68256, c = .72623. a a = 100, c = 125, 5 = 170° 16' [1 70.27°]. 9. & = 62.8, c = 47.2, A = 60°. 10. Given ^ = 29° 32' 16" [29.54°], &=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 parallelogj^m 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 deter- mine 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 viewecf from G. 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, DAG is the angle of depression of G as viewed from A. 89. I. To determine the Height of an Accessible Object above a Horizontal Plane. In Fig. 71 let AB be the object whose altitude is sought, and EF the o. horizontal plane, and G the point of y observation. X In the right triangle ABG, what / line shall we measure ? What angle ? / EG How then can AB be computed ? fig. 7i. 131 Fig. 72. 132 TRIGONOMETRY 90. II. To find the Distance on a Horizontal Plane to an In- accessible Object whose Height is Known. In Fig. 71, let AB be the inaccessible object whose height is known ; let SF be the horizontal plane and C the position of the observer. In the right triangle ABO, what side is known ? What angle can be measured ? How then can BC he computed ? 91. III. To determine the Height of an Inaccessible Object above a Horizontal Plane. Let AB, Fig. 72, be the altitude which is to be meas- ured, and JEF the horizontal plane. Place the transit in- strument at I) and measure the angle of elevation ABB. Measure the distance BC toward B, and measure the angle ACB. By solving the triangle ACB the line AC is found. By solving the right triangle ACB, AB is found. In case it is desired to compute AB by means of right tri- angles alone, the solution may be effected by dropping a per- pendicular CP from C to AB and solving the right triangles BCP, 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 BAB, BB = AB tan Z DAB, 'm^CAB, CB = ABtaiiZCAB. Subtracting, BB-CB, or DC = AB (tan Z BAB - tan Z CAB). BC Hence AB = tan Z BAB - tan Z CAB ' In case it is not possible to move directly from D toward B, we may proceed as follows: Measure Z ABB (F'm.'J^), PRACTICAL APPLICATIONS 130 Measure the line DO in the horizontal plane in any con- venient direction from D. Measure dBDC and DOB. Then in the triangle DOB, DB may be computed (How ?) . Afterward in the triangle ADB compute AB (How?). 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 A ADC and A CB, and the dis- tance DC, we may then compute AC (How ?). If we then measure Z BDF, we may compute z CAB (How?). Then AB may be com- FiQ. 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 meas- urements and computations are neces- sary in accordance with the figure. Fig. 75. 94. VI. To find the Distance between two Objects separated by an Impassable Barrier (and possibly invisible to each other). 134 TRIGONOMETRY Let it be required to find the dis- tance between A and B (Fig. 76), which are separated by a swamp or a mountain for instance. Take a sta- tion C from which both A and B are visible. Measure the angle C and the lines CA and CB. In the triangle 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 two islands off the shore CD). Measure the line CD and the ^ACD, BCD, ADC, BDC. In the triangle ACD, com- pute AC; in the triangle BCD, compute 5C; in the triangle ABC, commute AB. Fig. 77. 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 instru- ment 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 jingles 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 numer- ical 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 Pk- 1. The measurement of a base line AB (Fig. 78) at least 4 or 5 miles long, so accurately that the error shall not ex- ceed ^ of an inch per mile. 2. Tlie choice of a convenient station 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. 4. Proceeding in like manner from station to station till convenient points, C 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 with the result of the measurement. This final measurement of CD serves as a test of the accuracy of all the inter- vening vrork. 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 line, and observing from each end of AB the angle made by a line drawn to some convenient planet p. The distance of the planet may then be computed by Art. 93. The ratio of the dis- tance of the sun to that of the planet from the earth being Fig. 78. Fig. 79. 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 92,800,000 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 applications we will briefly indicate its method of use in navigation. If a ship should sail from R to B on the diagram (Fig 80), crossing each meridian at the same angle, for certain purposes the AARB {AB being the arc of a parallel of latitude) could be re- 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 par- allel of latitude comprehended between the two meridians. Thus, AB is a departure between PAP' and PBP'. Evi- dently 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 ES:&vc AB = OR:CA-=OA:CA = j^:l = 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 dis- tances (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 lormula ^^^ j^^^^ _ ^gp^^yiy^^g 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 67°. • 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 hori- zontal 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" [53.62°]. Find the height of the tower and its dis- tance 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 0, 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 60 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 12.6 and 12.8 ft. respec- tively, and their included angle is 62° 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 61° 16' [61.27°], 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., respec- tively, 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 V, on opposite sides of a river, a line TK and the angles TTK and TKT 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 TK on the prolongation of TT' through T, 89 yd. in length, and then take KP, 165 yd. in length, at right angles to KT. The angle TPT 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 north- west at the rate of 11.6 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 2? to a buoy B, dis- tances ILK and KP are measured to points K and P from which both rock and buoy can be seen, the distance RK being 2600 m., and KP being 3600 m. The following angles are then measured: ZJB^i2 = 38°48' [38.8°], Z5A''P = 75°64' [75.9°], and ZJSP/i =79°30' [79.6°]. 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 bill from the base of the tower ilO 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 is acted on by a force which gives a velocity of 1376 ft. per second along OA, and by another force which gives a velocity of 1135 ft. per second along OB. ZAOX= 30°, A BOX = 101°. What will be 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 fche 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 66° 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 complemen- tary. 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 embank- ment 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 B0= 121.6 yd. A point D is taken on the line AC produced through C, so that the angle BDG 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 miniites, 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 acciimulated. 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. A radian is the angle which, when its vertex is placed at the center of a circle, intercepts an arc equal to the radius of the circle. J.JQ 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 = , or, radius denoting the number of radians in an angle by p, the subtended arc by a, and the radius of the circle hy R, p= — . R Ex. 1. Find the number of radians ^'°- ^^• in an angle AOB whose arc is 13 and radius 5. We have, Z AOB = if = 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 = — , R n A 14 in. „ 14 in. t- ooj. • j 2.4= • .-. R = = 6.83+ m., Ans. R 2A ' 101. I. Converting Degrees into Radians. The number of radians about a point in a plane _ circumference radius 2nR ■ = 2 IT. R .: 360° = 2 TT, or 6.2832 radians, ^g^^^r^ ^^ ^^g^^ ^^^.^^^ 180° = TT, or 3.1416 radians. 4 90° = |, or 1.5708 radians. 30° = ^, or 0.5236 radians. 60° = ~, or 1.0472 radians. 1° = -^, or .01745 radians. 3 180 144 TRIGONOMETRY Hence to convert degrees into radians Multiply the given number of degrees by r-^ (or by .01745"^). Ex. 1. How many radians in 26° 17' 36"? 26° 17' 36" = 26.293+° = (26.293+) (.01745) radians. = 0.46882+ radians, Ans. Ex. 2. Simplify sin [Z +x sin(- + a;j = sin — cosa; + cos ^ sin a; (Art. 66) \6 / 6 6 = ■!■ cos a; + i-VS sin x, Ans. (Art. 33) Where the meaning is evident from the context, it is customary to abbreviate "it radians" into "tt." Thus also we abbreviate "sin^ 6 radians" into "sin-" and similarly for other expressions. 6 102. II. Converting Radians into Degrees. Since 2 it radians = 360° or 1 radian = 57.29579+° = 57° 17' 45" = 206265". Hence to convert radians into degrees 180° Multiply the given number of radians by (or 57.3°-). TT Ex. Convert 2.5 radians into degrees, minutes, and seconds. 2.5 radians = 2.5 x (57.2958°-) = 143.2395° = 143°14'22", ^ws. 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 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°N P^R ^""^ ^=K^- 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 = A (in radians) when ^ is a small angle, is an ap- proximation 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 = = no. radians in A approximately. 2000 ^^ ^ ... A = -i- X 206265" = 103+" = V 43", Ans. EXERCISE 42 ' 1. Keduce the following angles to circular measure, expressing the results as fractions of tt : 30°, 136", 60°, 90°, 210°, 270°, 225°, 72°, 315°. 2. Express the following angles in degrees : TT ir TT 2 ir iir Sir 1 ir 8 tt 6' 4' 3' T' "S"' T' T' 15" 3. What decimal part of a radian is 1° ? 16" ? 2' 15" ? 5° 14' ? 4. How 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 B. 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 ^ and their sum is 110°. Find the angles in degrees ; in radians. 9. Find both in radians and degrees the complement and supple- ment of the following angles : IT TT TT TT O IT 6' 3' 1' 9' 18" 10. Write out the trigonometric ratios of the following angles : TT TT IT TT 3 TT 7 TT 7 IT 6' 3' 4' 2' T' '¥' 4 ■ 11. How many radians in an angle whose arc is 12 and radius 10 ? How many degrees ? 12. Show that sin (aj + -^ ir) + sin {x—\ir) = sin x. Supply the two missing quantities in each of the following: 13 P a R A 3.5 10 in. 14 .35 50 in. 15 13 ft. 1 ft. 6 in. 16 43 in. rao' 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 - — 4 cos - tan - + 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 sliall 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 TEIGONOMETEIC FUNCTIONS 104. Graph of sin oc. To form what is called the graph of sin X use the equation y = sin x and also a pair of rectan- gular axes (see Art. 54). In the equation y = sin x, let x have convenient successive values and find the corresponding valiies 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 for I or .5236^ Thus, if we desire to make a graph of y = sin x we may take the following corresponding values of x and y : x = 0,y = 0. a; = |,2/ = ^=.5. a; = -|, y = - J = - .6. a; = |,2/=jV3 = .86- a; = -|, 2/ = - iV3 = - .86+ a; = ^,2/ = iV3 = .86^ a; = -^, 2/ = - jV3 = - .86- IT' ^~2-- • ^- ^ x = -^,y=\ = .5. x = — —,y = -\ = -.5. a; = TT, y = 0, etc. a; = — ir, 2/ = 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. i-T-rrT-T- |-+-+-+-4— |- -T-T-rT-T T rT-T"T-|-T-; -+-- f-+-+-+- 4--*— +-+-- 1-4—-|— +- i" T t t" 1 t ht |-t {-t" 1 1 1 1 1 1 p.<-f^TvJ J. 1 ' 1 1 1 . iB ,t-t t-t-T-r /fill l\,7r! 1 1 : 1 7\ X \i^ Q^^^j?P^^,ii_i^-i-. h-+-+-+-4-+- 1 1 1 *■ * r :.: i.:-..-i. l._i-i-I_i_t_l_i-j— J.-l.-i_i - Fig. 83. 105. Graphs of Other Trigonometric Functions- By treat- ing the equations y = cos x, y = tan x, y = sec x, etc, simi- larly, the graphs of the other trigonometric functions may be constructed. Fig. 84. It is important to observe in constructing the graph of For as we 77 tan x, that, as x= -, y = either -I- oc or — oc IT "r proceed from x = and make x= n, y = + az; but as we proceed from x = tt and make x = ^, ?/ = — oc. Hence we 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. 2/ = sin X. 9. y = tan ^ x. 2. 2/ = cos X. 10. y = sin x + cos x. 3. y = tan x. 11. y = sin x — cos x. 1. y = cot X. 12. y = Vsin a;. 5. y = sec X. 13. y = sin^ x. 6. y = CSC a;. U. y = l + sin a;. 7. y= sin J «. 15. 2/ = 1 — cos ». 8. y = sin 2 a;. 16. y = x + sin x. 106. Solutions of Trigonometric Equations. Answers not greater than 360°, i.e. than 2 tt radians. Ex. 1. Find the values of x less than 2 it radians which shall satisfy the equation sin x = ^. Since sin 30° = ^, and also sin 150° = ^, x = - or — ^ radians, Ans. 6 6 ' Ex. 2. Solve 4 cos a; — 3 sec 2; = for values of x less than 2 tt. 3 4 cos a; — . = 0. cos X 4 cos^ a; — 3 = 0. cos x= ±\ Vs. Hence, a; = 30°, 150°, 210°, 330°, or a! = -, -— , -—, -— radians, Ans. ODD 6 107. Answers Unlimited. Ex. 1. Solve the equation cos x = ^. 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 = 2mr ±^ radians, where n is zero or any positive or o negative integer. Ex. 2. Solve the equation sin x — esc x + f = 0. Solving the equation, we obtain, sin a; = — 2, ^. Since the sine of an angle cannot be greater than 1, no angle corre- sponds to the value — 2. For sin x = ^, x = 2mr + '^., (2n + l)7r-^, Ans. EXERCISE 44 Solve each of the following equations, expressing the answers in radians, by use of tt. 1. cot^ e = 3. 12. 52t^±l = cos 2 X. cot x — 1 2. tan^ 6 = 3. 13. 2 sin^ a; — sin a; = sin 2 35— cos a;. 3. cot'' 6 = 1. 14. cos 2 a; + cos a; = 0. 4. sin^ 0=3. 15. tan (45° + a;) + tan (45°-a;) =4. 5. cot e = 2 cos e. 16. 2 csc^ a; — V3 cot a; = 5. 6. cos e + sec e = f. 17. sin 3 x = sin 5 a; + sin x. 7. 3 sin^ X + cos^ * = f • 18. cos 3 x + cos a; = cos 2 a;. 8. 3 cot^ X 4- tan^ a; = 4. 19. sin 6 a; — sin a; = cos 3 x. 9. cos X = sin 2 x. 20. cos 3 a; — cos a; = — sin 2 a;. 10. cos 2 a; + sin a; = 4 sin^ x. 21. sin 6 a; + sin 3 a; + sin sc = 0. 11. sin 2 a; = tan'' x. 22. cos 5 a; + cos 3 a; + cos x=0. 108. Simultaneous Trigonometric Equations. Ex. 1. Solve X sin y=a] „ -, " , fior X and w. a;cos?/=6J ^ Dividing the first equation by the second, tan V = - • .•.y = Z. whose tan is - , Ans. (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. h Thusa!=- sm y or a; = - cos y Ex. 2. Solve for x and y the equations, X COS A + y sm. A = a X sin A — y ca& A = 'b Multiply equation (1) by cos A, then X cos^ A-\-y si-a. A cos A=a cos A. . . Multiply equation (2) by sin A, then X sin^ A — y sin A cos A = l)sui A . , Add (3) and (4), using the fact that sin^^ + cos^-4 = 1. then a; = a cos A+b sm A, and similarly, y= a sin A—b cos A. Ans. (1) (2) (3) (4) EXERCISE 45 Solve for x and 6, or for x and ?/ : ' X cos 6 = 86.65, X sin 5=50. f f X sin e = 118.96, I a; cose = 160.78. 3. 4. a; tan 61 = 816.95, a; sin 6 = 426.3. X sin y = 4, a; cos 2/ = 8. 5. 7. 8. 10. f a; sin 30° + y cos 45° = 53.28, I X cos 30° + y sin 46° = 71.68. f a; sin 48° + y cos 19° = 2634.1 , I X cos 48° + y sin 19° = 1320.3. f sin a; + sin y = 1.573, [Use Art. 71.] [ cos X + cos y = 1.207. C sin a; — sin y = .215i,, \ cos a; — cos 2/ = — .12 1231. fa: sin (5-21.5°) = 771.1, la; cos (5 -32.5°) = 766. ■ a; cos J. — 2/ sin ^ = a, xsmA + y cos 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 = sin y, or (2) y = sin~^ X, which reads " ?/ is the angle whose sine is x " or " y is the anti- FiG. 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 discus- sion. Thus if the angle, or y, is more prominent, a; = sin y is used; but if the sine, x, is more prominent, y = s'm~^x is used. The pupil should carefully discriminate between sin"^ x and the —1 power of sin x. The latter is expressed thus, (sin x)-\ Thus, = sin a; (sin x)-\ and not sin"' x. But (sin x)-' may be written sin"'' a;. 110. Other An ti- trigonometric Functions. Similarly cos~' x means " the angle whose cosine is cc " ; tan"^ x means " the angle whose tangent is x." Let the pupil state the meaning of cof^a;, csc"*a;, vers~'x. It is evident that sin (sin-' a;) = x, since the sine of the angle whose sine is x must be x. Similarly cos (cos~' x) = x, etc. _ Hence there is a similarity in form between a(a~^)x = x, and sin (sin~' 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 = /. 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 vahies which satisfy them. Thus, if j/ = sin 30°, y has a single vahie, ^; but if a; = sin"' J, X can have an indefinite number of vahies, viz. : 30°, 150°, 390°, 510°, etc.; or :c = 2 n IT + |, (2 n + l)7r - 1- (See Art. 107, Ex. 2.) For many purposes it is customary to hmit the values of an inverse circular function to the smallest value that will satisfy a given expression. Thus, if 6' = tan-' 1, we take ^ = 45°. 112. Given an Anti-trigonometric Function, to find the other Related Functions. Ex. 1. Given 9 — tan"' f, find sin 9 \ that is, find sin (tan~' -|). 6 = tan~' -f may be converted into the form tan Q = \ for which a diagram may be con- structed (Fig. 86). .-. sin (tan-^ |) = 3%Vl3 Ans. Ex. 2. Find sin 2(cos-' \). Let X be the angle whose cosine is \. Then cos x = \, sin a; = VI — -^ = | V2. .-. sin2a! = 2sina;cosx = 2(|V2)^ = |V2. Hence, sin 2(cos"' ^) = f V2, Ans. 154 TRIGONOMETRY Ex. 3. If 6 = tan ' a, express the direct and inverse func- tions of 6 in terms of a. tan 0=a, hence = tan~^ a. cot 5 = • « = cot-'-. sec 6 = Vl + a", cos 6 = vr+^' 6 = sec-Vl + a^. 6 = cos~^- vr+T^ sin 6 = - Fig. 87. CSC 6 = ^ = sin-1- Vl + a^' e = csc^-^^ — — — 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-^ ^. . sin a; = \, cosa;=^V3. Let y = cos~^ |. ^^^ cos 2/ = 1, . sin 2/ = ^ V5. ^y^ Fig. 88. Fig. 89. Then sin (sin~' \ + cos~' |) = sin (a; + y) = sin a; cos ?/ + cos x sin y = i-| + iV3.iV5 = ^(2+Vl5), ^«s. Ex. 2. Prove that sin~^ a + cos"^ '^—\' A Using the method of Ex. 1, show that sin (sin~^ a + cos~' a) = 1 = sin f . Ex. 3. Show that tan-^ a + tan-^ h = tan-^ .^LiA. \-ab Let a; = tan"^ a. .-. a = tan x, But y = tan~^ 6. , b = tan y. tan(a; + y)= ^ana^ + tany 1 — tan a; tan a INVERSE TRIGONOMETRIC FUNCTIONS 155 .-. taa (tan~^ a + tan-' 6) = ^^^-i — , or tan"' a + tan"' 6 = tan"' ""*" . 1 — ab 1 — ab 114. Solution of Trigonometric Equations by Use of In- verse Trigonometric Functions. It is sometimes useful to express the answer obtained by solving a trigonometric equa- tion in terms of an inverse function. Ex. Solve 6 cos^ x — cos x = 2. Factoring, (2 cos a; + 1)(3 cos a; — 2) = 0. ■ '■ ''OS a; = — J, ^. .-. X — COS"' (— ^), COS"' I, 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 |," and might be written in the form "find cosZ cotf " (see Art. 110). Express the following angles first in degrees and then in radians : 1. cos-'|-V2, tan-'V3, sin"' I, sec-'v'2, csc-'|V3, cot-'VS, cos-' ^, sec-' 2, sin-' J VS, cot"' | V3, tan -' ^ VS. Find the value of : 2. cos(cot-^l). 8. sin (2 tan-' ^). 3. tan (sin-' /^). 9. cos (2 sec-' -'-f). 4. sec (tan-' j^). 10. sin (^ cos"' ^). 5. sin (cot-' a). 11. cot (^ tan-' -'/). / ^a\ 12. sin (3 sin-' i). 6. cot cos '- . . , . , , V 6/ 13. sin (sin-' 1 — COS"' I). 7. tan (2 sin-' J). 14. tan (tan-' 2 + cot"' 3). Show that : 15. tan-'i-|-tan-'5 = ^. 16. tan-' 2 + tan"' i = ^. 17. sin"' J^ + sin"' | = sin"' ||-. 18. cos"'f 4-cos"'i^ = cos-'(— fl). 19. tan-' I + tan ^ = tan-' ||. 20. cot"' a + cot"' 6 = cot-' ^^-^ , 6-|-a 156 TRIGONOMETRY Prove that : 21. sin (sin-^ f + cot"' |) = 1. 22. cos-i if + tan-' -^ = sin-' ^. 23. sin (2 tan-' a;) = -. 24. sin-' X = cot ^.„,-iViEZ. 25. COS-' a — COS-' b = cos"' (a6 + VI — a^ — 6^ + a^&^). 26. 3 cos "' a; = cos-' (4 as' — 3 a;). 27. 3 sin-' a; = sin-' (3 a; — 4 k'). a — b 28. tan-' a — tan-' 6 = 1 +a&' 29. sin-' a + sin-' 6 = cos-'(Vl — a^ — b^ + a^b' — a6). Express the vahie of each of the following in its most general form : 30. sin-'f 35. cos-'|-V3. 31. tan-'|^V3. 32. cos-'^V2. 33. cot-' I VS. 34. sin"' ^ V3. 40. Prove that tan (2 tan-' a) = 41. Prove sin (2 tan-' a) = , 42. If cos a; = cos 2 x, find x. 43. Express the following angles in the inverse notation : 30°, 60°, 90°, 45°, 0°; »il80°, n90°. 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) ? 36. tan-' 00. 37. cot-' VS. 38. sec-'V2. 39. sin-'(-^) 2a CHAPTER X COMPUTATION OF TABLES TEIGONOMETEIO SERIES ,,- T- -i.- 1 , sin a; , tanas 115. Limiting values of ,. and X X It is important to determine the values vrhich — — and approach when X X x=0, X being the value of an angle expressed in circular measure (radians). Take any angle AOP (Fig. 92) less than 90° and denote it by x; construct the angle AOP' equal to AOP, and draw the tangents PT and P'T. These tangents will meet at J on OA produced. Draw PP'. Then OT' is ± to PP' at its middle point M. By geometry, arc PP' > chord PP' ; also Sire PP' PM, and arc PA sin X, and x < tan x. a; . 1 . 1 X 1 sm x > 1, and cos a; < sin X sinx < cos a; <1. x 157 1^8 TRIGONOMETRY As a; = 0, COS a; = 1, hence ?yL^= 1, since ?^5_^ lies between -I 1 Ou w cos X and i. Hence as a; = 0, limit f j = 1 . This result may also be stated thus, as a; = 0, sin x = x. . 1 tan X _ sin a: _ f sin x\ / 1 \ X X cos X ^ X ^ ^cos a/ But as X = 0, = 1, and = - or 1. X cos X 1 Hence = 1 x 1, or 1. X Or, as a; = 0, limit ( ) = 1. ^ X ^ Since the number of radians in a;= — — — — , it follows OA that as the angle a; = 0, the number of radians in x = sin x, and also = tan x. In practical work, when x < 2°, sin x and tan x may be taken as = p without appreciable error. 116. Computation of the Tables of Trigonometric Func- tions. Since, as a; = 0, sin x 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, r = ^"It-^^^y radians = .0002908882+ radians. 180 X 60 Hence sin 1' = . 0002908882 + COMPUTATION OF TABLES 159 But COS r= Vl - sin^ r= Vl - (.0002908882+)''' = .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 cosa; = sin (90° — a;). ■ The tangents of these angles may be computed by the use of the formula tanx= . To obtain the cotangents, the cos a; formula cot x — tan (90° — ic) may be used: The above method of computing sines and cosines may be abbreviated thus : sin (a; + ?/) + sin (cc — ?/) = 2 sin x cos y. (Art. 71) Let a; = a + 2 &, and y = h. Then, by substitution, sin (a + 3 &) + sin (a + &) = 2 sin (a + 2 &) cos &. Whence sin (a+ 3 &) = 2 sin(a + 26) cosJ — sin(a + &). . . (1) Tn like manner, cos (a+3 6) = 2 cos(a + 26)cos& — cos(a + &). . (2) Let&=l'in(l) and (2). sin (a + 3') = 2 sin (a + 2') cos 1'- sin (a + 1'). . . (3) cos(a + 3') = 2cos(a + 2') cosr-cos(a + r). . . (4) Letting a = — 1', 0, 1', 2', . . .in succession, we obtain f^«™(3) sin 2' = 2 sin r cos r. sin 3' = 2 sin 2' cos r - sin r. sin 4' = 2 sin 3' cos 1' — sin 2', etc. 160 TRIGONOMETRY Similarly from (4), cos2' = 2cosl'-l. 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 : a? , x' x' , , 01? ,2 x\ 17 a? , Unx = x+.- + — + ^^+... The student is aided in recalling these series by the fact that sin ( — a;) = — sin a; (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 ( — cc) = 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 cer- tain 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 V — 1 by the symbol i, it is shown that (cos x + i sin a;)" = cos nx + i sin nx (called De Moivre's Theorem). COMPUTATION OF TABLES 161 By the aid of this theorem and similar principles, trigo- nometry gains much additional power. This branch of the subject is termed analytical trigonometry (though it is some- times 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 (cos x-\- i sin a;)^ = cos 3 a; -|- « sin 3 x. But (cos X + i sin xf = cos' x + 3i sin x cos^ x + 3 i' siri' x cos x+ i' cos^ x. .". cos 3 a; + i sin 3 a; = cos' x — 3 sin^ x cos x -{- i {3 cos^ x sin x — sin' a;). 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 imagi- naries. Hence, cos 3 a; = cos' x—3 sin- x cos a; = 4 cos' a; — 3 cos x sin 3 a; = 3 cos^ x sin x — sin' a; = 3sina; — 4 sin' x. In like manner, by De Moivre's Theorem, prove : . sin 4 a; = 2 sin 2 a; (1 — 2 sin^ x), cos 4 a; = 8 cos'' a; — 8 cos^ a; + 1- sin 5 a; = 16 sin' a; — 20 sin' a; + 5 sin x, cos 5 a; = 16 cos° a; — 20 cos' x + 5 cos x. 4. sin 7 X = 7 sin a; — 56 sin' x + 112 sin' a; — 64 sin' x. nin — 1) 5. cos nx = cos" X r- — cos"-'' x sin^ x lA «(«. - l)(n - 2)(n - 3) , . , + -^ '-^^ ^-^ '- cos"-' X sm< a; -f- — . [± 1 • n(n — IVn — 2) „_, . <. 6. sin nx = n cos"-i x sin x ^^ r^^— — ' cos" ' x sin' x n(.-l)(.-2)(»-3)(n-4)^^^„_,^^.^3^_^...^ 7. tan 2 a,: ^tan.a; 1 — tan^ 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 begin- nings, or germs, of Trigonometry are found in the Rhind Papyrus, now preserved in the British Museum. This papy- rus, 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 co- sine 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. Hindoo, 450 a.d.-HOO 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 Trigonom- etry studied as a part of Geometry. III. (1700 A.D.- ) Trigonometry as an independent science. PEINOIPAL MAKERS OP TEIGONOMETRY 120. Hipparchus. The founder of trigonometry as a science was Hipparchus, a Greek, born about 180 B.C. in 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 trigo- nometry used by him was almost exclusively spherical. 121. Ptolemy (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 Miiller, 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 assas- sinated while in that city. The ephemerides calculated by * Ptolemy entitled his work t^eyhTr; fi,aBrjij,aTiKii irtivrd^is, or "Greatest Mathe- matical 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 neyiaTi] (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 trigonom- etry 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 con- tributions 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 THBIE NOTATION 124. Sine. During all the early history of trigonometry, the trigonometric functions were regarded as lines, not as ratios. V Hipparchus (120 B.C.) used but one trigono- metric function. This was the chord subtended -JA by double the angle, and it therefore corresponded j^ in a general way to the sine of an angle. Thus, the angle AOP was regarded as determined by ^"'- 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 HI'STORY 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 semi- chord 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 " saves the continual doubling " mentioned above. Rheticus (Germany, 1514-1576) 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, trigonom- etry became an independent science. I" 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 jiha, which resembled an Arabic word, jaih, 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 jaih was designated by the Latin word siniis (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 " comple- menti 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 trigo- nometric functions were first used by Thomas Finck (Den- mark, 1583). Gunter, who invented the word cosine, also invented the word cotangent- Girard (Holland, 1590-1633) was the first to use the ab- breviations 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 OP TEIGONOMETBICAL 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 sec- onds, 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 primae" and "partes minutse secundse." 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. Tlie inventors of the metric system of weights and measures at the time of the French Eevolution 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 frac- tions 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 arc 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 deter- mined 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 ^° of the quadrant, the table being remarkably accurate. Hero of Alexandria (110 B.C.) gives a table of cotangents calculated for cot f-^i when n=B, A, ... 12. (v) The Hindoos (530 a.d.) computed a table of sines for every 3f ° of the quadrant. The Arabs (Bagdad, 980 a.d.) formed a table of sines for every ^°, 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 com- puting very full and exact trigonometric tables, and to their industry we owe our tables essentially in their present form. Peuerbach (1423-1461), teacher of Regiomontanus, com- puted 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 com- pleting 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 10^^. By annexing tables of proportional parts, he facili- tated 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. Hip- parchus and Ptolemy in construct- ing their tables of chords used the theorem of geometry which reads " If a quadrilateral be inscribed in a circle, the product of the diagonals equals the sum of the products of the oppo- site sides;" i.e^ (Fig. 94) ACxBD = BCxAD + CDxAB. By means of this theorem, if the chords of two ^'°' ^' 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 com- 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 essen- tially 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 veri- fying and correcting the tables as at first computed. SOIilTTION OF TRIANGLES 131. Greeks (see Ptolemy's Almagest, Book 1) made spher- ical trigonometry primary and fundamental. Plane trigo- nometry 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 compara- tively 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 tri- angles. 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 Qne of the ex- amples solved in the Almagest, three sides of an oblique tri- angle 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 trigo- nometry, we may mention one of the problems occurring in their treat- ment 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=: a given angle (j), to find x and y so that = -• Stated with refer- sm y b ence to the triangle ABC, this problem becomes one in Case II of oblique plane triangles; for Z C =180°— (x + 2/) = 180° — j', /LA = 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 lead- ing place in the study of the subject. They simplified Ptolemy's method of solving spherical triangles, discovered ., , . u • 1 X • 1 A cos a — cos 6 cos c j , that m spherical triangles cos A = -. : , and to sm b sm 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 = cosb 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 trigo- nometry 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 isos- celes 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 solu- tion 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 nec- essarily 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 tri- angles 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 OP GONIOMETBY 138. Greeks. As has been stated (Art. 130), the geomet- rical 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^ ^. 139. The Hindoos knew the identical equation sin^ A + cos^ ^ = 1. They also used the formula sinl A = Vl719(3438 — cos J.), where 3438 is the radius of the circle. This is equivalent to the formula sin i A ^ ^^i-co^_ HISTORY OF TRIGONOMETRY 173 In computing trigonometric tables they appear to have used the formula sin (n + l) 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 , . sin 6 , , cos (b tan (b = ^, cot = — — 2: cos

= — • Vl + d' 141. Rheticus obtained the formulas sin 2 A = 2 sin A cos A, sin 3 A = 3 sin A — 4 sinM. Romanus discovered the formula for sin {A + B). The formulas for sin {A — B) and cos {A ± B) were published byPitiscus(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 PBOCESSBS 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 terms 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 alge- braic 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 tanrie^ in term of powers of tan <^. This use of i was followed up by Euler, who among other results obtained the formula (sin x + i cos xY = sin nx + i cos nx known as De Moivre's Theorem. EXERCISE 48. GENERAL REVIEW 1. Simplify logs 4 + 5 logs 9 + ^ logi„ .1 - logio VaM. 2. Compute the value of x from the equation 5 3?= ■V.2784 3. Also from cos x = (.9387)1 4. Also from tan x = >^-^ ^-^- — '- — • (27.32)t 5. If X is an angle in the first quadrant and cos x = Jy, find the 1 n sin X 4- tan x value 01 ■ • 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 = h, prove that (1 — a-) (1 + 6^^ = 1. 9. ABCD is a square. D is joined to E, the midpoint of AB. Eind 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 ^""^^-g^^'^^ . * ^cos J. + gsin^ 12. Which of the following statements are possible and which im- possible : (1) 16 sin a; = 1. (2) 4 sec 6 = 1. (3) 7 tan;/ = 30. GENERAL REVIEW EXERCISE 175 13. Prove that sec x + tan x = — — + tan x. tan X + sec x 14. Prove that I5£!!^ = ^^H1^- Sin a;. sm a; 1 + cos x 15. Find the numerical value of 3 tan" 30° sec'' 60° sin= 90° tan^ 45° + 5 cos 90°. 16. If tan^ 45° - cos^ 60° = y sin 45° cos 45° tan 60°, find y. cos^^sec^tan- 17. ' If X sin Z cos^- = - - -, find x. Solve each of the following right triangles, given : 18. ^ = 36° 1 8' 6" [36.3°], b = 217.9 ft. 19. 6 = 315.92 ft., c = 814.23 ft. 21. B = 12° 16' [12.25°], c = 1001.4. 20. c = 900, b = 887. 22. ^ = 1° 20' [1.33°], c = 872.56. 23. In a right triangle 6 = 426, A = 38° 45' [38.75°]. Pind 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. Pind 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. Pind 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 : 29. COS 2^ = ^. 30. sm3A=.^-^^-SzM. 31. (sin J. -sin 5)2+ (cos ^ + COS £/ = 2. 176 TRIGONOMETRY 32. rind the other trigonometric functions of A, when cos A= — ^ and A lies between 540° and 630°. 33. Given sec a; = — f and x in the third quadrant, find the value of sin X + tan x «os X + cot X 34. Find the trigonometric functions of 180° + a; 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 numerical value of 3 sin^ 225° + 4 sin (- 120°) tan 150° - 1 008^330° cot 750° + 5 sin^ 180°. 37. For each of the following angles state whicji of the three princi- pal trigonometric ratios are positive : (1) 460°. (2) -220°. (3) -1200°. (4) i|^. 38. Trace the changes in sign and magnitude of sin A between 0° and 360°. CSC A between and ir. cos x between tt and 2 tt. tan A between — 90° and — 270°. 39. If A is in the third quadrant and tan A = ^, 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, (6) the cosine o^the complement of the angle. 41. If sin ^ = xf ai^d sin -B = f , and A and B are both acute, find the numerical value of tan (A + B); also of tan (A — E). 42. If x is an angle in the second quadrant and sin a; = |, find the value of sin 2 a; + cos 2 x. 43. Express 2 cos — cos — as a sum or difference. o o 44. If sin ^ a; = \, find the numerical value of cos x. Also of tan x. Prove that : 45. sin^ {A + B)- sin^ {A-B) = sin 2 ^ sin 2 B. 46. ^HLi^±^HLl^ = cotAa;. 47. sin 50° + sin 10° = sin 70°. cos 3 a; — cos 4 a; GENERAL REVIEW EXERCISE 17 48. sinn5° + cos=15° = l. 49. cos 55° + sin 26° = sin 85°. __ sin ^ + sin 2 J. + sin 3 ^ , „ . 50. ■ — = tan 2 A. cos A + cos 2 A + cos 3 A „ l-tanV45°-a;) . „ 51- z ^tTt^ ■ = sin 2 X. l + tan2(46°-a;) cosC^-eVcosCl+e') Solve each of the following oblique triangles, given : 53. A = 30° 18' 12" £30.3°], & = 3294, c = 2846. 54. A = 76° 24' 36" [76.41°], B = 48° 42' [48.7°], c = 1012. 55. a = 850, 6 = 760, c = 590. 56. B = 46° 18' [46.3°], b = 213.76, a = 192.72. 57. h = 927, A = 79°, B = 21° 17' 12" [21.29°]. 58. a = V3, 6 = V2, c = V5. 59. A = 61° 30' [61.5°], a = 294.6, 6 = 301.7. 60. a = 926.8, b = 842.6, C= 46° 27' [46.45°]. 61. Solve the triangle in which K= 2060.2, a = 214.2, and 6 = 315.8. 62. The diagonals of a parallelogram are 347 and 264 ft., and the area of the parallelogram, is 4043.7 sq. ft. Find the sides and angles of the parallelogram. 63. The diagonals of a quadrilateral are 34 and 66, and they inter- sect 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 a; = ± V3. 67. 2 sin x sin 3 x — sin^ 2 a; = 0. 65. sec^ a; + cot^ x = y*-. 68. sm2 6 + sin 6 = cos 2 6 + cos 6. 66. sin 2 a; = V3 cos K. 69. sin2 y+VBcos 2 2/ = 1. 70. sin (60° - x) - sin (60° + x) = W3. 71. Give the answers to Exs. 64-70, in the unjimited form. 178 TKIGONOMETRY 72. If 2 cos^ x — 7 cos a; + 3 = 0, show that there is only one value for cos a;. 73. Find the least possible positive value of 6 which will satisfy the equation 2-^/3 cos^ 6 = sin 6. 74. Solve sin x + sin 2 x -{■ sin 3 a; = 1 + cos x + cos 2 x. 75. If sin 3 aj + sin 2 a; = 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 afe .6 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 (~ + x\- cos (^ - a;^ = 2 sin x. 80. Find the numerical value of - sin^ 5! + 4 cos^ ^ _ 1 tan^ — ■ 2 6 4 3 4 81. If sin (x + 1") sin {^-t)=\, find x. 82. Simplify tan (~ - x\^ tan (— + x\. 83. An angle of 30° at the centre of a circle C subtends an arc AB of length ^ ft. Find the length of the perpendicular dropped from A on o BC. 84. Express each of the following angles in degrees : sin-i|; cos-iiV2; tan-i(-l); sin-i(-l); cos"' (- -^VS) . 85. Find tan (cot-^i). 86. Prove that tan~^ 2 + tan~^|^ = ^ • o 87. Find the value of x, if tan~' a; + 2 cot~' x = -—- • o 88. How many degrees in sin-' (—iVS) ? How many radians ? 89. Prove sin-^a = sec"'^ — • VI -a" GENERAL REVIEW EXERCISE 179 90. Solve the following for x and y : sin~' X + sin-^ y = 120°. cos~' x — cos~' y = 60°. 91. At a point 50 ft. from the base of a tower the angle of eleva- tion of the top of the tower was found by the use of a transit instru- ment 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 8f 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°. Pind, 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 sub- tends 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 th* 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 horizon- tal projection of the cornice being 1 ft. 100. 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 a; = — f ; find the value of CSC a;, tana;, sin -la;, tan (180° — a;), and sin(— a;). 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 diagonal 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 a; in terms of each of the functions of x. 110. Express cos A cos 5 as a sum. 111. If cos A = h, and tan A = 1c, find the equation connecting h and Ic. . 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"^ f| + 2 tan"' i = sin-^ f . ,,.. Tx • 2 o J tan^a; + cos^a; 114. If sin a; = -, find — 3 tan^ X — cos^ x 115. In the isosceles righj triangle ABC, D is the midpoint of AC. Prove without the use of tables that cot Z ABD : cot Z DBC = 3:2. 116. If e lies between 180° and 270°, and 3 tan 6 = 4, find the value of 2 cot 6 — 5 cos 6 + sin 9. 117. Is it possible to have an angle whose tan is 603 ? Whose cos is f ? Whose secant is ^ ? Whose sine is 23 ? 118. Show that cos 80° + cos 40° — cos 20° = 0. 119. That 2 sin fx + -\ sin ( x - -^ = sin^ a; - cos^ x. GENERAL REVIEW EXERCISE 181 120. If sin (60° - a;) - sin (60° + x) = ^Vs, 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-'^ + 3tan-^^V3 — 2cot-^l + sec-'l, using values between 0° and 90°- 123. If tan 2 a; = ^, 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 (10° + A) cos (10° -A) + sin (10° + .4) 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 tan A = ^ and sin B = \^, and A is in the third quadrant and S 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 seo^ ^ csc^ ^ — 4 = 0, find the numerical value of cot^. 131. If 6 is an angle in the third quadrant, and sec^ Q = 2-\-2 tan 0, find sin 6. 132. Find all the values of x between 0° and 500° which satisfy the equation tan (45° — a;) + cot (45° — a;) = 4. 133. Graph y = sin-' X. 134. Also, ?/ = tan~'a;. 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 a; + 3 sin a; = 20 ? Why ? 137. Find the numerical value of tan^ -^ + cos^ — ^ + sin^ J • 3 4 6 138. Find the sines of all angles less than 2 tt whose tangents are equal to cos 135°. 182 TRIGONOMETRY 139. Given cos(^ + x\—a, find cot[-^ + a! )■ 140. What is the most general value of x which satisfies both of the equations cot x = — V3 and esc a; = — 2 ? 141. Show that 2 sin (j + A\ cos (j + ^)= cos (A + E)+ sin (A — B). 142. Find the length of a circular arc whose radius is 5 ft. ard whose subtending angle is 3 units of circular measure. 143. In the triangle ABC, B is 45°, and 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 sina; + sin2a; ^ ^^^ ^_ 1 + cos X + cos 2 X 145. When y = — '—, find the numerical value of ^ 4 ' sin^ y — cos^ y + 2 tan y — sec^ y. 146. Prove the identity sin~^ y + tan"^ y = sin~^ ^^ ' ~ y / • VI +y^ 147. Is sin x — 2 cos a; + 3 sin a; — 6 = 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 icot2Z: + 3sin2l_2csc''5:_§tan2T = 15. 3 6 3 3 4 6 3 151. Find the value of sin x, if tan ( ^ + a; j tan ( ^ — a; J + 2 = 0. 152. What sign has sin a; cos a; for the following values of x: 140°, 278°, -366°, -1125°? 153. If 1 + sin^ a; = 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 — — ^ = 1.423, find r when j = 34.37°. ^^^'^ GENERAL REVIEW EXERCISE 183 155. When is sin x = ^J^ possible, and when impossible? 2ab 156. Show that sin (2 a -13) cos (a — 2 ;8) - cos (2a-l3) sin (a - 2 18) = sin (a + j3). 157. Solve sin 2 a; — cos 2 0! — sin ic + cos a; = 0. 158. Solve a; = sin"^-|- + tan-^l. 159. Trace the changes in sign and magnitude of ^^^ as x in- „ _ cos 2^ creases from 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 cos2^ + cos2^ ^ ^^^ ,^ ^. ^^^ (A-B). cos25-cos2^ V -r y V ; 2. In a right triangle show that \l^^ — h \/' 162 ^" " ''''"•i^J- ^-«^. 1" "T ' j-T,«j- ^ jwy-pf I . 1^ ^ J sm vi a-6 ^a + b Vcos25 ^I + l^Vtan^^-i^ tanf , „ , , , ■,«^, -D U 2 y \A 2 y 163. Prove ) ^ — ( ) ^ — (- = esc A. tang + |^)-tang-^^) 164. In any triangle prove that c = a cos B+b cos ^, and hence show that sin (A + B)= sin A cos 5 + cos J. sin B. 165. Determine the angles in a right triangle in which a>b, and c — a = a — 6. 166. Prove cos^ (a; — .y) — 2 cos (x — ?/) cos x cos ?/ = sin^ x — cos^ y. 167. If sin X — COS a; + 4 cos^ a; = 2, find the ratio of tan a; to sec x. 168. liA + B = 225°, prove that f^^^iA\ (^^^B_\ ^ 1 _ '^ ^1+cot^yu+cotBy 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 yl is a geometric mean between sin B and cos 5, prove cos 2 ^ = 2 sin (45° - B) cos (45°+S). 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 sub- tends 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. 174. If 2 tan-i X = cos-^ ^^^=^ - cos-^ t^, find x. 175. If tan2 (180° -x)- sec (180° + a;) = 5, find cos x. 176. In orHer to fix the distance between two islands and D, a base line, AB, 900 ft. long, is measured on the shore. Also, Z. BAC was found to be 110° 50' [110.8.S°], Z DAB, 67° 61' [67.85°], Z GBA, 49° 51' [49.85°], Z ABD, 85° 19' [85.32°]. What was the distance between the islands? LOGARITHMIC AI^D TRIGONOMETRIC TABLES EDITED BY FLETCHER DURELL, Ph.D-. HEAD OP THE MATHEMATICAL DEPARTMENT THE LAWKENCEVILLB SCHOOL NEW YORK CHARLES E. MERRILL CO. Durell's Mathematical Series School Algebra 607 pages, 12mo, cloth $1.10 Plane Geometry 341 pages, 12mo, cloth 75 cents Solid Geometry 213 pages, 12mo, cloth 75 cents Plane and Solid Geometry 514 pages, 12mo, cloth $1.25 Plane Trigonometry 184 pages, 8vo, cloth $1.00 Plane Trigonometry and Tables 298 pages, 8vo, cloth $1.25 Plane and Spherical Trigonometry, with Tables 351 pages, 8vo, cloth ... . $1.40 Plane Trigonometry with Surveying and Tables 390 pages, Svo, cloth $1.40 Plane and Spherical Trigonometry, with Surveying and Tables' 448 pages, Svo, cloth $1.50 Logarithmic and Trigonometric Tables 114 pages, Svo, cloth 75 cents Copyright, 1910, by Charles E. Merrill Co. [4] CONTENTS j . PA5E Introduction to Tables 5 Tables : T. Five-place Logarithms of Numbers 1-10,000 ... 21 II. Logarithms and Cologarithms of Much-used Numbers 40 in. Five-place Logarithms of the Sine, Cosine, Tangent, and Cotangent for Each Minute of the Quadrant j^ 41 IV. Auxiliary Five-place Table for Small Angles . . 87 V. Four-place Table op the Natural Sine, Cosine, Tan- gent, 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 Func- tions 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 INTRODUCTION TO TABLES 1. Number of Decimal Places in Tables. All trigonometric work ^s based on the results of measurements. But no measurement is accurate beyond the sixth or seventh figure ; this is Sowing 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 mthin 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 y^-^ or -j-^-^ 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 6,643,786, if the figure 6 to the right is ^ 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 or- dinary work (such as is done by a land surveyor, or in a physi- cal 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 LOGAEITHMS OP NUMBEES 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 these for all numbers from 1 to 10,000. 6 TRIGONOMETRY In using Part II the characteristic of each logarithm must be deter- mined and supplied in accordance with the methods stated in Arts. 4 and 6 of Durell's Plane Trigonometry. DiEECT Use of Table I 3. To find the mantissa for a number containing four figures. In the given table the left-hand column (headed JST) is a column of ordinary numbers. The first three figures of the given niimber 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 figm-es. 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 loga- rithms 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 1682 = .19921 Mantissa of 1581 = .19893 ra. for 1681 = .19893 .00028 x .47 = .00013 Dife. for 1 = .00028 Mantissa of 1581.47 = .19906, Ans. Tor 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 6 or a larger number, the figure in the fifth place of the correction is to be increased by 1 ; if less than 6, 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. Hence, in general, to find a mantissa for a number contain- ing 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 the mantissa for the given figures. TRIGONOMETRY Ex. 1. Find log 3.62057. o m. oi 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. in. of 7855 = .89515 ' log .07854 = 8.89509 - 10 m. 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 man- tissa of the given logarithm; Use the characteristic of the given logarithm to fix the deci- mal 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. (&) In case the given mantissa does not occur in the table : Obtain from the table the next lower mantissa with the corre- sponding four figures of the antilogarithm; INTRODUCTION TO TABLES 9 Subtract the tabular mantissa from the given mantissas- 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 j-\ 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 a; = 7.26323 -10, find a;. Nearest less mantissa = .26316, whose number is 1833. Tab. dife. = 24. 7 + 24 = .29+- Hence x = .00183329, A7is. The first part of Exercise 4 of Durell's Plane Trigonom- etry should be worked at this point. TABLE II. LOGARITHMS AND COLOGARITHMS OP MUCH-USED NUMBERS (p. 40) This table explains itself. TABLE III. FIVE-PLACE LOGARITHMS OP TRIGONOMETRIC PUNO- TIONS POR EVERY MINUTE OP THE QUADRANT (pp. 41-86) 7. Description of Table III. This table gives the loga- rithms 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 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 left- hand 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' = 9.66130 - 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 |^ of 1' 'will make an increase of |3. of 16 in the log of the function. Hence we have log sin 37° 42'= 9.78642 - 10 Diff. for 53" = f f 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" = 1 « 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" = ^ 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 Dife. for 43" = 28 x ff= 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 func- tions; multiply this difference by — '—— ; 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 (com- plementary functions). 12 TRIGONOMETRY 9. Log Secants. To find the log secant of an angle, use the formula sec x = .•. log sec a; = + colog cos x. cosx 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. Inveese Use op Table III 11. Given the logarithm of a function to find the correspond- ing 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 func- tion, noting the corresponding number of degrees and minutes; subtract this logarithm from the given logarithm; Divide the difference 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 sub- tracted 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 ^ of 60", or 41", in the angle. .-. 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 P of 60" = 48" 6 antilog cos 9.96623 - 10 = 22° 18' 12", Ans. Ex. 3. Find antilog cot 0.57603. antilog cot 0.57603 = 14° 52' - 0.57601 -^of 60" = 2" 61 antilog cot 0.67603 = 14° 61' 58", Ans. Ex. 4. Find antilog cos 9.60172 - 10. antilog cos 9.60172 - 10 = 66° 27'- 9.60167-10 — 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 Tan- gent 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 num- ber 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 num- bers 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 ac- cording as the log sin or log tan is desired. Ex. Find log sin 1° 26' 13". r 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 fable; 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 = 6098" = 1° 24' 58", Ans. The reason for the above process is seen from the fact that . , . , 5098" sm of required Z = 205265^/ .-. 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 ^os 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 ob- tained 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. POUB-PLACE TABLE OP THE NATURAL SINE, COSINE, TANGENT, AND COTANGENT FOB EVERY TEN MINUTES OP THE QUADRANT (pp. 91-96) 16. Method of using Table V. By natural trigonometric functions are meant the actual numerical (not logarithhiic) values of these functions. Thus ^ is the natural sine of 30°. Interpolation for this table is made in the same general way as for Table III. Ex. Find natural sine 27° 48'. N". Sine 27° 40' = 0.4643 Jj of 26 = 21 N. Sine 27° 48' = 0.4664, Ans. TABLE VI. POUB-PLACE TABLE OF LOGARITHMS OP NUMBERS 1-2000 (pp. 97-101) 17. Method of using Table VI. In using the four-place log of a number, when the first signifi- cant 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. POUB-PLACB LOGARITHMIC TABLE OP THE TRIGONO- METRIC FUNCTIONS FOE 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 Intro- duction apply ha 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° = 9.8781 - 10 log sin 48.3° = 9.8731 - 10 -^ oi 7 = 3 7 log sin 48.34° = 9.8734 - 10, Ans. Ex. 2. Find the antilog tan 0.2165. Zlog tan 0.2165 = 68.7°+ 2161 -i of 10 = 2+ 17 Z 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 vdJy small Z x J. . . Zxin degrees = no. radians in Z x, or = "„ I .ZvO V 18 TRIGONOMETRY .-. log sin (or tan) of small Za3 = log a; + colog 57.296 = log a; + 8.2419 -10. ., , • ' n + 1 57.296° Also when % is small cot x = = — -. — = tan X X m degrees .•. log cot small Zx= 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 J. = sin (90°-^). .-. log cos A = log sin (90° -A) = 8.2419 - 10 + log (90° - A). cot ^ = tan {90° -A). .-. log cot A = log tan (90° - J.) = 8.2419 - 10 + log (90° - A). tan A = -^—■. .-. log tan ^ = 1.7581 -log (90°- J.). ■ cot A Ex. 1. Find sin 0.876°. log 0.876° = 9.9425 - 10 colog 67.296° = 8.2419 - 1 .-. 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 ifHE 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°, J.WS. TABLE rx. TABLE FOE CONVERTING THE DECIMAL PARTS OP A DEGREE INTO MINUTES AND SECONDS (p. 114) 21. The method of using Table IX is also evident from the table itself. TABLE I COMMON LOGARITHMS OF NUMBERS PAET I Logarithms (with Characteristics) op Numbers ] -100 N. Log. N. log. N. Log. N. Log. 1 2 3 — Infinity 30 31 32 33 1.47 712 60 61 62 63 1.77 815 90 91 92 93 1.95 424 0.00 000 0.30 103 0.47 712 1.49 136 1.50 51B 1.51 851 1.78 533 1.79 239 1.79 934 1.95 904 1.96 379 1.96 848 4 5 6 0.60 206 0.69 897 0.77 815 34 35 36 1.63 148 1.54 407 1.55 630 64 65 66 1.80 618 1.81 291 1.81 954 94 95 96 1.97 313 1.97 772 1.98 227 7 8 9 10 11 12 13 0.84 510 0.90 309 0.95 424 37 38 39 40 41 42 43 1.56 820 1.57 978 1.59 106 67 68 69 70 71 72 73 1.82 607 1.83 251 1.83 885 97 98 99 100 1.98 677 1.99 123 1.99 564 1.00 000 1.60 206 1.84 510 2.00 000 1.04 139 1.07 918 1.11 394 1.61 278 1.62 325 1.63 347 1.85 126 1.85 733 1.86 332 14 15 16 1.14 613 1.17 609 1.20 412 44 45 46 1.64 345 1.65 321 1.66 276 74 75 76 1.86 923 1.87 506 1.88 081 17 18 19 20 21 22 23 1.23 045 1.25 527 1.27 875 47 48 ■ 49 50 51 52 53 1.67 210 1.68 124 1.69 020 77 78 79 80 81 82 83 1.88 649 1.89 209 1.89 763 1.30 103 1.69 897 1.90 309 1.32 222 1.34 242 1.36 173 1.70 757 . 1.71 600 1.72 428 1.90 849 1.91 381 1.91 908 24 25 26 1.38 021 1.39 794 1.41 497 54 55 56 1.73 239 1.74 036 1.74 819 84 85 86 1.92 428 1.92 942 1.93 450 27 28 29 30 1.43 136 1.44 716 1.46 240 57 58 59 60 1.75 587 1.76 343 1.77 085 87 88 89 90 1.93 952 1.94 448 1.94 939 1.47 712 1.77 815 1.95 424 [211 Part II Mantissas of Numbers 1-10,000 N. 1 2 3 4 5 6 7 8 9 100 01 02 03 00 000 043 087 130 173 217 260 303 346 389 432 860 01 284 475 903 326 518 945 368 561 988 410 604 *030 452 647 *072 494 689 *115 536 73r *1F7 578 775 *199 620 817 *242 662 04 05 06 703 02 119 531 745 160 572 787 202 612 828 243 653 870 284 694 912 325 735 953 366 776 995 407 816 *036 449 857 *078 490 898 07 08 09 110 11 12 13 938 03 342 743 979 383 782 *019 423 822 *060 463 862 *100 503 902 *141 543 941 *181 583 981 *222 623 *021 *262 663 *060 *302 703 *100 04 139 179 218 258 297 336 376 415 454 493 532 922 05 308 571 961 346 610 999 385 650 *038 423 689 *077 461 727 *115 500 766 *154 538 805 *192 576 844 *231 614 883 *269 652 14 15 16 690 06 070 446 729 108 483 767 145 521 805 183 558 843 221^ 59S 881 258 633 918 296 670 956 333 707 994 371 744 *032 408 781 17 18 19 120 21 22 23 819 07 188 555 856 225 591 893 262 628 930 298 664 967 335 700 *004 372 737 *041 408 773 *078 445 809 *115 482 846 *151 518 882 918 954 990 *027 *063 *099 *135 *17i *207 *243 08 279 636 991 314 672 *026 350 707 *061 386 743 *096 422 778 *132 458 814 *167 493 849 *202 529 884 *237 565 920 *272 600 955 *307 24 25 26 09 342 691 10 037 377 726 072 412 760 106 447 795 140 482 830 175 517 864 209 552 899 243 587 934 278 621 96S«i 3J',2 656. 346 27 28 29 130 31 32 33 380 721 11 059 415 755 093 449 789 126 483 823 160 517 857 193 551 890 227 585 924 261 619 958 294 653. 992 327 687 *025 361 394 428 461 494 528 561 594 628 661 694 727 12 057 385 760 090 418 793 123 450 826 156 483 860 189 516 893 222 548 926 254 581 959 287 613 992 320 646 *024 352 678 34 35 36 710 13 033 354 743 066 386 775 098 418 808 ' 130 450 840 162 481 872 194 513 905 226 545 937 258 577 969 290 609 *001 322 640 37 38 39 140 41 42 43 672 988 14 301 704 *019 333 735 *051 364 767 *082 395 799 *114 426 830 *145 457 862. *176 489 893 *208 520 925 *239 551 956 *270 582 613 644 675 706 737 768 799 829 860 891 922 15 229 534 953 259 564 983 290 594 *014 320 625 *045 351 655 *076 381 685 *106 412 715 *137 442 746 *168 473 776 *198 503 806 44 45 46 836 16 137 435 866 167 465 897 197 495 927 227 524 957 256 554 987 286 584 *017 316 613 *047 346 643 *077 376 673 *107 4t)6 702 47 48 49 150 732 17 026 319 761 056 348 791 085 377 820 114 406 850 143 435 879 173 464 909 202 493 938 231 522 967 260 551 997 289 580 609 638 667 696 725 754 782 811 840 869 N. 1 2 3 4 6 6 7 8 9 r22i N. 1 2 3 4 S 6 7 8 9 160 51 52 53 17 609 638 667 696 725 754 782 811 840 869 898 18 184 469 026 213 41.18 955 241 526 984 270 554 *013 298 583 *041 327 611 *070 355 639 *099 384 667 *127 412 696 *156 441 724 54 55 56 752 19 033 312 780 061 340 808 089 368 837 117 396 865 145 424 893 173 451 921 201 479 949 229 507 977 257 535 *005 285 562 57 58 59 160 61 62 63 590 866 20 140 618 893 167 645 921 194 673 948 222 700 976 249 728 *003 276 756 *030 303 783 *058 330 811 *085 358 838 *112 385 412 439 466 493 520 548 575 602 629 656 683 952- 21 219 710 978 245 737 *005 272 763 *032 299 790 *059 325 817 *085 352 844 *112 378 871 *139 405 893 *165 431 925 *192 458 64 65 66 484 748 22 Oil 511 775 037 537 801 063 564 827 089 590 854 115 617 880 141 643 906 167 669 932 194 696 958 220 722 985 246 67 68 69 170 71 72 73 272 531 789 298 557 814 324 583 840 350 608 866 376 634 891 401 660 917 427 686 .943 453 712 968 479 737 994 505 763 *019 23 045 070 096 121 147 172 198 223 249 274 300 553 805 325 578 830 350 603 855 376 629 880 401 654 905 426 679 930 452 704 955 477 729 980 502 754 *005 528 779 *030 74 75 76 24 055 304 551 080 329 576 105 353 601 130 378 625 155 403 650 180 428 674 204 452 699 229 477 724 254 502 748 279 527 773 77 78 79 180 81 82 83 797 25 042 285 822 066 310 846 091 334 871 115 358 895 139 382 920 164 406 944 188 431 969 212 45^ 993 237 479 *018 261 503 527 . 551 575 600 624 648 672 696 720 744 768 26 007 245 792 031 269 816 055 293 840 079 316 864 102 340 888 126 364 912 150 387 935 174 411 959 198 435 983 221 458 84 85 86 482 717 951 505 741 975 529 764 998 553 788 *021 576 811 *045 goo 834 *068 623 858 *091 647 881 *114 670 905 *138 694 928 *161 87 88 89 190 91 92 93 27 184 416 646 207 439 669 231 462 692 254 485 715 277 508 738 300 531 761 323 554 784 346 577 807 370 600 830 393 623 852 875 898 921 944 967 989 *012 *035 *058 *081 28 103 330 556 126 353 578 149 375 601 171 398 623 194 421 646 217 443 668 240 466 691 262 488 713 285 511 735 307 533 758 94 95 96 780 29 003 226 803 026 248 825 048 270 847 070 292 870 092 314 892 115 336 914 137 358 937 159 380 959 181 403 981 203 425 97 98 99 200 447 667 885 469 688 907 491 710 929 513 732 951 535 754 973 557 776 994 579 798 *016 601 820 *038 623 842 *060 645 863 *081 30 103 125 146 168 190 211 233 255 276 298 N. 1 2 3 4 5 6 7 8 9 [23] N. 1 2 3 4 5 6 7 8 9 200 01 . 02 03 30 103 125 146 168 190 211 233 255 276 298 320 535 750 341 557 771 363 578 792 384 600 814 406 621 835 428 643 856 449 664 878 . ■471 685 899 492 707 920 514 728 942 04 05 06 963 31 175 387 984 197 408 *006 218 429 *027 239 450 *048 260 471 *069 281 492 *091 307 513 *112 323 534 *133 345 555 *154 366 576 07' 08 09 210 11 12 13 597 806 32 015 618 827 035 639 848 056 660 869 077 681 890 098 702 911 118 723 931 139 744 952 160 765 973 181 785 994 201 222 243 263 284 305 325 346 366 387 408 428 634 838 449 654 858 469 675 879 490 695 899 510 715 919 531 736 940 552 756 960 572 777 980 593 797 *001 613 818 *021 14 15 16 33 041 244 445 062 264 465 082 284 486 102 304 506 122 325 526 143 345 546 163 365 566 183 385 586 203 405 606 224 425 626 17 18 19 220 21 22 23 646 846 34 044 666 866 064 686 885 084 706 905 104 726 925 124 746 945 s 143 , 766 965 163 786 985 183 806 *005 203 826 *025 223 242 262 282 301 321 341 361 380 400 420 439 635 830 459 655 850 479 674 869 498 694 889 518 713 908 537 733 928 557 753 947 577 772 967 596 792 986 616 811 *005 24 25 26 35 025 218 411 044 238 430 064 257 449 083 276 468 102 295 488 122 315 507 141 334 526 160 353 545 180 372 564 199 392 583 27 28 29 230 31 32 33 603 793 984 622 813 *003 641 832 *021 660 851 *040 679 870 *059 698 889 *078 717 908 *097 736 927 *116 755 946 *135 774 965 *154 36 173 192 211 229 248 267 286 305 324 342 361 549 736 380 568 754 399 586 773 418 605 791 436 624 810 455 642 829 474 661 847 493 680 866 511 698 884 530 717 903 34 35 36 922 37 107 291 940 125 310 959 144 328 977 162 346 996 181 365 *014 199 383 *033 218 401 *051 236 420 *070 254 438 *088 273 457 37 38 39 240 41 42 43 475 658 840 493 676 858 511 694 876 530 712 894 548 731 912 566 749 931 585 767 949 603 785 967 621 803 985 639 822 *003 38 021 039 057 075 093 112 130 148 166 184 202 382 561 220 399 578 238 417 596 256 435 614 274 453 632 292 471 650 310 489 668 328 507 686 346 525 703 364 543 721 44 45 46 739 .917 39 094 757 934 111 775 952 129 792 970 146 810 987 164 828 *005 182 846 *023 199 863 *041 217 881 *058 235 899 *076 252 47 48 49 250 270 445 620 287 463 637 305 480 655 322 498 672 340 515 690 358 533 707 375 550 724 393 568 742 410 585 759 428 602 777 794 811 829 846 863 881 898 915 933 950 N. 1 1 ^ 3 4 5 6 7 8 9 [24] N. O 1 2 3 4 5 6 7 8 9 260 51 52 53 39 794 811 829 846 863 881 898 915 933 950 967 40 140 312 985 157 329 *002 175 346 *019 192 364 *037 209 381 *054 226 398 *071 243 415 *088 261 432 *106 278 449 *123 295 466 54 55 56 483 654 824 500 671 841 518 688 85.8 535 705 875 552 722 892 569 739 909 586 . 756 926 603 ,773 943 620 790 960 637 807 976 57 58 59 260 61 62 63 993 41 162 330 *010 179 347 *027 196 363 *044 212 380 *061 229 397 *078 246 414 *095 263 430 *111 280 447 *128 296 464 *145 313 481 497 514 531 547 564 581 597 614 631 647 664 830 996 681 847 *012 697 863 *029 714 880 *045 731 896 *062 747 913 *078 764 929 *095 780 946 *111 797 963 *127 814 979 *144 64 65 66 42 160 325 488 177 341 504 193 357 521 210 374 537 226 390 553 243 406 570 259 423 586 275 439 602 292 455 619 308 472 635 67 68 69 270 71 72 73 651 813 975 667 830 991 684 846 *008 700 862 *024 716 878 *040 732 894 *056 749 911 *072 765 927 *088 781 943 *104 797 959 *120 43 136 152 169 185 201 217 233 249 265 281 297 457 616 313 473 632 329 489 648 345 505 664 361 -521 680 377 537 696 393 553 712 409 569 727 425 584 743 441 600 759 74 75 76 775 933 44 091 791 949 107 807 965 122 823 981 138 838 996 154 854 *012 170 870 *028 185 886 *044 201 902 *059 217 917 *b75 232 77 78 79 280 81 82 83 248 404 560 264 420 576 279 436 592 295 451 607 311 467 623 326 483 638 342 498 654 358 514 669 373 529 685 389 545 700 .716 731 747 762 778 793 809 824 840 855 871 45 025 179 886 040 194 902 056 209 917 071 225 932 086 240 948 102 255 963 117 271 979 133 286 994 148 301 *010 163 317 84 85 86 332 484 637 347 500 652 362 515 667 378 530 682 393 545 697 408 561 712 423 576 728 439 591 743 454 606 758 469 621 773 87 88 89 290 91 92 93 788 939 46 090 803 954 105 818 969 120 834 984 135 849 *000 150 864 *015 165 879 *030 180 894 *045 195 909 *060 210 924 *075 225 240 255 270 285 300 315 330 345 359 374 389 538 687 404 553 702 419 568 716 434 583 731 449 598 746 464 613 761 479 627 776 494 642 790 509 657 805 523 672 820 94 95 96 835 982 47 129 850 997 144 864 *012 159 879 *026 173 894 *041 188 909 *056 202 923 *070 217 938 *085 232 953 *100 246 967 *114 261 ^ 97 98 99 300 276 422 567 290 436 582 305 451 596 319 465 611 334 480 625 349 494 640 363 509 654 378 524 669 392 538 683 407 553 698 712 727 741 756 770 784 799 813 828 842 If. 1 2 3 4 5 6 7 8 9 [25] ». 1 2 3 4 5 6 7 8 9 300 01 02 03 47 712 727 741 756 770 784 799 813 828 842 857 48 001 144 871 015 159 885 029 173 900 044 187 914 058 202 929 073 216 943 087 230 958 101 244 972 116 259 986 130 273 04 05 06 287 430 572 302 444 586 316 458 601 330 473 615 344 487 629 359 501 643 373 515 657 387 530 671 401 544 686 416 558 700 07 08 09 310 11 12 13 714 855 996 728 869 *010 742 883 *024 756 897 *038 770 911 *052 785 926 *066 799 940 *080 813 954 *094 827 968 *108 841 982 *122 49 136 150 164 178 192 206 220 234 248 262 276 415 554 290 429 568 304 443 582 318 457 596 332 471 610 346 485 624 360 499 638 374 513 651 388 527 665 402 541 679 14 15 16 693 831 969 707 845 982 721 859 996 734 872 *010 748 886 *024 762 900 *037 776 914 *051 790 927 *065 803 941 *079 817 955 *092 17 18 19 320 21 22 23 50 106 243 379 120 256 393 133 270 406 147 284 420 161 297 433 174 311 447 188 325 461 202 338 474 215 352 488 229 365 501 515 529 542 556 569 583 596 610 623 637 651 786 920 664 799 934 678 813 947 691 826 961 705 840 974 718 853 987 732 866 *001 745 880 *014 759 893 *028 772 907 *041 24 25 26 51 055 188 322 068 202 335 081 215 348 095 228 362 108 242 375 121 255 388 135 268 402 148 282 415 162 295 428 175 308 441 27 28 29 330 31 32 33 455 587 720 468 601 733 481 614 746 495 627 759 508 640 772 521 654 786 534 667 799 548 680 812 561 693 825 574 706 838 851 865 878 891 904 917 930 943 957 970 983 52 114 244 996 127 257 *oa9 140 270 *022 153 284 *035 166 297 *048 179 310 *061 192 323 *075 205 336 *088 218 349 *101 231 362 34 35 36 375 504 634 388 517 647 401 530 660 414 543 673 427 556 686 440 569 699 453 582 711 466 595 724 479 608 737 492 621 750 37 38 39 340 41 42 43 763 892 53 020 776 905 033 789 917 046 802 930 058 815 943 071 827 956 084 840 969 097 853 982 110 866 994 122 879 *007 135 148 161 173 186 199 212 224 237 250 263 275 403 529 288 415 542 301 428 555 314 441 567 326 453 580 339 466 593 352 479 605 364 491 618 377 504 631 390 517 643 44 45 46 656 782 908 668 794 920 681 807 933 694 820 945 706 832 958 719 845 970 732 857 983 744 870 995 757 882 *008 769 895 *020 47 48 49 350 54 033 158 283 045 170 295 058 183 307 070 195 320 083 208 332 095 220 345 108 233 357 120 245 370 133 258 382 145 270 394 407 419 432 444 456 469 481 494 506 518 F 1 2 3 4 5 6 7 8 9 [26] N. 1 2 3 4 5 6 7 8 9 350 51 52 53 54 407 419 432 444 456 469 481 494 506 518 531 654 777 543 667 790 555 679 802 568 691 814 580 704 827 593 716 839 605 728 851 617 741 864 630 753 876 642 765 888 54 55 56 900 55 023 145 913 035 157 925 047 169 937 060 182 949 072 194 962 084 206 974 096 218 986 108 230 998 121 242 *oii 133 255 57 58 59 360 61 62 63 267 388 509 279 400 522 291 413 534 303 425 546 315 437 558 328 449 570 340 461 582 352 473 594 364 485 606 376 497 618 630 642 654 666 678 691 703 715 727 739 751 871 991 763 883 *003 775 895 *015 787 907 *027 799 919 *038 811 931 *050 823 943 *062 835 955 *074 847 967 *086 859 979 *098 64 65 66 56 110 229 348 122 241 360 134 253 372 146 265 384 158 277 396 170 289 407 182 301 419 194 312 431 205 324 443 217 336 455 67 68 69 370 71 72 73 467 585 703 478 597 714 490 608 726 502 620 738 514 632 750 526 644 761 538 656 773 549 667 785 561 679 797 573 691 808 820 832 844 855 867 879 891 902 914 926 937 57 054 171 949 066 183 961 078 194 972 089 206 984 101 217 996 113 229 *008 124 241 *019 136 252 *031 148 264 *043 159 276 74 75 76 287 403 519 299 415 530 310 426 542 322 438 553 334 449 565 345 461 576 357 473 588 368 484 600 380 496 611 392 507 623 77 78 79 380 81 82 83 634 749 864 646 761 875 657 772 887 669 784 898 680 795 910 692 807 921 703 818 933 715 830 944 726 841 955 738 852 967 978 990 *001 *013 *024 *035 *047 *058 *070 *081 58 092 206 320 104 218 331 115 229 343 127 240 354 13"8 252 365 149 263 377 161 274 388 172 286 399 184 297 410 195 309 422 84 85 86 433 546 659 444 557 670 456 569 681 467 580 692 478 591 704 490 602 715 501 614 726 512 625 737 524 636 749 535 647 760 87 88 89 390 91 92 93 771 883 995 782 894 *006 794 906 *017 805 917 *028 816 928 *040 827 939 *051 838 950 *062 850 961 *073 861 973 *084 872 984 *095 59 106 118 129 140 151 162 173 184 195 207 218 329 439 229 340 450 240 351 461 251 362 472 262 373 483 273 384 494 284 395 506 295 406 517 306 417 528 318 428 539 94 95 96 550 660 770 561 671 780 572 682 791 583 693 802 594 704 813 605 715 824 616 726 835 627 737 846 638 748 857 649 759 868 97 98 99 400 879 988 60 097 890 999 108 901 *010 119 912 *021 130 923 *032 141 934 *043 152 945 *054 163 956 *065 173 966 *076 184 977 *086 195 206 217 228 239 249 260 271 282 293 304 N. 1 2 3 4 5 6 7 8 ^ [27] N. 1 2 3 4 5 6 7 8 9 400 01 02 03 60 206 217 228 239 249 260 271 282 293 304 314 423 531 325 433 541 336 444 552 347 455 563 358 466 574 369 477 584 379 487 595 390 498 606 401 509 617 412 520 627 04 05 06 638 746 853 649 756 863 660 767 874 670 778 885 681 788 895 692 799 906 703 810 917 713 821 927 724 831 938 735 842 949 07 08 09 410 11 12 13 959 61 066 172 970 077 183 981 087 194 991 098 204 *002 109 215 *013 119 225 *023 130 236 *034 140 247 *045 151 257 *055 162 268 278 289 300 310 321 331 342 352 363 374 384 490 595. 395 500 606 405 511 616 416 521 627 426 532 637 437 542 648 448 553 658 458 563 669 469 574 679 479 584 690 14 15 16 700 805 909 711 815 920 721 826 930 731 836 941 742 847 951 752 857 962 763 868 972 773 878 982 784 888 993 794 899 *003 17 18 19 420 21 22 23 62 014 118 221 024 128 232 034 138 242 045 149 252 055 159 263 066 170 273 076 180 284 086 190 294 097 201 304 107 211 , 315 32B 335 346 356 366 377 387 397 408 418 428 531 634 439 542 644 449 552 655 459 562 665 469 572 675 480 583 685 490 593 696 500 603 70fe 511 613 716 521 624 726 24 25 26 737 839 941 747 849 951 757 859 961 767 870 972 778 880 982 788 890 992 798 900' *002 808 910 *012 818 921 *022 829 931 *033 27 28 29 430 31 32 33 63 043 144 246 053 155 256 063 165 266 073 175 276 083 185 286 094 195 296 104 205 306 114 215 317 124 225 327 134 236 337 347 357 367 377 387 397 407 417 428 438 448 548 649 458 558 659 468 568 669 478 579 679 488 589 689 498 599 699 508 609 709 518 619 719 528 629 729 538 639 739 34 35 36 749 849 949 759 859 959 769 869 969 779 879 979 789 889 988 799 899 998 809 909 *008 819 919 *018 829 929 *028 839 939 *038 37 38 39 440 41 42 43 64 048 147 246 058 157 256 068 167 266 078 177 276 088 187 286 098 197 296 108 207 306 118 217 316 128 227 326 137 237 335 345 355 365 37B 385 395 404 414 424 434 444 542 640 454 552 650 464 562 660 473 572 670 483 582 680 493 591 689 503 601 699 513 611 709 523 621 719 532 631 729 44 45 46 738 836 933 748 846 943 758 856 953 768 865 963 777 875 972 787 885 982 797 895 992 807 904 *0C2 816 914 *011 826 924 *021 47 48 49 460 65 031 128 225 040 137 234 050 147 244 060 157 254 070 167 263 079 176 273 089 186 283 099 196 292 108 205 302 118 215 312 321 331 341 350 360 369 379 389 398 408 N. 1 2 3 4 5 6 7 8 9 [28] N. 1 2 3 4 5 6 7 8 9 460 51 52 53 65 321 331 341 350 360 369 379 389 398 408 418 514 610 427 523 619 437 533 629 447 543 639 456 552 648 466 562 658 475 571 667 485 581 677 495 591 686 504 600 696 54 55 56 706 801 896 715 811 906 725 820 916 734 830 925 744 839 935 753 849 944 763 858 954 772 868 963 782 877 973 792 887 982 57 58 59 460 ei 62 63 992 66 087 181 *001 096 191 *011 106 200 *020 115 210 *030 124 219 *039 134 229 *049 143 238 *058 153 247 *068 162 257 *077 172 266 276 285 295 304 314 323 332 342 351 361 370 464 558 380 474 567 389 483 577 398 492 586 408 502 596 417 511 605 427 521 614 436 530 624 445 539 633 455 549 642 64 65 66 652 74.5 839 661 755 848 671 764 857 680 773 867 689 783 876 699 792 885 708 801 894 717 811 904 727 820 913 736 829 922 67 68 69 470 71 72 73 932 67 025 117 941 034 127 950 043 136 960 052 145 969 062 154 978 071 164 987 ■ 080 173 997 089 182 *006 099 191 *015 108 201 210 219 228 237 247 256 265 274 284 293 302 394 486 311 403 495 321 413 504 330 422 514 339 431 523 348 440 532 357 449 541 367 459 550 376 468 560 385 477 569 74 75 76 578 669 761 587 679 770 596 688 779 605 697 788 614 706 797 624 715 806 633 724 815 642 733 825 651 742 834 660 752 843 77 78 79 480 81 82 83 852 943 68 034 861 952 043 870 961 052 879 970 061 888 979 070 897 988 079 906 997 088 916 *006 097 925 *015 106 934 *024 115 124 133 142 151 160 169 178 187 196 205 215 305 395 224 314 404 233 323 413 242 332 422 251 341 431 260 350 440 269 359 449 278 368 458 287 377 467 296 386 476 84 85 86 485 574 664 494 583 673 502 592 681 511 601 690 520 610 699 529 619 708 538 628 717 547 637 726 556 646 735 565 655 744 87 88 89 490 91 92 93 753 842 931 762 851 940 771 860 949 780 869 958 789 878 966 797 886 975 806 895 984 815 904 993 824 913 *002 833 922 *011 69 020 028 037 046 055 064 073 082 090 099 108 197 285 117 205 294 126 214 302 135 223 311 144 232 320 152 241 329 161 249 338 170 258 346 179 267 355 188 276 364 94 95 96 373 461 548 381 469 557 390 478 566 399 487 574 408 496 583 417 504 592 425 513 601 434 522 609 443 531 618 452 539 627 97 98 99 500 636 723 810 644 732 819 653 740 827 662 749 836 671 758 845 679 767 854 688 775 862 697 784 871 705 793 880 714 801 888 897 906 914 923 932 940 949 958 966 975 If, 1 2 3 4 5 6 7 8 9 [29] N. O 1 2 3 4 5 6 7 8 9 600 01 02 03 69 897 906 914 923 932 940 949 958 966 975 984 70 070 157 992 079 165 *001 088 174 *010 096 183 *018 105 191 *027 114 200 *036 122 209 *044 131 217 *053 140 226 *062 148 234 04 05 06 243 329 415 252 338 424 260 346 432 269 355 441 278 364 449 286 372 458 295 381 467 303 389 475 312 398 484 321 406 492 07 08 09 610 11 12 13 501 586 672 509 595 680 518 603 689 526 612 697 535 621 706 544 629 714 552 638 723 561 646 731 569 655 740 578 663 749 757 766 774 783 791 800 808 817 825 834 842 , 927 71 012 851 935 020 859 944 029 868 952 037 87B 961 046 885 969 054 893 978 063 902 986 071 910 995 079 919 *003 088 14 15 16 096 181 265 105 189 273 113 198 282 122 206 290 130 214 299 139 223 307 147 231 315 155 240 324 164 248 332 172 257 341 17 18 19 620 21 22 23 349 433 517 357 441 525 366 450 533 374 458 542 383 466 550 391 475 559 399 483 567 408 492 575 416 500 584 425 508 592 600 609 617 625 634 642 650 659 667 675 684 767 850 692 775 858 700 784 867 709 792 875 717 800 883 725 809 892 734 817 900 742 825 908 750 834 917 759 842 925 24 25 26 933 72 016 099 941 024 107 950 032 115 958 041 123 966 049 132 975 057 140 983 066 148, 991 074 156 999 082 165 *008 090 173 27 28 29 630 31 32 33 181 263 346 189 272 354 198 280 362 206 288 370 214 296 378 222 304 387 230 313 395 239 321 403 247 ■1329 411 255 337 419 428 436 444 452 460 469 477 485 493 501 509 591 673 518 599 681 526 607 689 534 616 697 542 624 705 550 632 713 558 640 722 567 648 730 575 656 738 583 665 746 34 35 36 754 835 916 762 843 925 770 852 933 779 860 941 787 868 949 795 876 957 803 884 965 . 811 892 973 819 900 981 827 908 989 37 38 39 640 41 42 43 997 73 078 159 *006 086 167 *014 094 175 *022 102 183 *030 111 191 *038 119 199 *046 127 207 *054 135 215 *062 143 223 *070 151 231 239 247 255 263 272 280 288 296 304 312 320 400 480 328 408 488 336 416 496 344 424 504 352 432 512 360 440 520 368 448 528 376 456 536 384 464 544 392 472 552 44 45 46 560 640 719 568 648 727 576 656 735 484 664 743 592 672 751 600 679 759 608 687 767 616 695 775 624 703 783 632 711 791 47 48 49 660 799 878 957 807 886 965 815 894 973 823 902 981 830 910 989 838 918 997 846 926 *005 854 933 *013 862 941 *020 870 949 *028 74 036 044 052 060 068 076 084 092 ■ 099 107 N. 1 2 3 4 5' 6 7 8 9 [30] N. 1 2 8 4 5 6 7 8 9 550 51 52 53 74 036 044 052 060 068 076 084 092 099 107 115 194 273 123 202 280 131 210 288 139 218 296 147 225 304 155 233 312 162 241 320 170 249 327 178 257 335 186 265 343 54 55 56 351 429 507 359 437 515 367 445 523 374 453 531 382 461 539 390 468 547 398 476 554 406 484 562 414 492 570 421 500 578 57 58 59 660 61 62 63 586 663 741 593 671 749 601 679 757 609 687 764 617 695 772 624 702 780 632 710 788 640 718 796 648 726 803 656 733 811 819 827 834 842 850 858 865 873 881 889 896 974 75 051 904 981 059 912 989 066 920 997 074 927 *005 082 935 *012 089 943 *020 097 950 *028 105 958 *035 113 966 *043 120 64 65 66 128 205 282 136 213 289 143 220 297 151 228 305 159 236 312 166 243 320 174 251 328 182 259 335 189 266 343 197 274 351 67 68 69 570 71 72 73 358 435 511 366 442 519 374 450 526 381 458 534 389 465 542 397 - 473 549 404 481 557 412 488 565 420 496 572 427 504 580 587 595 603 610 618 626 633 641 648 656 664 740 815 671 747 823 679 755 831 686 762 838 694 770 846 702 778 853 709 785 861 717 793 868 724 800 876 732 808 884 74 75 76 891 967 76 042 899 974 050 906 982 057 914 989 065 921 997 072 929 *005 080 937 *012 087 944 *020 095 952 *027 103 959 *035 110 77 78 79 680 81 82 83 118 193 . 268 125 200 275 133 208 283 140 215 290 148 223 298 155 230 305 163 238 313 170 245 320 178 253 328 185 260 335 343 350 358 365 373 380 388 395 403 410 - 418 492 567 425 500 574 433 507 582 440 515 589 448 522 597 455 530 604 462 537 612 470 545 619 477 552 626 485 559 634 84 85 86 641 716 790 649 723 797 656 730 805 664 738 812 671 745 819 678 753 827 686 760 834 693 768 842 701 775 849 708 782 856 87 88 89 590 91 92 93 864 938 77 012 871 945 019 879 953 026 886 960 034 893 967 041 901 975 048 908 982 056 916 989 063 923 997 070 930 *004 078 085 093 100 107 115 122 129 137 144 151 159 232 305 166 240 313 173 247 320 181 254 327 188 262 335 195 269 342 203 276 349 210 283 357 217 291 364 225 298 371 94 95 96 379 452 525 386 459 532 393 466 539 401 474 546 408 481 554 415 488 561 422 495 568 430 503 576 437 510 583 444 517 590 97 98 99 597 670 743 605 677 750 612 685 757 619 692 764 627 699 772 634 706 779 641 714 786 648 721 793 656 728 801 663 735 808 600 815 822 830 837 844 851 859 866 873 880 Jf. 1 2 3 4 5 6 7 8 9 [31] If. 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 49 660 N. 77 815 887 960 78 032 104 176 247 319 390 462 533 604 675 746 817 888 958 79 029 099 169 239 309 379 449 518 588 657 727 796 865 934 80 003 072 140 209 277 346 414 482 550 618 686 754 821 956 81 023 090 158 224 291 822 895 967 039 111 183 254 326 398 469 540 611 682 753 824 895 i965 036 106 176 246 316 386 456 525 595 664 734 803 872 830 902 974 046 118 190 262 333 405 476 547 618 689 760 831 902 972 043 113 183 253 941 010 079 147 216 284 353 421 489 557 625 693 760 828 895 963 030 097 164 231 298 323 393 463 532 602 671 741 810 879 837 909 981 053 125 197 269 340 412 483 554 948 017 085 154 223 291 359 428 496 564 632 699 767 835 902 969 037 104 171 238 305 625 696 767 838 909 979 050 120 190 260 330 400 470 539 609 678 748 817 886 844 916 988 061 132 204 276 347 419 490 561 955 024 092 161 229 298 366 434 502 570 638 706 774 841 909 976 043 111 178 245 311 633 704 774 845 916 986 057 127 197 851 267 337 407 477 546 616 685 754 824 893 962 030 099 168 236 305 373 441 509 577 924 996 068 140 211 283 355 426 497 569 640 711 781 852 923 993 064 134 204 274 344 414 484 553 623 692 761 831 900 859 931 *003 075 147 219 290 362 433 504 576 647 718 789 859 930 *000 071 141 211 281 969 645 713 781 848 916 983 050 117 184 251 318 037 106 175 243 312 380 448 516 584 652 720 787 855 922 990 057 124 191 258 325 351 421 491 560 630 699 768 837 906 866 938 *010 082 154 226 297 369 440 512 583 654 725 796 866 937 *007 078 148 218 288 975 044 113 182 250 318 387 455 523 591 659 726 794 862 929 996 064 131 198 265 331 358 428 498 567 637 706 775 844 913 873 945 *017 089 161 233 305 376 447 519 590 661 732 803 873 944 *014 085 155 225 295 982 051 120 188 257 325 393 462 530 598 665 365 435 505 574 644 713 782 851 920 880 952 *025 097 168 240 312 383 455 526 597 739 810 880 951 *021 092 162 232 989 058 127 195 264 332 400 468 536 604 733 801 868 936 *003 070 137 204 271 338 672 740 808 875 943 *010 077 144 211 278 345 302 372 442 511 581 650 720 789 858 927 996 065 134 202 271 339 407 475 543 611 679 747 814 882 949 *017 084 151 218 285 351 [32] N. 1 2 3 4 5 6 7 8 9 650 51 52 53 81 291 298 305 311 318 325 331 338 345 351 358 425 491 365 431 498 371 438 505 378 445 511 385 451 518 391 458 525 398 465 531 405 471 538 411 478 544 418 485 551 54 55 56 558 624 690 564 631 697 571 637 704 578 644 710 584 651 717 591 657 723 598 664 730 604 671 737 611 677 743 617 684 750 57 58 59 660 61 62 63 757 823 889 763 829 895 770 836 902 776 842 908 783 849 915 790 856 921 796 862 928 803 869 935 809 875 941 816 882 948 954 961 968 974 981 987 994 *000 *007 *014 82 020 086 151 027 092 158 033 099 164 040 105 171 046 112 178 053 119 184 060 125 191 066 132 197 073 138 204 079 145 210 64 65 66 217 282 347 223 289 354 230 295 360 236 302 367 243 308 373 249 315 380 256 321 387 263 328 393 269 334 400 276 341 406 67 68 69 670 71 72 73 413 478 543 419 484 549 426 491 556 432 497 562 439 504 569 445 510 575 452 517 582 458 523 588 465 530 595 471 536 601 607 614 620 627 633 640 646 653 659 666 672 737 802 679 743 80S 685 75C) 814 692 756 821 698 763 827 705 769 834 711 776 840 718 782 847 724 789 853 730 795 860 74 75 76 866 930 995 872 937 *001 879 943 *008 885 950 *014 892 956 *02ff 898 963 *027 905 969 *033 911 975 *040 918 982 ♦■046 924 988 *052 77 78 79 680 81 82 83 83 059 123 187 065 129 193 072 136 200 078 142 206 085 149 213 091 155 219 097 161 225 104 168 232 110 174 238 117 181 245 251 257 264 270 276 283 289 296 302 308 315 378 442 321 385 448 327 391 455 334 398 461 340 404 467 347 410 474 353 417 480 359. 423 487 366 429 493 372 436 499 84 85 86 506 569 632 512 575 639 518 582 645 525 588 651 531 594 658 537 601 664 544 607 670 550 613 677 556 620 683 563 626 689 67 88 89 ^90 91 92 93 696 759 822 702 765 828 708 771 835 715 778 841 721 784 847 727 790 853 734 797 860 740 803 866 746 809 872 753 816 879 885 891 897 904 910 916 923 929 935 942 948 84 Oil 073 954 017 080 960 023 086 967 029 092 973 036 098 979 042 105 985 048 111 992 055 117 998 061 123 *004 067 130 94 95 96 136 198 261 142 205 267 148 211 273 155 217 280 161 223 286 167 230 292 173 236 298 180 242 305 186 248 311 192 255^ 317 97 98 99 700 323 386 448 330 392 454 336 398 460 342 404 466 348 410 473 354 417 479 361 423 485 367 429 491 373 435 497 379 442 504 510 516 522 528 535 541 547 553 559 566 N. O 1 2 3 4 5 6 7 8 9 [33] N. 1 2 3 4 5 6 7 8 9 700 01 02 03 84 510 516 522 528 535 541 547 553 559 566 572 634 696 578 640 702 584 646 708 590 652 714 597 658 720 603 665 726 609 671 733 615 677 739 621 683 745 628 689 751 04 05 06 757 819 880 763 825 887 770 831 893 776 837 899 782 844 905 788 850 911 794 856 917 800 862 924 807 868 930 813 874 936 07 08 09 710 11 12 13 942 85 003 065 948 009 071 954 016 077 960 022 083 967 028 089 973 034 095 979 040 101 985 046 107 991 052 114 997 058 120 126 132 138 144 150 156 163 169 - 175 181 187 248 309 193 254 315 199 260 321 205 266 327 211 272 333 217 278 339 224 285 345 230 291 352 236 297 358 242 303 364 14 15 16 370 431 491 376 437 497 382 443 503 388 449 509 394 455 516 400 461 522 406 467 528 412 473 534 418 479 540 425 485 546 17 18 19 720 21 22 23 552 612 673 558 618 679 564 625 685 570 631 691 576 637 697 582 643 703 588 649 709 594 655 715 600 661 721 606 667 727 733 739 745 751 757 763 769 775 781 788 794 854 914 800 860 920 806 866 926 812 872 932 818 878 938 824 884 944 830 890 950 836 896 956. 842 902 962 848 908 968 24 25 26 974 86 034 094 980 040 100 986 046 106 992 052 112 998 058 118 *004 064 124 *010 070 130 *016 076 136 *022 082 141 *028 088 147 27 28 29 730 31 32 33 153 213 273 159 219 279 165 225 285 171 231 291 177 237 297 183 243 303 189 249 308 195 255 314 201 261 320 207 267 326 332 338 344 350 356 362 368 374 380 386 392 451 ■ 510 390 457 516 404 463 522 410 469 528 415 475 534 421 481 540 427 487 546 433 493 552 439 499 558 445 504 564 34 35 36 570 629 688 576 635 694 581 641 700 587 646 705 593 652 711 599 658 717 605 664 723 611 670 729 617 676 735 623 682 741 37 38 39 740 41 42 43 747 806 864 753 812 870 759 817 876 764 823 882 770 829 888 776 835 894 782 841 900 788 847 906 794 853 911 800 859 917 923 929 935 941 947 953 958 964 970 976 982 87 040 099 988 046 105 994 052 111 999 058 116 *005 064 122 *011 070 128 *017 075 134 *023 081 140 *029 087 146 *035 093 151 44 45 46 157 216 274 163 221 280 169 227 286 175 233 ■291 181 239 297 186 245 303 192 251 309 198 256 315 204 262 320 210 268 326 47 48 49 750 332 390 448 338 396 454 344 402 460 349 408 466 355 413 471 361 419 477 367 425 483 373 431 489 379 437 495 384 442 500 506 512 518 523 529 535 541 547 552 558 N. 1 2 3 4 5 6 7 8 9 [34] N. 1 2 3 4 5 6 7 8 9 750 51 52 53 87 506 512 518 523 529 535 541 547 552 558 564 622 679 570 628 685 576 633 691 581 639 697 587 645 703 593 651 708 599 656 714 604 662 720 610 668 726 616 674 731 54 55 56 737 795 852 743 800 858 749 806 864 754 812 869 760 818 875 766 823- 881 772 829 887 777 835 892 783 841 898 789 846 904 57 58 59 760 61 62 63 910 967 88 024 915 973 030 921 978 036 927 984 041 933 990 047 938 996 053 944 *001 058 950 *007 064 955 *013 070 961 *018 076 081 087 093 098 104 110 116 121 127 133 138 195 252 144 201 258 150 207 264 156 213 270 161 218 275 167 224 281 173 230 287 178 235 292 184 241 298 190 247 304 64 65 66 309 366 423 315 372 429 321 377 434 326 383 440 332 389 446 338 395 451 343 400 457 349 406 463 355 412 468 360 417 474 67 68 69 770 71 72 73 480 536 593 485 542 598 491 547 604 497 553 610 502 559 615 508 564 621 513 570 627 519 576 632 525 581 638 530 587 643 649 655 660 666 672 677 683 689 694 700 705 762 818 711 767 824 717 773 829 722 779 835 728 784 840 734 790 846 739 795 852 745 801 857 750 807 863 756 812 868 74 75 76 874 930 986 880 936 992 885 941 997 891 947 *003 897 953 *009 902 958 *014 908 964 *020 913 969 *025 919 975 *031 925 981 *037 77 78 79 780 81 82 83 89 042 098 154 048 104 159 053 109 165 059 115 170 064 120 176 070 126 182 076 131 187 081 137 193 087 143 198 092 148 204 209 215 221 226 232 237 243 248 254 260 265 321 376 271 326 382 276 332 387 .282 337 393 287 343 398 293 348 404 298 354 409 304 360 415 310 365 421 315 371 426 84 85 86 432 487 542 437 492 548 443 498 553 448 504 559 454 509, 564 459 515 570 465 520 575 470 526 581 476 531 586 481 537 592 87 88 89 790 91 92 93 597 653 708 603 658 713 609 664 719 614 669 724 620 675 730 625 680 735 631 686 741 636 691 746 642 697 752 647 702 757 763 768 774 779 785 790 796 801 807 812 818 873 927 823 878 933 829 883 938 834 889 944 840 894 949 845 900 955 851 905 960 856 911 966 862 916 971 867 922 977 94 95 96 982 90 037 091 988 042 097 993 048 102 998 053 108 *004 059 113 *009 064 119 *015 069 124 *020 075 129 *026 080 135 *031 086 140 97 98 99 800 146 200 255 151 206 260 157 211 266 162 217 271 168 222 276 173 227 282 179 233 287 184 238 293 189 244 298 195 249 304 309 314 320 325 331 336 342 347 352 358 N. 1 2 3 4 s 6 7 8 9 [35] N. 800 01 02 03 O 1 2 3 4 S 6 7 8 9 90 309 314 320 325 331 336 342 347 352 358 363 417 472 369 423 477 374 428 482 380 434 488 385 439 493 390 445 499 396 450 504 401 455 509 407 461 515 412 466 520 04 05 06 526 580 634 531 585 639 536 590 644 542 596 650 547 601 655 553 607 660 558 612 666 563 617 671 569 623 677 574 628 682 07 08 09 810 li 12 13 687 741 795 693 747 800 698 ■ 752 806 703 757 811 709 763 816 714 768 822 720 773 827 725 779 832 730 784 838 736 789 843 849 854 859 865 870 875 881 886 891 897 902 956 91 009 907 961 014 913 966 020 918 972 025 924 977 030 929 982 036 934 988 041 940 993 046 945 998 052 950 *004 057 14 15 . 16 062 116 169 068 121- 174 073 126 180 078 132 185 084 137 190 089 142 196 094 148 201 100 153 206 105 158 212 110 164 217 17 18 19 820 21 22 23 222 275 328 228 281 334 233 286 339 238 291 344 243 297 350 249 302 355 254 307 360 259 312 365 265 318 371 270 323 376 381 387 392 397 403 408 413 418 424 429 434 487 540 440 492 545 445 498 551 450 503 556 455 508 561 461 514 566 466 519 572 471 524 577 477 529 582 482 535 587 24 25 26 593 645 698 598 651 703 603 656 709 609 661 714 614 666, 719 619 672 724 624 677 730 630 682 735 635 687 740 640 693 745 27 28 29 830 31 32 33 751 803 855 756 808 861 761 814 866 766 819 871 772 824 876 777 829 882 782 834 887 787 840 892 793 845 897 798 850 903 908 913 918 924 929 934 939 944 950 955 960 92 012 065 965 018 070 971 023 075 976 028 080 981 033 085 986 038 091 991 044 096 997 049 101 *002 054 106 *007 059 111 34 35 36 117 169 221 122 174 226 127 179 231 132 184 236 137 189 241 143 195 247 148 200 252 153 205 257 158 210 262 163 215 267 37 38 39 840 41 42 43 273 324 376 278 330 381 283 335 387 288 340 392 293 345 397 298 350 402 , 304 355 407 309 361 412 314 366 418 319 371 423 428 433 438 443 449 454 459 464 469 474 480 531 583 485 536 588 490 542 593 495 547 598 500 552 603 505 557 609 511 562 614 516 567 619 521 572 624 526 578 629 44 45 46 634 686 737 639 691 742 645 696 747 650 701 752 655 706 758 660 711 763 665 716 768 670 722 773 675 727 778 681 732 783 47 48 49 850 788 840 891 793 845 896 799 850 901 804 855 906 809 860 911 814 865 916 819 870 921 824 875 927 829 881 932 834 886 937 942 947 952 957 962 967 973 978 983 988 1 2 3 4 5 6 7 S 9 [36] If. 1 2 3 4 5 6 7 8 9 850 51 52 53 92 942 947 952 957 962 967 973 978 983 988 993 93 044 095 998 049 100 *003 054 105 *008 059 110 *013 064 115 *018 069 120 *024 075 125 *029 080 131 *034 085 136 *039 090 141 54 55 56 146 197 247 151 202 252 156 207 258 161 212 263 166 217 268 171 222 273 176 227 278 181 232 283 186 237 288 192 242 293 57 58 59 860 61 62 63 298 349 399 303 354 404 308 359 409 313 364 414 318 369 420 323 374 425 328 379 430 334 384 435 339 389 440 344 394 445 450 455 460 465 470 475 480 485 490 495 500 551 601 505 556 606 510 561 611 515 566 616 520 571 621 526 576 626 531 581 631 536 586 636 541 591 641 546 596 646 64 65 66 651 702 752 656 707 757 661 712 762 666 717 767 671 722 772 676 727 777 682 732 782 687 737 787 692 742 792 697 747 797 67 68 69 870 71 72 73 802 852 902 807 857 907 812 862 912 817 867 917 822 872 922 827 877 927 832 882 932 837 887 937 842 892 942 847 897 947 952 957 962 967 972 977 982 987 992 997 94 002 052 101 007 057 106 012 062 111 017 067 116 022 072 121 027 077 126 032 082 131 037 086 136 042 091 141 047 096 146 74 75 76 151 201 250 156 206 255 161 211 260 166 216 265 171 221 270 176 226 275 181 231 280 186 236 285 191 240 290 196 245 295 77 78 79 880 81 82 83 300 349 399 305 354 404 310 359 409 315 364 414 320 369 419 325 374 424 330 379 429 335 384 433 340 389 438 345 394 443 448 453 458 463 468 473 478 483 488 493 498 547 596 503 552 601 507 557 606 512 562 611 517 567 616 522 571 621 527 576 626 532 581 630 537 586 635 542 591 640 84 85 86 645 694 743 650 699 748 655 704 753 660 709 758 665 714 763 670 719 768 675 724 773 680 729 778 685 734 783 689 738 787 87 88 89 890 91 92 93 792 841 890 797 846 895 802 851 900 807 856 905 812 861 910 817 866 915 822 871 919 827 876 924 832 880 929 836 885 934 939 944 949 954 959 963 968 973 978 983 988 95 036 085 993 041 090 998 046 095 *002 051 100 *007 056 105 *012 061 109 *017 066 114 *022 071 119 *027 075 124 *032 080 129 94 95 96 134 182 231 139 187 236 143 192 240 148 197 245 153 202 250 158 207 255 163 211- 260 168 216 265 173 221 270 177 226 274 97 98 99 900 279 328 376 284 332 381 289 337 386 294 342 390 299 347 395 303 352 400 308 357 405 313 361 410 318 366 415 323 371 419 424 429 434 439 444 448 453 458 463 468 N. O 1 2 3 4 5 6 7 8 9 [37] N. 1 2 3 4 5 6 7 8 9 900 01 02 03 95 424 429 434 439 444 448 453 458 463 468 472 521 569 477 525 574 482 530 578 487 535 583 492 540 588 497 545 593 501 550 598 506 554 602 511 559 607 516 564 612 04 05 06 617 665 713 622 670 718 626 674 722 631 679 727 636 684 732 641 689 737 646 694 742 650 698 746 655 703 751 660 708 756 07 08 09 910 11 12 13 761 809 856 766 813 861 770 818 866 775 823 871 780 828 875 785 832 880 789 837 885 794 842 890 799 847 895 804 852 899 904 909 914 918 923 928 933 938 942 947 952 999 96 047 957 *004 052 961 *009 057 966 *014 061 971 *019 066 976 *023 071 980 *028 076 985 *033 080 990 *038 085 995 *042 090 14 15 16 095 142 190 099 147 194 104 152 199 109 156 204 114 161 209 118 166 2-13 123 171 218 128 175 223 133 180 227 137 185 232 17 18 19 920 21 22 23 237 284 332 242 289 336 246 294 341 251 298 346 256 303 350 261 308 355 265 313 360 270 317 365 275 322 369 280 327 374 379 384 388 393 398 402 407 412 417 421 426 473 520 431 478 525 435 483 530 440 487 534 445 492 539 450 497 544 454 501 548 459 506 553 464 511 558 468 515 562 24 25 26 567 614 661 572 619 666 577 624 670 581 628 675 586 633 680 591 638 685 595 642 689 600 647 694 605 652 699 609 656 703 27 28 29 930 31 32 33 708 755 802 713 759 806 717 764 811 722 769 816 727 774 820 731 778 825 736 783 830 741 788 834 745 792 839 750 797 844 848 853 858 862 867 872 876 881 886 890 895 942 988 900 946 993 904 951 997 909 956 *002 914 960 *007 918 965 *011 923 970 *016 928 974 *021 932 979 *025 937 984 *030 34 35 36 97 035 081 128 039 086 132 044 090 137 049 095 142 053 100 146 058 104 151 063 109 155 067 114 160 072 118 165 077 123 169 37 38 39 940 41 42 43 174 220 267 179 225 271 183 230 276 188 234 280 192 239 285 197 243 290 202 248 294 206 253 299 211 257 304 216 262 308 313 317 322 327 331 336 340 345 350 354 359 405 451 364 410 456 368 414 460 373 419 465 377 424 470 382 428 474 387 433 479 391 437 483 396 442 488 400 447 493 44 45 46 497 543 589 502 548 594 506 552 598 511 557 603 516 562 607 520 566 612 525 571 617 529 575 621 534 580 626 539 585 630 47 48 49 950 635 681 727 640 685 731 644 690 736 649 695 740 653 699 745 658 704 749 663 708 754 667 713 759 672 717 763 676 722 768 772 777 782 786 791 795 800 804 809 813 N. 1 2 3 4 6 6 7 8 9 [38] If. 1 2 3 4 5 6 7 8 9 960 51 52 53 97 772 777 782 786 791 795 800 804 809 813 818 864 909 823 868 914 827 873 918 832 877 923 836 882 928 841 886 932 845 891 937 850 896 941 855 900 946 859 905 950 54 55 56 955 98 000 046 959 005 050 964 009 055 968 014 059 973 019 064 978 023 068 982 028 073 987 032 078 991 037 082 996 041 087 57 53 59 960 61 62 63 091 137 182 096 141 186 100 146 191 105 150 195 109 155 200 114 159 204 118 164 209 123 168 214 127 173 218 132 177 223 227 232 236 241 245 250 254 259 263 268 272 318 363 277 322 «67 281 327 372 286 331 376 290 336 381 295 340 385 299 345 390 304 349 394 308 354 399 313 358 403 64 65 66 408 453 498 412 457 502 417 462 507 421 466 511 426 471 516 430 475 520 435 480 525 439 484 529 444 489 534 448 493 538 67 68 69 970 71 72 73 543 588 632 547 592 637 552 597 641 556 601 646 561 605 650 565 610 655 570 614 659 574 619 664 579 623 668 583 628 673 677 682 686 691 695 700 704 709 713 717 722 767 811 726 771 816 731 776 820 735 780 825 740 784 829 744 789 834 749 793 838 753 798 843 758 802 847 762 807 851 74 75 76 856 900 945 860 905 ,949 865 909 954 869 914 958 874 918 963 878 923 967 883 927 972 887 932 976 892 936 981 896 941 985 77 78 79 980 81 82 83 989 99 034 078 994 038 083 998 043 087 *003 047 092 *007 052 096 *012 056 100 *016 061 105 *021 065 109 *025 069 114 *029 074 118 123 127 131 136 140 145 149 154 158 162 167 211 255 171 216 260 176 220 264 180 224 269 185 229 273 189 233 277 193 238 282 198 242 286 202 247 291 207 251 295 84 85 86 300 344 388 304 348 392 308 352 396 313 357 401 317 361 405 322 366 410 326 370 414 330 374 4'19 335 379 423 339 383 427 87 88 89 990 91 92 93 432 476 520 436 480 524 441 484 528 445 489 533 449 493 537 454 498 542 458 502 546 463 506 550 467 511 555 471 515 559 564 568 572 577 581 585 590 594 599 603 607 651 695 612 656 699 616 660 704 621 664 708 625 669 712 629 673 717 634 677 721 638 682 726 642 686 730 647 691 734 94 95 96 739 782 826 743 787 830 747 791 835 752 795 839 756 800 843 760 804 848 765 808 852 769 813 856 774 817 861 778 822 865 97 98 99 1000 870 913 957 874 917 961 878 922 965 883 926 970 887 930 974 891 935 978 896 939 983 900 944 987 904 948 991 909 952 996 00 000 004 009 013 017 022 026 030 035 039 If. 1 2 3 4 5 6 7 8 9 [39] TABLE II LOGS AND COLOGS OF CERTAIN MUCH-USED NUMBERS Nttmber Logarithm Co LOGARITHM 2 0.3010300 9.6989700-10 3 0.4771213 9.5228787-10 V2 0.1505150 9.8494850-10 V3 0.2385607 9.7614393-10 IT 0.4971499 9.5028501-10 ^2 0.9942997 9.0057003-10 -2ir 0.7981799 9.2018201-10 V^ 0.2485749 9.7514251-10 57.2957795 1.7581226 8.2418774-10 206264.806 5.3144251 4.6855749-10 EivE Place 2 0.30103 9.69897-10 3 0.47712 9.52288-10 V2 0.15052 9.84948-10 V3 0.23856 9.76144-10 IT 0.49715 9.50285-10 T^ 0.99430 9.00570-10 27r 0.79818 9.20182-10 V^ 0.24857 9.75143-10 57.2957795 1.75812 8.24188-10 206264.806 5.31443 4.68557-10 EouK Place 2 0.3010 9.6990-10 3 0.4771 9.5229-10 ^/2 0.1505 9.8495-10 V3 0.2386 9.7614-10 IT 0.4971 9.5029-10 ff2 0.9943 9.0057-10 27r 0.7982 9.2018-10 V^ 0.2486 9.7514-10 57.2956695 1.7581 8.2419-10 206264.806 5.3144 4.6856-10 [40] TABLE III FIVE-PLACE LOGARITHMS SINE, COSINE, TANGENT, AND COTANGENT EACH MINUTE OF THE QUADRANT [411 1 L. Sin. L. Tang. L. Cotg. L. Cos. CO 00 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 2.63 318 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 60 n 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 0° 29 7.92 612 7.92 613 2.07 387 9.99 998 31 89° 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.b6 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 ^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 L. Cog. L. Cotg. L. Tang. L. Sin. t [421 ( L. Sin. L. Tang. L. Cotg. L. Cos. "^ 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 60 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 8.39 818 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 1° 29 8.41 307 8.41 321 1.58 679 9.99 985 31 88° 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 9.99 984 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 60 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 L. Cos. L. Cotg. L. Xang. I. Sin. 1 [43] 1 L. Sin. L. Tang. L. Cotg. L. Cos. 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 60 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 8.62 497 8.62 535 1.37 465 9.99 961 35 26 8.62 795 8.62 834 1.37 166 9.99 961 34 27 8.63 091 8.63 131 1.36 869 9.99 960 33 28 8.63 385 8.63 426 1.36 574 9.99 960 32 2° 29 8.63 678 8.63 718 1.36 282 9.99 959 31 ST 30 8.63 968 8.64 009 1.35 991 9.99 959 30 31 8.64 256 8.64 298 1.35 702 9.99 958 29 32 8.64 543 8.64 585 1.35 415 9.99 958 28 33 8.64 827 8.64 870 1.35 130 9.99 957 27 34 8.65 110 8.65 154 1.34 846 9.99 956 26 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 5? 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 L. Cos. L. Cotg. L. Tang. L. Sin. ; [44] / L. Sin. L. Tang. L. Cotg. 1. Cos. 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 1.23913 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 1.21978 9.99 921 33 28 8.78 152 8.78 232 1.21 768 9.99 920 32 3° 29 8.78 360 8.78 441 1.21 559 9.99 920 31 86° 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.gl 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 8^2 610 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 L. Cos. I. Cotg. L. Tang. L. Sin. / [45] 1 L. Sin. L. Tang. 1. Colg. L. Cos. ^^ 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 60 ' 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 r 29 8.89 304 8.89 437 1.10 563 9.99 867 31 85° 30 8.89 464 8.89 598 1.10 402 9.99 866 30 31 8.89 625 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 ' 8.91029 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 60 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 L. Cos. 1. Cotg. L. Tang. L. Sin. / [46] 1 L. Sin. L. Tang. L. Cotg. L. Cos. 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 60 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 5° 29 8.98 026 8.98 225 1.01 775 9.99 801 31 84° 30 8.98 157 8.98 358 1.01 642 9.99 800 30 31 8.98 288 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 60 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 L. Cos. 1. Colg. L. Tang. L. Sin. / [47] / L. Sin. L. Tang. I. Cotg. L. Cos. 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 9.0? 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 9.99 753 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 25 9.04 828 9.05 101 0.94 899 9.99 727 35 26 9.04 940 9.05 214 0.94 786 9.99 726 34 27 9.05 052 9.05 328 0.94 672 9.99 724 33 28 9.05 164 9.05 441 0.94 559 9.99 723 32 6° 29 9.05 275 9.05 553 0.94 447 9.99 721 31 83° 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 60 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 L. Cos. L. Cotg. L. Tang. L. Sin. / [48] 1 L. Sin. L. Tang. I. Cotg. L. Cos. "^ 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 9.08 795 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 60 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 33 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 T 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 0.87 475 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 0.83 616 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 9.99 595 12 49 9.13 355 9.13 761 0.86 239 9.99 595 11 60 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 9.14 780 0.85 220 9.99 575 1. Cos. L. Cotg. L. Tang. L. Sin. 1 [49] 1 L. Sin. L. Tang. L. Cotg. L. Cos. 9.14 3b6 9.14 780 0.85 220 9.99 575 bU 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 60 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 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 8° 29 9.16 886 9.17 363 0.82 637 9.99 522 31 81° 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 60 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 L. Cos. L. Cotg. L. Tang. L. Sin. / [50] 1 L. Sin. L. Tan. L. Cotg. 1. Cos, 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 9° 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 L. Cos. L. Cotg. L. Tang. L. Sin. 1 [51] / L. Sin. L. Tang. L. Cotg. L. Cos. "^r 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 10 9.24 677 9.25 365 0.74 635 9.99 313 bO 11 9.24 748 9.25 437 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 4U 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 10^ 29 9.25 995 9.26 726 0.73 274 9.99 269 31 79° 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 60 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 1 L. Cos. L. Cotg. L. Tang. L. Siu. r [52] / L. Sin. I. Tang. L. Cotg. L. Cos. "T" 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 0X9 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 ir 29 9.29 903 9.30 782 0.69 218 9.99 122 31 78° 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 60 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 L. Cos. L. Cotg. L. Tang. L. Sin. 1 [53] 1 L. Sin. L. Tan?. L. Cotg. L. Cos. □ 9.31 788 9.32 747 0.b7 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 Oil 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 12° 29 9.33 477 9.34 516 0.65 484 9.98 961 31 IT 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 60 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 Oil 9.98 890 6 55 9.34 934 9.36 047 0.63 953 9.98 887 5 58 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 L. Cos. L. Cotg. L. Tang. L. Sin. 1 [54] t L.Sin. L. Tang. L. Cotg. L. Cos. 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 60 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 13° 29 9.36 766 9.37 980 0.62 020 9.98 786 31 76° 30 9.36 819 9.38 035 0.61 965 9.98 783 30 31 9.36 871 9.38 091 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 9J37 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 60 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 Oil 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 L. Cos. I. Cotg. L. Tang. L. Sin. ' [55] 1 L. Sin. L. Tang. L. Col^. L. Cos. 9.38 3b8 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 eo H 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 M 9.39.369 9.40 742 0.59 258 9.98 627 4U 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 w 29 9.39 811 9.41 214 0.58 786 9.98 597 31 75^ ao 9.39 860 9.41 266 0.58 734 9.98 594 au 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 L. Cos. L. Cotg. 1. Tang. L. Sin. 1 [661 / L. Sin. L. Taug. L. Cotg. L. Cos. u 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 15° 29 9.42 644 9.44 250 0.55 750 9.98 395 31 74= 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 L. Cos. L. Cotg. L. Tang. L. Sin. / r57i t L. Sin. L. Tang. L. Cotg. L. Cos. 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 10 9.44 472 9.46 224 0.53 776 9.98 248 60 H 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 4U 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 16° 29 9.45 292 9.47 114 0.52 886 9.98 177 31 n° 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 Oil 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 Oil 9.98 106 12 49 9.46 136 9.48 035 0.51 965 9.98 102 11 60 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 i- L. Cos. L. Cotg. L. Tang. L. Sin. "^^" K rssi » L.Siii. L. Tang. I. Cotg. L. Cos. 9.46 594 9.48 534 0.51 466 9.98 060 ^r 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 60 n 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 17° 29 9.47 774 9.49 828 0.50 172 9.97 946 31 72° 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 23 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 m 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 60 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 9.4.8342 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 _6g_ 9.48 998 9.51 178 0.48 822 9 97 821 ^_ L. Cos. L. Cotg. L. Tang. LSin. F [69] t L. Sin. L. Tang. 1. Cotg. L. Cos. 9.48 998 9.51 178 0.48 822 9.97 821 60 1 9.49 037 9.51 221 0.48 779 9.97 817 59 2 9.49 076 9.51 264 0.48 736 9.97 812 58 3 9.49 lis 9.51 306 0.48 694 9.97 808 57 4 9.49 153 9.51 349 0.48 651 9.97 804 56 5 9.49 192 9.51 392 0.48 608 9.97 800 55 6 9.49 231 9.51 435 0.48 565 9.97 796 54 7 9.49 269 9.51 478 0.48 522 9.97 792 53 8 9.49 308 9.51 520 0.48 480 9.97 788 52 9 9.49 347 9.51 563 0.48 437 9.97 784 51 10 9.49 385 9.51 606 0.48 394 9.97 779 50 11 9.49 424 9.51 648 0.48 352 9.97 775 49 12 9.49 462 9.51 691 0.48 309 9.97 771 48 13 9.49 500 9.51 734 0.48 266 9.97 767 47 14 9.49 539 9.51 776 0.48 224 9.97 763 46 15 9.49 577 9.51 819 0.48 181 9.97 759 45 16 9.49 615 9.51 861 0.48 139 9.97 754 44 17 9.49 654 9.51 903 0.48 097 9.97 750 43 18 9.49 692 9.51 946 0.48 054 9.97 746 42 19 9.49 730 9.51 988 0.48 012 9.97 742 41 ^0 9.49 7fa8 9.52 031 0.47 969 9.97 738 40 21 9.49 806 9.52 073 0.47 927 9.97 734 39 22 9.49 844 9.52 115 0.47 885 9.97 729 38 23 9.49 882 9.52 157 0.47 843 9.97 725 37 24 9.49 920 9.52 200 0.47 800 9.97 721 36 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 18° 29 9.50 110 9.52 410 0.47 590 9.97 700 31 ir 30 9.50 148 9.52 452 0.47 548 9.97 696 30 31 9.50 185 9.^2 494 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.'^2 620 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:F3 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 9.9-/ ?84 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 L. Cos. 1. Cotg. L. Tang. 1. Sin. t [60] 1 L. Sin. L. Tang. L. Ootg. 1. Cos. 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 w 29 9.52 314 9.54 875 0.45 125 9.97 439 31 w 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 bO 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 Oil 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 L. Cos. L. Cotg. L. Tang. I. Sin. ; [61] ; LSin. L. Tang. 1. Cotg. L. Cos. ^^ 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 60 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 20° 29 9.54 399 9.57 235 0.42 765 9.97 163 31 fiQ° 30 9.54 433 9.57 274 0.42 726 9.97 159 30 \ju 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 60 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 L. Cos. L. Cotg. L. Tang. L. Sin. 1 [62] ; L. Sin. L. Tang. L. Cotg. L. Cos. 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 9.68 719 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 60 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 46 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 C83 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 2r 29 9.56 375 9.59 503 0.40 497 9.96 873 31 68° 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 1 L. Cos. L. Cotg. I. Tang. L. Sin. 1 [63] / L. Sin. L. Tang. 1. Cotg. L. Cos. "^T 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 e>o 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 22° 29 9.58 253 9.61 687 0.38 313 9.96 567 31 67° 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 9.96 530 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 L. Cos. L. Cotg. 1. Tang. L. Sin. / [64J 1 L. Sin. L. Tan^. L. Cotg. L. Cos. 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 9.02 961 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 9.9ti 349 60 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.C3 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 23° 29 9.60 041 9.63 796 0.36 204 9.96 245 31 66° 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 43 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 eo 9.60 931 9.64 858 0.35 142 9.96 073 L. Cos. I. Cotg. L. Tang. I. Sin. t [65] 1 LSin. L. Tang. L. Cotg. L. Cos. 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 9.61214 9.65 197 0.34 803 9.96 017 60 11 9.61 242 9.65 231 0.34 769 9.96 Oil 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 24° 29 9.61 74S 9.65 837 0.34 163 9.95 908 31 65° 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 60 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 L. Cos. L. Cotg. L. Tang. L. Sin. / [66] > L. Sin. L. Xang. L. Cotg. L. Cob. 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 5S 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 60 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 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 25° 29 9.63 372 9.67 817 0.32 183 9.95 555 31 64° 30 9.63 398 9.67 850 0.32 150 9.95 549 no 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 L. Cos. L. Colg. L. Tang. 1. Sin. ; [67] / L. Sin. L. Tang. 1. Cotg. L. Cos. 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 9.64 800 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 26° 29 9.64 927 9.69 742 0.30 258 9.95 185 31 fi.^° 30 9.64 953 9.69 774 0.30 226 9.95 179 30 HO 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 40 9:65 205 9.70 089 0.29 911 9.95 116 20 41 9.65 230 9.70 121 0.29 879 9.95 110 19 42 9.65 255 9.70 152 0.29 848 9.95 103 18 43 9.65 281 9.70 184 0.29 816 9.95 097 17 44 9.65 306 9.70 215 0.29 785 9.95 090 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 60 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 9.94 996 1 60 9.65 705 9.70 717 0.29 283 9.94 988 L. Cog. L. Cotg. L. Tang. L. Sin. 1 resi ; L. Sin. L. Tang. 1. Cotg. L. Cos. ~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 27° 29 9.66 416 9.71 617 0.28 383 9.94 799 31 62° 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 60 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 L. Cos. L. Cotg. L. Tang. L. Sin. 1 [69] / I. Sin. L. Tang. L. Cotg. L. Cos. 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 60 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 28° 29 9.67 843 9.73 446 0.26 554 9.94 397 31 6r 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 60 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 L. Cos. L. Cotg. L. Tang. L. Sin. 1 [70] L. Sin. L. Tan^. L. Cotg. L. Cos. 9.68 557 9.74 375 0.25 625 9.94 182 60 1 9.68 580 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° 29 9.69 212 9 75 235 0.24 765 9.93 977 31 60° 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 L. Cos. L. Cotg. L. Tang. L. Sin. ( [71] / L. Sin. L. Tang. L. Cotg. L. Cos. 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 5.9 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 48 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 30° 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 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 60 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 C77 0.22 323 9.93 360 7 54 9.71 058 9.77 703 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 bO 9.71 184 9.77 877 0.22 123 9.93 307 L. Cos. L. Cotg. L. Tang. L. Sin. / [72] / L. Sin. L. Tang. L. Cotg. L. Cos. 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 60 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 31° 29 • 9.71 788 9.78 704 0.21 296 9.93 084 31 58° 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 L. Cos. 1. Cotg. 1. Tang. L. Sin. 1 [73] 1 L. Sin. L. Tang. L. Cotg. L. Cos. 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 n 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 32° 29 9.73 002 9.80 391 0.19 609 9.92 611 31 57° 30 9.73 022 9.80 419 0.19 581 9.92 603 30 tM 9 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 60 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 L. Cos. L. Cotg. L. Tang. L. Sin. » [74] 1 L. Sin. L. Tang. L. Cotg. L. Co8. 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 60 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 33° 29 9.74 170 9.82 051 0.17 949 9.92 119 31 56° 30 9.74 189 9.82 078 0.17 922 9.92 111 30 v\w 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 60 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 L. Cos. L. Cotg. L. Tang. 1. Sin. 1 [75] ( L. Sin. L. Tang. L. Cotg. L. Cos. "~o~ 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 60 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 34° 29 9.75 294 9.83 686 0.16 314 9.91 603 31 55° 30 9.75 313 9.83 713 0.16 287 9.91 599 30 WW 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 Oil 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 9.54 3G1 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 I. Cos. L. Cotg. L. Tang. L. Sin. f r76i ; L. Sin. L. Tang. L. Cotg. L. Cos. 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 60 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 35° 29 9.76 378 9.85 300 0.14 700 9.91 078 31 sr 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 33 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 62S 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 60 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 L. Cos. L. Cotg. L. Tang. L. Sin. / [77] / L.Sin. L. Tang. L. Cotg. L. Cos. 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 36° 29 9.77 422 9.86 894 0.13 106 9.90 527 31 53° 30 9.77 439 9.86 921 0.13079 9.90 518 30 Ut* 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 60 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 L. Cos. L. Cotg. L. Tang. L. Sin. r [78] / L. Sin. L. Tang. L. Cotg. L. Cos. 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 37° 29 9.78 428 9.88 472 0.11 528 9.89 956 31 ft2° 30 9.78 445 9.88 498 0.11 502 9.89 947 30 VfW 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 0.11293 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 60 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 L. Cos. L. Cotg. L. Tang. L. Sin. [79] 1 L. Sin. L. Tang. L. Cotg. L. Cos. ^" 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 60 n 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 38° 29 9.79 399 9.90 035 0.09 965 9.89 364 31 Sl° 30 9.79 415 9.90 061 0.09 939 9.89 354 30 «x 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 9.89 244 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 60 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 L. Cos. L. Cotg. L. Tang. L. Sin. f rsoi / L. Sin. L. Tang. L. Cotg. L. Cos. ~~o~ 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 9.79 950 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 60 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 39° 29 9.80 336 9.91 585 0.08 415 9.88 751 31 50° 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 I. Cos. L. Cotg. L. Tang. L. Sin. 1 [81] / L. Sin. 1. Tang. L. Cotg. L. Cos. 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 40° 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 *tf 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 96'4 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 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 60 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 L. Cos. L. Cotg. L. Tang. L. Sin. / r82i / L. Sin. L. Tang. L. Cotg. L. Cos. 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 4r 29 9.82 112 9.94 655 0.05 345 9.87 457 31 18° 30 9.82 126 9.94 681 0.05 319 9.87 446 30 to 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 L. Cos. I. Cotg. L. Tang. L. Sin. 1 [83] ; L. Sin. L. Tang. L.Cotg. L. Cos. ~T" 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 ^7 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 42^ 29 9.82 955 9.96 180 0.03 820 9.86 775 31 4-T 30 9.82 968 9.96 205 0.03 795 9.86 763 30 * f 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 0.03 440 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 60 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 L. Cos. L. Cotg. L. Tang. L. Sin. / r84i t L. Sin. L. Tang. L. Cotg. L. Cos. 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 9.83 648 9.97 472 0.02 528 9.86 176 40 21 9.83 661 9.97 497 0.02 503 9.86 164 39 22 9.83 674 9.97 523 0.02 477 9.86 152 38 23 9.83 688 9.97 548 0.02 452 9.86 140 37 24 9.83 701 9.97 573 0.02 427 9.86 128 36 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 43° 29 9.83 768 9.97 700 0.02 300 9.86 068 31 46° . 30 9.83 781 9.97 725 0.02 275 9.86 056 30 TVF 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 9.35 888 16 45 9.83 980 9.98 104 0.01 895 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 6G8 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 L. Cos. 1. Cotg. L. Tang. L. Sin. ( [85] / L. Sin. L. Tang. L. Cotg. L. Cos. 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 60 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 'AO 9.84 437 9.98 989 0.01 Oil 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 44° 29 9.84 553 9.99 217 0.00 783 9.85 337 31 45° 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 L. Cos. L. Cotg. L. Tang. L. Sin. 1 [861 TABLE IV AUXILIARY FIYE-PLAOE TABLE SMALL ANGLES [87] II 1 S T S' X' L. Sin. 4.68557 4.68557 5.31443 5.31443 60 il .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 .7190a 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 0° 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] II ' s T iS' X' L. Sin. . 3600 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 5.31445 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 1° 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 4.68549 4.68575 5.31451 5.31425 8.54282 [89] TABLE V FOUR-PLACE TABLE NATURAL SINE, COSINE, TANGENT, AND COTANGENT EVERY 10' OF THE QUADEANT [91] , — M T ™ ^^\ ■ f o 1 N. Sin. If, Tan. N. Cot. N. Cos. « 00 .0000 .0000 CX) 1.0000 00 90 V 10 .0029 .0029 343.77 1.0000 50 20 .0058 . .0058 171.89 1.0000 40 i, 30 11 30 .0087 • .0087 114.59 1.0000 ^ 40 .0116 .0116 85.940 .9999 20 50 .0145 .0145 68.750 .9999 10 1 00 .0175 .0175 57.290 .99^ 00 89 10 .0204 .0204 49.104 .9998 50 i 20 .0233 .0233 42.96'" .9997 40 f 30 .026^ .0262 38.18i .9997 30 . 40 .0291 .0291 34.568 .9996 20 r 50 .0320 .0320 31.242 .9995 '° / 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 6 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 .9949 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 B3 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 .9&90 30 40 .1507 .1524 6.5606 .988^ 20 50 .1536 .1554 6.4348 .9881 10 9 00 .1564 .1584 6.3138 .9877 00 81 If. Cos. N. Cot. N. Tan. N. Sin. 1 o [921 -•i O 1 N. 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 5.5764 .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 .255^ 3.9136 .9689 40 30 .2504 .2586 3.8667 .9681 30 40 .2532 .2617 3.8208 .9674 20 50 .2560 .2648 3.7760 .9667 10 16 00 .2588 .2679 3.7321 .9659 00 75 10 .2616 .2711 3.6891 .9652 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 .9688 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 1 5. Cos. N. Cot. N. Tan. X. Sin. » [93] o f N.Sin. N. Tan. S. 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 .334,6 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 2.6279 .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 i 20 i .3800 .4108 2.4342 .9250 40 30 ' ^ .3827 .4142 ; 2.4142 .9239 30 40 i .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 y. Cos. N. Cot. N. Tan. N. Sin. ; o [94] ) / N. Sin. N. Tan. N. Cot. .Cos. 17 00 .4540 .5095 1.9626 .8910 00 63 / 10 .4566 .5132 1.9486 .8897 50 / 20 / 30 .4592 .5169 1.9347 .8884 40 .4617 .5206 1.9210 .8870 30 40 .4643 .5243 1.9074 .8857 20 • 50 .4669 .5280 1.8940 .8843 10 1 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 69 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 68 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 67 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 66 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 1 X. Cos. N. Cot. N. Tan. N. Sin. 1 o [95] a ( N. Sin. ^ N. Tan. N. Cot. N. Cos. _ 36 00 .9878 .7265 1.3764 .8090 00 64 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 61 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 5b .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 .7509 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 1.0599 .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 46 N. Cos. N. Cot. N. Tan. N. Sin. 1 [96] TABLE VI FOUR-PLACE LOaARITHMS NUMBERS 1-2000 [97 1 N. 1 2 3 4 5 6 7 8 9 1 2 3 0000 0000 3010 4771 6021 6990 7782 8451 9031 9542 0000 3010 4771 0414 3222 4914 0792 3424 5051 1139 3617 5185 1461 3802 5315 1761 3979 5441 2041 4150 5563 2304 4314 5682 2553 4472 5798 2788 4624 5911 4 5 6 6021 6990 7782 6128 7076 7853 6232 7160 7924 6335 7243 7993 6435, 7324 8062 6532 7404 8129 6628 7482 8195 6721 7559 8261 6812 7634 8325 6902 7709 8388 7 8 9 10 11 12 13 8451 9031 9542 8513 ,9085 9590 8573 9138 9638 8633 9191 9685 8692 9243 9731 8751 9294 9777 8808 9345 9823 8865 9395 9868 8921 9445 9912 8976 9494 9956 0000 0043 0086 0128 0170 0212 0253 0294 '0334 0374 0414 0792 1139 0453 0828 1173 0492 0864 1206 0531 0899 1239 0569 0934 1271 0607 0969 1303 0645 1004 1335 0682 1038 1367 0719 1072 1399 0755 1106 1430 14 15 16 1461 1761 2041 1492 1790 2068 1523 1818 2095 1553 1847 2122 1584 1875 2148 1614 1903 2175 1644 1931 2201 1673 1959 2227 1703 1987 2253 1732 2014 2279 17 18 19 20 21 22 23 2304 2553 2788 2330 2577 2810 2355 2601 2833 2380 2625 2856 2405 2648 2878 2430 2672 2900 2455 2695 2923 2480 2718 2945 2504 2742 2967 2529 2765 2989 3010 3032 3054 3075 3096 3118 3139 3160 3181 3201 3222 3424 3617 3243 3444 3636 3263 3464 3655 3284 3483 3674 3304 3502 3692 3324 3522 3711 3345 3541 3729 3365 3560 3747 3385 3579 3766 3404 3598 3784 24V. 25 26 3802 ^979 4150 3820 3997 4166 3838 4014 4183 3856 4031 4200 3874 4048 4216 3892 4065 4232 3909 4082 4249 3927 4099 4265 3945 4116 4281 3962 '4133 4298 27 28 29 30 31 32 33 4314 4472 4624 4330 4487 4639 4346 4502 4654 4362 4518 4669 4378 4533 4683 4393 4548 4698 4409 4564 4713 4425 4579 4728 4440 4594 4742 4456 4609 4757 4771 4786 4800 4814 4829 4843 4857 4871 4886 4900 4914 5051 5185 4928 5065 5198 4942 5079 5211 4955 5092 5224 4969 5105 5237 4983 5119 5250 4997 5132 5263 5011 5145 5276 5024 5159 5289 5038 5172 5302 34 35 36 5315 5441 5563 5328 5453 5575 5340 5465 5587 5353 5478 5599 5366 5490 5611 5378 5502 5623 5391 5514 5635 5403 5527 5647 5416 5539 5658 5428 5551 5670 37 38 39 40 41 42 43 5682 5798 5911 5694 5809 5922 5705 5821 5933 5717 5832 5944 5729 5843 5955 5740 5855 5966 5752 5866 5977 5763 5877 5988 5775 5888 5999 5786 5900 6010 6021 6031 6042 6053 6064 6075 6085 6096 6107 6117 6128 6232 6335 6138 6243 6345 6149 6253 6355 6160 6263 6365 6170 6274 6375 6180 6284 6385 6191 6294 6395 6201 6304 6405 6212 6314 6415 6222 6325 6425 44 45 46 6435 6532 6628 6444 6542 6637 6454 6551 6646 6464 6561 6656 6474 6571 6665 6484 6580 6675 6493 6590 6684 6503 6599 6693 6513 6609 6702 6522 6618 6712 47 48 49 50 6721 6812 6902 6730 6821 6911 6739 6830 6920 6749 6839 6928 6758 6848 6937 6767 6857 6946 6776 6866 6955 6785 6875 6964 6794 6884 6972 6803 6893 6981 6990 6998 7007 7016 7024 7033 7042 7050 7059 7067 N. 1 2 3 4 5 6 7 8 P r98i N. 1 2 3 i 5 6 7 8 9 50 51 52 53 6990 6998 7007 7016 7024 7033 7042 7050 7059 7067 7076 7160 7243 7084 7168 7251 7093 7177 7259 7101 7185 7267 7110 7193 7275 711'! 720 i 7284 7126 7210 7292 7135 7218 7300 7143 7226 7308 7152 7235 7316 54 55 56 7324 7404 7482 7332 7412 7490 7340 7419 7497 7348 7427 7505 7356 7435 7513 7364 7443 7520 7372 7451 7528 7380 7459 7536 7388 7466 7543 7396 7474 7551 57 58 59 60 61 62 63 7559 7634 7709 7566 7642 7716 7574 7649 7723 7582 7657 7731 7589 7664 7738 7597 7672 7745 7604 7679 7752 7612 7686 7760 7619 7694 7767 7627 7701 7774 7782 7789 7^96 7803 7810 7818 7825 7832 7839 7846 7853 7924 7993 7860 7931 8000 7868 7938 8007 7875 7945 8014 7882 7952 8021 7889 7959 8028 7896 7966 8035 7903 7973 8041 7910 7980 8048 7917 7987 8055 64 65 66 8062 8129 8195 8069 8136 8202 8075 8142 8209 8082 8149 8215 8089 8156 8222 8096 8162 8228 8102 8169 8235 8109 8176 8241 8116 8182 8248 8122 8189 8254 67 68 69 70 71 72 73 8261 8325 8388 8267 8331 8395 8274 8338 8401 8280 8344 8407 8287 8351 8414 8293 8357 8420 8299 8363 8426 8306 8370 8432 8312 8376 8439 8319 8382 8445 8451 8457 8463 8470 8476 8482 8488 8494 8500 8506 8513 8573 8633 8519 8579 8639 8525 8585 8645 8531 8591 8651 8537 8597 8657 8543 8603 8663 8549 8609 8669 8555 8615 8675 8561 8621 8681 8567 8627 8686 74 75 76 8692 8751 8803 8698 8756 8814 8704 8762 8820 8710 8768 8825 8716 8774 8831 8722 8779 8837 8727 8785 8842 8733 8791 8848 8739 8797 8854 8745 8802 8859 77 78 79 . 80 81 82 83 8865 8921 8976 8871 8927 8982 8876 8932 8987 8882 8938 8993 •8887 8943 8998 8893 8949 9004 8899 8954 9009 8904 8960 9015 8910 8965 9020 8915 8971 9025 9031 9036 9042 9047 9053 9058 9063 9069 9074 9079 9085 9138 9191 9090 9143 9196 9096 9149 9201 9101 9154 9206 9106 9159 9212 9112 9165 9217 9117 9170 9222 9122 9175 9227 9128 9180 9232 9133 9186 9238 84 85 86 9243 9294 9345 9248 9299 9350 9253 9304 9355 9258 9309 9360 9263 9315 9365 9269 9320 9370 9274 9325 9375 9279 9330 9380 9284 9335 9385 9289 9340 9390 87 • 88 89 90 91 92 93 9395 9445 9494 9400 9450 9499 9405 94'55 9504 9410 9460 9509 9415 9465 9513 9420 9469 9518 9425 9474 9523 9430 9479 9528 9435 9484 9533 9440 9489 9538 9542 9547 9552 9557 9562 9566 9571 9576 9581 9586 9590 9S38 9685 9595 9643 9689 9600 9647 9694 9605 9652 9699 9609 9657 9703 9614 9661 9708 9619 9866 9713 9624 9671 9717 9628 9675 9722 9633 9680 9727 94 95 96 9731 9777 9823 9736 9782 9827 9741 9786 9832 9745 9791 9836 9750 9795 9841 9754 9800 9845 9759 9805 9850 9763 9809 9854 9768 9814 9859 9773 9818 9863 97 98 99 100 9868 9912 9956 9872 9917 9961 9877 9921 9965 9881 9926 9969 9886 9930 9974 9890 9934 9978 9894 9939 9983 9899 9943 9987 9903 9948 9991 9908 9952 9996 0000 0004 0009 0013 0017 0022 0026 0030 0035 0039 N. 1 2 3 4 5 6 7 S 9 [99] N. 1 2 3 4 5 6 7 8 9 100 101 102 103 0000 0004 0009 0013 0017 0022 0026 0030 0035 0039 0043 0086 0128 0048 0090 0133 0052 0095 0137 0056 0099 0141 0060 0103 0145 0065 0107 0149 0069 0111 0154 0073 0116 0158 0077 0120 0162 0082 0124 0166 104 105 106 0170 0212 0253 0175 0216 0257 0179 0220 0261 0183 0224 0265 0187 0228 0269 0191 0233 0273 0195 0237 0278 0199 0241 0282 0204 0245 0286 0208 0249 0290 107 108 109 110 111 112 113 0294 0334 0374 0298 0338 0378 0302 0342 0382 0306 0346 0386 0310 0350 0390 0314 0354 0394 0318 0358 0398 0322 0362 0402 0326 0366 0406 0330 0370 0410 0414 0418 0422 0426 0430 0434 0438 0441 0445 0449 0453 0492 0531 0457 0496 0535 0461 0500 0538 0465 0504 0542 0469 0508 0546 0473 0512 0550 0477 0515 0554 0481 0519 0558 0484 0523 0561 0488 0527 0565 114 115 116 0569 0607 0645 0573 0611 0648 0577 0615 0652 0580 0618 0656 0584 0622 0660 0588 0626 0663 0592 0630 0667 0596 0633 0671 0599 0637 0674 0603 0641 0678 117 118 119 120 121 122 123 0682 0719 0755 0686 0722 0759 0689 0726 0763 0693 0730 0766 0697 0734 0770 0700 0737 0774 0704 0741 0777 0708 0745 0781 0711 0748 0785 0715 0752 0788 0792 0795 0799 0803 0806 0810 0813 0817 0821 0824 0828 0864 0899 0831 0867 0903 0835 0871 0906 0839 0874 0910 0842 0878 0913 0846 0881 0917 0849 0885 0920 0853 0888 0924 0856 0892 0927 0860 0896 0931 124 125 126 0934 0969 1004 0938 0973 1007 0941 0976 1011 0945 0980 1014 0948 0983 1017 0952 0986 1021 0955 0990 1024 0959 0993 1028 0962 0997 1031 0966 1000 1035 127 128 129 130 131 132 133 1038 1072 1106 1041 1075 1109 1045 1079 1113 1048 1082 1116 1052 1086 1119 1055 1089 1123 1059 1092 1126 1062 1096 1129 1065 1099 1133 1069 1103 1136 1139 1143 1146 1149 1153 1156 1159 1163 1166 1169 1173 1206 1239 1176 1209 1242 1179 1212 1245 1183 1216 1248 1186 1219 1252 1189 1222 1255 1193 1225 1258 1196 1229 1261 1199 1232 1265 1202 1235 1268 134 135 136 1271 1303 1335 1274 1307 1339 1278 1310 1342 1281 1313 1345 1284 1316 1348 1287 1319 1351 1290 1323 1355 1294 1326 1358 1297 1329 1361 1300 1332 1364 137 138 139 140 141 142 143 1367 1399 1430 1370 1402 1433 1374 1405 1436 1377 1408 1440 1380 1411 1443 1383 1414 1446 1386 1418 1449 1389 1421 1452 1392 1424 ' 1455 1396 1427 1458 1461 1464 1467 1471 1474 1477 1480 1483 1486 1489 1492 1523 1553 1495 1526 1556 1498 1529 1559 1501 1532 1562 1504 1535 1565 1508 1538 1569 1511 1541 1572 1514 1544 1575 1517 1547 1578 1520 1550 1581 144 145 146 1584 1614 1644 1587 1617 1647 1590 1620 1649 1593 1623 1652 1596 1626 1655 1599 1629 1658 1602 1632 1661 1605 1635 1664 1608 1638 1667 1611 1641 1670 147 148 149 150 1673 1703 1732 1676 1706 1735 1679 1708 1738 1682 1711 1741 1685 1714 1744 1688 1717 1746 1691 1720 1749 1694 1723 1752 1697 1726 1755 1700 1729 1758 1761 1764 1767 1770 1772 1775 1778 1781 1784 1787 l" 1 2 3 4 5 6 7 8 9 ;ioo] N. O 1 2 8 4 5 6 7 8 9 150 151 152 153 1761 1764 1767 1770 1772 1775 1778 1781 1784 1787 1790 1818 1847 1793 1821 1850 1796 1824 1853 1798 1827 1855 1801 1830 1858 1804 1833 1861 1807 1836 1864 1810 1838 1867 1813 1841 1870 1816 1844 1872 154 155 156 1875 1903 1931 1878 1906 1934 1881 1909 1937 1884 1912 1940 1886 1915 1942 1889 1917 1945 1892 1920 1948 1895 1923 1951 1898 1926 1953 1901 1928 1956 157 158 159 160 161 162 163 1959 1987 2014 1962 1989 2017 1965 1992 2019 1967 1995 2022 1970 1998 2025 1973 2000 2028 1976 2003 2030 1978 2006 2033 1981 2009 2036 1984 2011 2038 2041 2044 2047 2049 2052 2055 2057 2060 2063 2066 2068 2095 2122 2071 2098 2125 2074 2101 2127 2076 2103 2130 2079 2106 2133 2082 2109 2135 2084 2111 2138 2087 2114 2140 2090 2117 2143 2092 2119 2146 164 165 166 2148 2175 2201 2151 2177 2204 2154 2180 2206 2156 2183 2209 2159 2185 2212 2162 2188 2214 2164 2191 2217 2167 2193 2219 2170 2196 2222 2172 2198 2225 167 168 169 170 171 172 173 2227 2253 2279 2230 2256 2281 2232 2258 2284 2235 2261 2287 2238 2263 2289 2240 2266 2292 2243 2269 2294 2245 2271 2297 2248 2274 2299 2251 2276 2302 2304 2307 2310 2312 2315 2317 2320 2322 2325 2327 2330 2355 2380 2333 2358 2383 2335 2360 2385 2338 2363 2388 2340 2365 2390 2343 2368 2393 2345 2370 2395 2348 2373 2398 2350 2375 2400 2353 2378 2403 174 175 176 2405 2430 2455 2408 2433 2458 2410 2435 2460 2413 2438 2463 2415 2440 2465 2418 2443 2467 2420 2445 2470 2423 2448 2472 2425 2450 2475 2428 2453 2477 177 178 179 180 181 182 183 2480 2504 2529 2482 2507 2531 2485 2509 2533 2487 2512 2536 2490 2514 2538 2492 2516 2541 2494 , 2519 2543 2497 2521 2545 2499 2524 2548 2502 2526 2550 2553 2555 2558' 2560 2562 2565 2567 2570 2572 2574 2577 2601 2625 2579 2603 2627 2582 2605 2629 2584 2608 2632 2586 2610 2634 2589 2613 2636 2591 2615 2639 2594 2617 2641 2596 2620 2643 2598 2622 2646 184 185 186 2648 2672 2695 2651 2674 2697 2653 2676 "2700 2655 2679 2702 2658 2681 2704 2660 2683 2707 2662 2686 2709 2665 2688 2711 2667 2690 2714 2669 2693 2716 187 188 189 190 191 192 193 2718 2742 2765 2721 2744 2767 2723 2746 2769 2725 2749 2772 2728 2751 2774 2730 2753 2776 2732 2755 2778 2735 2758 2781 2737 2760 2783 2739 2762 2785 2788 2790 2792 2794 2797 2799 2801 2804 2806 2808 2810 2833 2856 2813 2835 2858 2815 2838 2860 2817 2840 2862 2819 2842 2865 2822 2844 2867 2824 2847 2869 2826 2849 2871 2828 2851 2874 2831 2853 2876 194 195 196 2878 2900 2923 2880 2903 2925 2883 2905 2927 2885 2907 2929 2887 2909 2931 2889 2911 2934 2891 2914 2936 2894 2916 2938 2896 2918 2940 2898 2920 2942 197 198 199 200 2945 2967 2984=V^, sec^=-'^, tan^='VS V5 Exercise 10 cos ^=3^, COS A=^-^, cos J.=j\, cot ^ = 1*5, C0t.4 = y^5, cot A=:fV, CSC .4 = 5:1. CSC J. = *J. sinJ.=----^, tan^ = — , sec^= 3 2 ' cot J.=|v'5, csc.4 = |V5. sin^= ^'f+\ tan.l = l, cos^= '"^r^+^ esc 4 = Vm* + 1. sec .4= ^/^ J ANSWERS 6. sinA=^' tan^ = i, 8ecA = ^, cobA = ^, cotA = 2. 6 2 7. tan^ = 0, sec 4 = 1, cos 4 = 1, cot4 = Qo, csc4 = ao. 8. sin 4 = 1, tan 4 =00, sec4 = oo, cot 4 = 0, esc 4 = 1. 9. sin 4 = 0, sec 4 = 1, cos 4 = 1, cot4 = co, csc4 = oo. 10. tan 4 = 00, sec 4 = 00, cos 4 = 0, cot 4 = 0, cso4 = l. 11. sin 4 = 1, tan 4 = 00, cos 4 = 0, cot 4 = 0, esc 4 = 1. 12. tana: = — ^^ . secg = - ^ . cos a; = VI - 25p2, . Vl-25p2 Vl - 25 p2 cot^ = ^lH112, csc,. = f. 5p op 13. sin4 = |, sec4 = |, cos4 = ^, cot4 = f, csc4=|, 14. sin4 = }f, tan4 = J^, sec4 = Y, cot^=^, esc 4 = ^. 15. sin4 = ^, tan4 = J/, sec4 = J^, cos4 = ^, cot4 = -j^. 16. sin 4 = ^^, tan 4 = J, sec 4 = 41, cos4 =-1^, c.cA=^ 13 3 13 2 17. tan4 = -J\/3, sec4 = |V3, cos4 = ^, cot 4 = V3, csc4 = 2. 18. sin4 = ^^-i2^ tan4=\/T5, cos4 = i, cot4 = ^Vl5, cso4 = t*jV15. 4 19. sin4 = "'^"''+ -l, 008 4= ^"^' + ^ cot4=l, sec4 = V^r^ + T, to2 + 1 m^ + 1 m CSC 4 = ■ ■ — m 20. sin 4 = , tanA=l, sec4=V2, cos 4= , cot 4=1, csc4 = \/2. 2 2 21. sinx=0, tanx = 0, seca; = l, cot a; = 00, csca; = oo. 22. sin 4 =^a, tan 4 = ^, sec 4 = ^, cos 4 = 5^, cot 4 = ,',,■, 23. sm4 = , cos 4 = — ; -, cot 4 = sec4 = -— — i — - nfi + IV m^ + iv 2 m« m' — n' CSC4: ™'+"' 2mn 24. sin4 = iV2 -Vi, tan4=V2-l, cos 4 = J V2 + v^, sec4=V4-2v'2, esc 4 = V4 + 2 V2. 25. tan 4 =00, sec 4 = 00, cos 4 = 0, cot 4 = 0, cso4=l. 26. 6. sin22J° = ^V2- V2, cos 22^° = ^^ + ^ , cot 22^° = V2 + 1, sec 22J° = V4 - 2 \/2, csc22i° = V4 + 2n/2. 27. sin4 = ^^, tan4=.:^, sec4 = f, cot4 = 3\V39, csc4 = ^^VSO. 5 28. sin4 = ^+"^^ . tan4=2 -fv/S, cos4 = :^^^— , cot4 = 2-\/3. 2 cso4=2V2- V3 ANSWERS 7 30. sin 15° = ^^-^^^, tanl5° = 2-V3, cos 15° = ^^ + ^^ , 2 ' 2 secl5° = 2V2 -V3, CSC 15° = 2 V2 + Vs. 31. cos A = Vl- sm2 A, tan ^ = ^ ^'" ^ , csc^=— ?— , Vl - sin2 vl sin A cot^= ^^T^'"' ^, seo^ sin^ Vl-sin^^ 32. sin J. = Vl — cos2 ^, tan A = Vl — cos^ A ^ . cos A cot A ■ sec^ = , csc^ = - COS^ V1-00S2 4 cos A Vl - cos2 A 33. sm^=--^5^A__, cos A = — ^ cot^= ^ Vl + tan2 A Vl + tan2 ^ tan 4' sec^= Vl + tan2^, esc ^ = :!:l±35s!J tan^ 34. tan A = , esc ^ = Vl + cot^^, sin A = . cot A VTT cot2 A . cot A „„ . Vl + cot^ A cos .4 = — -, sec .4 = ■ Vl + cot2 A cot A 35. cos .4= , tan A = Vsec^ A— I, eot^ = — -, sec A Vsec^ A-1 „„„ A sec A „. . Vseo^ .4 — 1 CSC A = — - , sm A = — ■ Vsec2 A—1 sec .4 36. sin.4 = -^, cos ^ = ^^S^ "^ ^ ~ ^ . tan^ = - — "^ CSC A CSC J. Vcso2 A — 1 sec ^ = y^" ^ , cot^ = Vcsc2.4-l. Vcsc^ ^ — 1 37. cos ^ = 1 — vers ^, sec4=— , 1 — vers A . . V2 vers A-i vers^ A „ ,. , 1 - vers A tan 4 = , cot .4 = ' 1 — vers A V2 vers A — vers^ A sin A = V2Ters A — vers'^ A, esc ^ = — ^r==^=^ ■ V2 vers A — vers^ A 38- ~ «■ A- ^«- ^• 39. ^^^879. 43. jViwI. 47. -TTH^- 40. JV3. 44. iV42. .„ o • 9 , ■ 1 ^ ^_ ' 48. 2 sin2 X + sni x = l. 41. 5«5V39. 45. l-cos24 + cosA ^„ . o „^ , " " 49. tan2 a; — 2 tan a; = 1, ANSWERS Ezercise 12 13. ^^ 17. -1-v^. 22 1. iVe. 36. 150; 259.8. 14. iv^(6+c). 18. -6f 23 . 5. 38. 961.3+. 15. 2+v^. 20. KV2-1). 35 . 86.6, 39. 165. 16. l-2v'3. 21. f. Iizercise 13 1. 60°. 4. 60» 7. 45°. 10. 60° 13. 60". 16. 30°. 19. 60° 2. 60°. 5. 0°. 8. 45°. 11. 45° 14. 30°, 17. 45°. 20. 90°. 3. 30°. 6. 45° 9. 30°. 12. 30°, ,90°. 16. 45' ) 18. 45°. 21. 0°. 22. 27° 13' 12". 26. 22^°. 28. 18°. 33. 30°. 23. 15°. 27. 90° . n + 1 29. 45°. 34. 60°. 24. 10°. 30. 38° 50' • 35. 30°. 25. 60°. Xizercise 14 1. 9.64647-10. 9. 8.85590-10. 19. 6.1493. 26. 9.9523- •10. 2. 9.98997 - 10. 10. 9.97991-10. 20. 14.991. 27. 0.3076. 3. 9.86603 - 10. 11. 0.11532. 21. 9.4214 - 10. 28. 0.6489. 4. 9.38699-10. 12. 9.99194-10. 22. 9.8297 - 10. 29. 9.8832- 10. S. 0.15908. 13. 1.24820. 23. 0.1759. 30., 0.2522. 6. 9.43707 - 10. 14. 8.91931-10. 24. 0.7033. 31. 0.6413. 7. 8.73767 - 10. 15. 9.84324-10, 25. 9.6622 - ■10. 32. (° = 1^ 1-6 = 8.1 i.24. 8. 9.86126 - 10. 16. 9.74610-10. 898. Exercise 15 1. 28° 15'. 8. 85° 5' 15". 15. 28.7". 21. 61.07°. 2. 28° 40'. 9. 65° 10' 20". 16. 18.5°. 22. 0.541°. 3. 35° 43'. 10. .5° 20' 29". 17. 56.?6°. 23. 88.465°. 4. 40° 23'. 11. 4°0'47". 18. 70.14°. 24. 65.67°, S. 66° 15' 24". 12. 85° 59' 13". 19. 64.43°. 25. 78.14°, 6. 70° 16' 21". 13. 26.5°. 20. 46.11°. 26. 14.47°, 7. 70°0'26". 14. 50.2°. Ezeroise 16 1. 8.21421 - 10. 14. 0°4'31". 27. 8.1238 - 10. 40. 4.662°. 2. 8.34812 - 10. 15. 0°2'39". 28. 8.1070 - 10. 41. 89.436°. 3. 8.49128 - 10. 16. 89° 45' 6". 29. 8.2701 - 10. 42. 8.3638- 10. 4. 1.72219. 17. 42° 5' 26". 30. 1.6657. 43. 1.6362. 5. 1.64078. 18. 82° 52' 2'. 31. 1.8744. 44. 89.266°. 6. 8.18538-10. 19. 83° 24' 25". 32. 8.3446 - 10. 45. .613°. 7. 8.28456 - 10. 20. 0° 17' 7.3". 33. 7.9686- 10. 46. 89.285°. 8. 8.47866-10. 21. 0° 17' 7.1". 34. 89.267°, 47. .624°. 9. 0°26'10". 22. 89° 54' 15", 35. 1.036°. 48. 1.6375. 10. 88° 63' 6". 23. 8.245. 36. 89.216°, 49. 2.792. 11. 0°42'53". 24. .1504. 37, .634°. 50. 112.82. 12. 89° 32' 27". 25. 1.6687. 38. 89.553°. 51. .7348. 13 89° 57''. 26. 8.3353-10. 39. .507°. 52. .0267. ANSWERS s Exercise 17 1. Sine A = ^y- Cosine A = {f. Cotangent A = ^. Secant A = if Cosecant .4 = Y- 6 = :30, c = 34. 2. -w^- 8 i. cot 37° > tan 37°. 22. 1, 6. sin 49° > cos 49°. 19 '. a; = 45°. 23. |V3_Jv^_|. 6. ^<45°. 20 '. a; = 60°. „. U-8V3 7. ^>60°. 21 . a; = 45°. 2 26. cot A = ^S CSC A =¥■ 26. ^- 27. r = 17° r Exercise 18 47' 38". 28. 300. 29. 270.1. 270.06. 4. B = 62°. 7 . B = 61°43'. 10. 5 = 51° 43' 36". a = 6.3804. = 11.448. a = 2.2478. c = 13.591. 6 = 21.276. 6 = 2.849. 6. B = 12°. 8 . .4 = 35° 17'. 11. .4 = 17° 43' 18". a = 26.15. a = 648.67. 6 = 70.985. 6 = 5.5585. 6 = 916.7. c = 74.5217. 6. £ = 43° 42'. 9 . ^ = 52° 41'. 12. .23661. a = 50.78. a = 385.436. 13. .282726. c = 70.24. c.= 484.644. 14. 5 = 26° 31' 20". 15 , ^ = 2° 43' 30". 16. .B = 38° 50' 54". 6 = 127.976. a = 18.85129. a = .153254. c = 286.5875. - 6 = .65927. 6 = .12343. 17. B = 63° 41' 24". 6 = 256.406. c = 286.033. 18. .96565. 19. 164.93. 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. 6 = 41.65. 41. .5=60°. c = 21.37. c = 44.71. a = JV3 = 4.0425. 25. B = 34°- 31. A = 23.73°. c = ^VS = 8.083. a = 2.22. a = .003824. 42. ffi = 6 = 6 V2 = 8.484, 6 = 1.4976. c = .009504. 43. a = ^/V3 = 14.43. 26. A = 51.8°. 32. .3907. c = 4v^ = 28.86. a = .604. 6 = .4753. 33. 34. .11388. 50.933. 44-. &=WjOj!vf = 577.4. c = iiyiiV3 = 1154.7. 27. 28. A = 7.5°. 6 = 95.42. c = 96.225. B = 52.33°. a = 77.43. 6 = 100.3. 36. 36. B = 1.83°. a = 13.125. 6 = .4194. A = 47.84». & = .4757. 45. 6 = M^V3 = 1154.8. c-^VS- 2309.6. 46. a = 600V3 = 1039.251 6 = 600. 47. a = 200. 29 A = 13.75°. 6 = 3.7845. 37 c = .7086. 129.15. c = 200\/2 = 282.8. 48. a=10d. c = 3.89583. 38. 1.081. b = WdV3=n.32d. 10 ANSWERS 49. Same as the respective answers for numbers 6 and 7 in this exercise. 51. DB = hO. BC = 2b. DC = ^« V3 = 21.65. 52. OQ = cosy, PQ = siiiy, OiV = sin a; cos y, OJV = cos a; cos y. Exercise 19 1. A = 35° 33' 27". 16. .B=17°56'5". 31. 50.43°. 6 = 14.969. 6 = 8.6188. 32. ^ = 18.96°. 2. A = 33° 18' 3". 17. 13° 7' 18". a = 50.91. 6 = 31.147. 18. Z = 67° 22' 48", 33. ^ = 7.812°. 3. A = 42= 24' 39". .-. 7' 12" too smaU. 6 = 117.166. 6-29.257. 21. A = 41.49°. 34. 57.26°. 4. A = 39° 48' 21". 6 = 17.755. 35. 26.77°. c = 7.81016. 22. A = 45.17°. 37. A = B = 4:6". 5. A = 49° 44' 5". a = .39855. c = 13V2 = 18.384. 6 = .579587. 23. A = 60.66°. 38. ^ = 30°. 6. A = 49°. c = 43.04. 6 = 9V3 = 15.588. a = 16.3608. 24. A = 32.01°. 39. JS = .30°. 7. A = 52° 12' 25". c = 9.434. a = 100V3.= 173.2. c = .079471. 25. A = 46.31°. 40. 5 = 30°. 8. A = 43° 52'. a = 7.015. c = 2. 6 = .184875. 26. A = 48.43°. 41. ^ = 60°. 9. 23° 57' 46". c = .19107. 6 = 8. 10. 21° 53' 58". 27. A = 40.67°. 42. A = 46°. 11. 42° 24' 39". a = 86.64. ' 6 = 1. 12. c = 8.48. 28. A = 40.95°. 43. ^ = 60°. 13. 25° 48' 40". 6 = .0839. 6 = 50. 14. B = 16° 11' 7". 29. A = 52.33°. 44. ^ = 30°. 6 = 32.702. c = 2990. a = 6. 15. ^ = 8° 31' 31". 30. A = 43.44°. c = 12. = 53.666. Iizercise 20 1. Leg = 120. Vertex Z = 60°. 8. Base Z = 46° 16' 41". Vertex Z = 87° 26' 38". 2. Base = 353.87. Leg = 6690.16. 3 Base = 9.6837. Vertex.^ =67° 24'. 9 r = 8.2583. B = 10.208. 4 Leg = 50.699. Base = 79.578. Perimeter = 60. Area = 247.75. Vertex Z = 103° 24' 20". 10. r = 1.5388. 5. VertexZ = 69°23' 12". J? = 1.618. Leg = 927.84. Perimeter = 10. Base = 1056.225. Area = 7.694. 6. Leg = 8.8204. Base Z = 62° 10'. Vertex Z = 55° 40". U- Side = 8.282. r = 15.455. Area = 768. 7. BaseZ = 33°21'30". 12. Side = 9.1102. Leg = .075978. r = 17. Area = 929.24. ANSWEES n 13. 14. 15. 16. 31. 32. 33. Side = 8.6524. r = 5.9546. Perimeter = 43.262. Area = 128.8. Perimeter = 4.70498. Area = 1.6417. A = IsinD. m = 21 cos D. C = 180° - 2 D. 2h tan D = - 17. 18. le. m -V'^^H?)^ sinD = cosic = 2 m : 0-. h-- 1-- D-- h-- -.2VP-hK -. 180= -2D. : i m tan Z>. 1 = tan J (7 = 21. 12.6447. 22. 95.94. 23. 15.1848. 26. 8.1182. 27. 48.2055. 28. Base : Vertex^ = 29. Leg = Vertex Z = Leg: 90" -iO. ^cotlc. 2 2 ^cscic. 2 2 2/i 30. Base : BaseZ = Base Z ■- Leg: BaseZ : Base : Base : Vertex Z ■■ 61.86. : 114.8° 2081.5. : 45.2°. : 34.48. 59.026. : 31.14°. : 52.86°. : .61014. :61.1°. : 124.4. : 114.2. : 114.54°. 10. 11. 12. 12560.57. 5911.7. 6 = 3.416. c = 4.2331. A = 36° 11' 53". a = 2.67813. 6 = 5.41875. c = 6.0445. a = 13.1945. b = 8.4405. A = 57° 23' 36". 42.847 Base BaseZ Vertex Z 34. 35. 36. 37. 39. 40. Leg = BaseZ = Leg = Base = Base Z = r — Area = Perimeter = Side = r = Side = r = It = Area = Perimeter = .05746. .12027. 54.275°. 26.77. .8462. 14.15°. 16.9. 946.5. 143.166. 1.0878. 1.6737. 20.22. 21. 23.3. 1486.34. 141.54. 41. 42. 43. 44. 45. 46. 47. 48. 49. = 42° 15' 34". = 95° 28' 52". Side = 9.318. r = 17.387. Area = 972. 22.025. 111.4. Altitude = %5V3 = 14.435. Base Z = 30°. Base Z = 30°. Base = 100 VS. = 173.2. Y V3= 11.547 = leg = base Base Z = 45°. Vertex Z = 90°. Altitude = 6. 120°. 7.07. Exercise 21 172.756. 545.44. 13. 14. 15. 16. 17. 18. .0287326. 244.79. 300.96. h = 5.2496. I = 6.1403. .A = 58° 45' 17". I = 1.5084. c = 2.6808. h = .69183. 3122. 20519.5. 19. 20. 21. 23. 24. 7. .19936. 8. 202281.818 T, = 7.1773. c = 12.299. h = 3.7011. .7723. 9.58675. 1.5458. .8874. S = 3.22046. c = 2.2029. r = 3.0263. 12 ANSWERS 25. Perimeter = 21.265. 42. 151.4. 54. B = 18.34. 26. p = 23.187. 43. 80.8. c = 10.3332. E = 3.9448. 44. .20845. r = 17.6. 28. 3331.54. 45. h = 8.828. 55. B = 4.031. 29. 47577. A = 22.03°. c = 2.7575. 30. 882. I = 23.54. r = 3.788. 31. .01618. 46. I = 1.235. 68. 101.15. 32. 31.47. h = -7508. 67. 2886.8 = i^g-wi V3. 33. 137.33. c = 1.9612. 68. 180000 -n/S =311778. 84. 6000000. 47. I = 54.575. 69. 278.78. 35. .00003529. c = 91.16. 60. 4050-^3 = 7014.6. 36. a = 8.283. ft = 30.01. 61. 3200^3 = 5542.4. A = 52.44°. 48. c = .8598. 62. 800. c = 10.46. h = .2384. 63. 2000000 V3= 3464000. 37. c = 77.22. A=29°. 64. 7200. = 68.9. 49. 58.75. 65. 2500V3 = 4330. 6 = 34.84. 50. .8308. 38. Impossible. 51. 36950. 66. iaoao vf _ 5773.3. 39. .13833. 52. 15.192. 67. 400V3 = 692.8. 40. 149.07. 53. M = 2.262. 68. 80,000. 41. 4816.6. c = 1.9625. r = 2.039. Exercise 22 In this exercise, where two answers are given to an examp] le, the first is the result obtained by use of flve-place log tables, and the second i answer is the result obtained by use of four-place tables. 1. 389.7 = Ht. 9. 695.414. 19. 23.013. 2. 474.788. 695.35. 23.012. 474.8. 10. 17° 31' 7". 20. 5246.25. 3. 114.1. 17.52°. 5246.6. 4. 10° 33' 26". 11. 82.056. 21. 43.3 = ht. of tree. 10.56°. 82.06. 25 = width of river. 5. 491.511. 12. 287.25. 22. KB = 12. 491.44. 287.2. JJP = 6V3 = 10.392. 6. Base = 76.79. 13. 231.7. JJ ,„ ^ ,. 24. —cos 10° 16'. 16. tan 5°. 34. a cos a; +& sin a;— c tana;. 35. psinaicosx. 36 — (a + 6) cos a; — (a — 6) sin x. Exercise 27 1. JV2. 2. Vs. 3. -I. 4. -Vs. 5. -Vs. 6. 0. 7. -2. 8. jVS. 9. -Jv^ 10. sin = — cos 29°. 13. sin = — sin 15°. 16. sin = sin 0°. cos = — sin 29°. cos = cos 15°. cos = — cos 0°. tan = cot 29°. tan = — tan 15°. tan = tan 0°. cot = tan 29°. cot = — cot 15°. cot = cot 0°. sec = — CSC 29°. sec = sec 15°. sec = — sec 0°. cso = — sec 29°. esc = — cso 15°. cso = esc 0°. 11. sin =— cos 9°. 14. sin = cos 17°. 17. sin = sin 36° 43'. cos = sin 9°. cos = — sin 17°. cos = — cos 36° 4S'. tan = - cot 9°. tan = — cot 17°. tan = — tan 36° 48'. cot = — tan 9°. cot = — tan 17°. cot = — cot 36° 43'. CSC = — sec 9°. sec = — cso 17°. sec = — sec 36° 43'. sec = CSC 9°. cso = sec 17°. esc = esc 36° 43'. 12. sin = sin 15°. 15. sin = cos 10°. 18. sin = cos 37.24°. cos = — cos 15°. cos = sin 10°. cos = sin 37.24°. tan = — tan 15°. tan = cot 10°. tan = cot 37.24°. cot = — cot 15°. cot = tan 10°. cot = tan 87.24°. sec = — sec 15°. sec = esc 10". sec = esc 37.24°. CSC = CSC 15°. cso = sec 10°. esc = sec 37.24°. ANSWERS 15 21. —cos a. 23. — sinx. 25. — seco!. 27. — 3oosx. 22. —cos a;. 24. tana;. 26. —sec a;. 28. — o cos a; + 6 sin a; — c tan x. 30. sin^ x cos x. 29. — TO cos .4— p cot .4 — g cot .4. 31. —cos a;. Exercise 28 1. 30°, 150°. 5. 30", 150°. 9. 46°, 225°. 2. 80°, 150°, 210°, 330°. 6. 60°, 300°, 180°. 10. 60°, 240°. 3. 45°, 135°, 225°, 315°. 7. 30°, 150°. 11. 45°, 225°. 4. 30°, 150°, 210°, 330°. ' 8. 45°, 225°. 12. 45°, 135°, 225', SIS". 13. 30°, 150°, 45°, 225°. 16. 60°, 150°, 240°, 330°. 14. 60°, 120°, 240°, 300°, 16. 30", 150°, 210°, 330°. 45°, 135°, 225°, 315°. 17. 30°, 120°. 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 = 217.389. 217.4. Distance of the spring from the barn = 229.12. 229.16. Exercise 29 1. sin(a5 + y) = ||. 4. (x + y')=oo. 2. sin(a;-s/)=||. 5. cot(a;-y) = 0. coslx + y) = ll. g ■V/6-V2 cos (a; - 3/) = if 4 ' 8. sin(x + 45°) = — (sina; + cosa;). 7. 2+y/3. cos(30°-a;)=:i^l52i|+^H«. sin(a;-60°) = ^'"^-°°^^^ - ^- i 9. y3-2. 10. 11. - cos90°=0 14. tan(45°H.,) = l^. 15. cot (60°■^.)= -^3cot^.-4c_ot^^ V3 . *-(^^"-^) = i^ cotC.30°.J..) = )^i-^^-iff^±^. Exercise 30 1. sin 60° = i Vs. 10. 4cosSa;-3cos!e. cos60° = i. 3 tan a; — tan' a; _ 2. tan60° = V3. ' l-Stan^x 3. sin 120° = J V3. 13. - ¥• tan 120° = -Vs. 14. --i^. 9. 3 sin a; — 4 sin' a:. 91. icos4a; + J cos2a! + f. 16 ANSWERS 2. sin 15° = jV2-V3 = .2588. tan 15° = 2 - Vg = .2 679. cos 15° = iV2+V3 = .9659. 3. cot 22i° = V2 + 1 = 2 .4142. cos 22J = ^V 2 + V2 = .9239. sin 22J = J V2 -V2 = .3827. 4. sin45° = cos45° = J\/2 = .7071. Exercise 31 6. 9 1, cos - = - v2+2 a. 2 2 cot - = — 2 1- -vn^. tan2 = -i-Vl- 2 l + a 12. . /I + cos 2 .4 '^=>/ 2 tan45° = oot45° = l._ sec 45° = CSC 45° = V2: : 1.4142. 6. cosi4 = jVl8 + 6V5. „„. 1 , 3 + V5 cot - A = — ■ 2 2 _ ^ 1 , 3-V5 tan - A = • 2 2 16. ^ = 79° 36' 40". A = 79.61°. 6 = 22. 5 = 10° 23' 20" (10.39°). 13. sm(A + B)-. .\/l5 + V3 sin(^-B) = cos (^ + JB) = cos (^ - B) = V15-V3 3V5-I 3V5 + 1 8 sin2^ = iV3^ sin2B = ^Vl5. cos 2 A = i. cos2 5 = J. 1. 13. sin .4 cot^ -15 cos 2 A cos 2 .4 cos 2 .4 14. 3 yg + 25 21 17. 17° 35' 42". 17. 59^ Exercise 32 14. sin (60° + 30°) = sin 60° + sin 30° = 15. - 1. V3 + 1 . 2 sin 29.5° cos 7.5° 16. sin 27° sin 11° 2 cos 3 A sin A cos6A 17. sin (4 + B) sin ( J. - 18. 3.44. .2136. -B). Exercise 34 6. cot e = i. sini9 = |V6. tan (180°-e) = -f. sin (- «) = |. 2. f. 3. 2 + Vs. 4. sin2x = ± ^1-^/7, tlie sign depending on whetlier J a; is taken in the first or fourth quadrants. In like manner { tan2a: = T^V7. 15° = iV2+V3. (6) (c) cos CSC 15° = 2 V2 + Vs. tan 15° = 2 - V3. ^ 3-4^3 10 ^ 4-3V§ 10 _ 4 + 3V3 . 10 ANSWERS 17 («) (/) (?) (ft) (0 (i) J. (o) (6) (c) (d) 8. (o) = -i- = -V3. = A- _ 25 V3 - 48 11 _ 25\/3-48 . 39 = iV3. 2 = i- = -2. -|V5. cos = sm K. (-1)=- tan (a;— - 1 = — cot a:, cot lx — — ] = — tan K. (6) (<:) W sin (tt — 9) = sinfl. cos (tt — fl) = — cos ff. tan (tt — fl) = — tan ff. cot (tt — 9) = — cot S. sin (a; IWcosk. cos (re — ^^ 1 =— sin; (-t)=- sin (tt + a;) = — sin a;, cos (tt + a;) = — cos x. tan (ir + a:) = tan a;, cot {tt + x) = cot a;. tan cot cot a;. tana;. 37. -|V3- 34. -i. 36. -|. 36. 39. tane = |. 41. - J|. sin 9 = — f . 64. T^f (35 — 48 cos 2 a; + 28 cos 4 a; — 16 cos 2 x cos 4 a; + cos 8 a;). 38. ■2 6, 63. 3 — 4cos4a; + cosSa; 128 Ezercise 35 3. 7. 120°. 9. a =c cos S. (I) = tan (A — 45-') and a right triangle. a + b (II) a + 6 = (a — 5) (2 + V3) an isosceles triangle with the angles 30'', 30°, 6 sin 5 = 8injl = c = 9.1226. C = 4r7'. 6 = 13.288. A = 134° 20'. 5 = 74.9916 c = 242.755 A = 57° 52'. a = 1116.98. c = 1260.26. Exercise 36 A = 109° 19'. a = 4899.56. 6 = 4106. C = 69° 57' 36". a = .85442. b = .64497. 4 = 29° 1' 2". a = 56.541. 6 = 90.164. 7. A = A = 99° 29' 12". 5 = 1.0943. c = .488667. C = 68°26'4". 6 = 1.3487. c = 1.8285. C = 68° 15' 30". a = .182095. 6 = .188745. 18 ANSWERS iO. 6 = 5.267 VI. 16. c = 38.52. 23. o = 20.34a = 7.4486. b = 57.412. c = 28.66. c=2.6335(V6+V5) '. A = 79.9°. .8 = 27.77°. = 10.175. 17. a = 13283.34. 24. = 838.67. G = 105°. c = 13346.67. 6 = 595.1. u. 0=75°. A = 80° 46'. = 56.6°. = 500(3^/2 -V6), ,18. a = 600.4. 26. 6 = c = a = 100. = 896.55. 6 = 602. .B = = 4 = 60°. h = 500(2 y/S - 2). 0=.75°. 26. = 30°. = 732.1. 19. c = 7.278. a = 200\/3 = 346.42. 12. 4.0954. 11.697. & = 14.793. 6 = c = 200. 13. 6 = 17.08. A = 117.67°. 27. 0=45°. c = 15.097. 20. 6 = .2592. & = 250(3^- V6) =448.a 0=56.73°. a = .2181. c = 250(2v^-2) = 365.7. 14. a = 634.3. 0=55.87°. 28. B = 30°. 6=632.86. 21. a = 186.25. c = 200-\/2 = 282.8. A = 81.32°. c = 32.47. a = 100( V6 + V2) = 386.4. 15. c = 1.022. A = 101.96°. 29. 925.8. a = 1.4815. 22. c = 4377. 5 = 25.57°. & = 5641.43. A = 55.69°. 30. Distance of balloon from first point = 2033 yd Distance of balloon from second point = 2363 yd • Height of balloon = 1740 yd Exercise 37 1. c = 26.9675. 7. 8.185 = c. 13. 5 = 141.99°. B = 39° 45' 17". 8. = 109° 36' 3". A = 25.89°. A = 72° 14' 43". B = 38° 5' 27". c = 3.972. 2. a = 385.43. a = 14.961. 14. ^ = 79.82°. B = 74° 38' 19". 9. = 6° 49' 41". O = 21.56°, = 37° 3' 41". 6 = 317.74. & = 1712.3. 3. 0=110° 22' 10". ^ = 4° 51' 41". 15. = 7.93. B = 39° 25' 30". 10. A = 49.06°. 16. 5 = 6.23°. = .1912. c = 208.1. = 4.97°. 4. A = 48° 42' 12". B = 79.12°. o = 5.934. 0=67° 42' 18". 11. a = .9418. 17. c = 102.425. 6 = .0748566. B = 117.99°. .4 = 65.83°. 6. = 34° 6' 36". = 33.85°. B = 45.93°. B = 22° 36' 54". 12. A = 32.26°. 18. 4 = 33.84°. a =4.70166. 0=35.56°. B = 102.98°. 6. a = 336.446. B = 99° 55' 36". = 27° 58' 24". 6 = .6563. c = 1474.67. 19. 6 = 10.7. Where two answers are given, the first answer is obtained by using the five- place tables, and the second answer is obtained by the use of the four-place tables. ANSWERS 19 20. 21. 22. Distance = 234.34 ft. Distance = 234.32 ft. 4.36 mi. Resultant = 14.986. Eesultant= 14.983. Zwith 0.4 = 77° 12' 51". Zwith 0^ = 77.2°. Z with OB = 43" 30' 9". ^■with 0^ = 43.52°. 3.21. 152.31. 152.3. 238.31. 238.28. Exercise 38 1. ^ = 78° 5' 36". 78.09°. 3 = 58° 23' 28". 58.39°. C = 43° 30' 58". 43.52°. 8. 2. vl = 44°32'4". 44.53°. .B = 86°25'. 86.41°. C = 49°2'58". 49.05°. 9. 3. A = 26° 19' 54". 26.33°. £ = 98° 18' 54'.'. 98.32°. C = 65° 21' 14". 55.36°. 10. 4. A = 45° 11' 50". 45.19°. S = 101° 22' 18". 101.38°. C = 33° 25' 58". 33.44°. 11. 5. A = 43° 53' 14". 43.88°. 5 = 60° 3' 36". 60.06°. C = 76°3'18". 76.06°. 12. 6. ^ = 61° 53' 38". 61.88°. B = 72° 46' 4". 72.78°. C= 45° 20' 20". 45.34°. 13. A = 91° 48'. 91.81°. B = 47° 36' 56". 47.61°. C = 40° 35' 10". 40.58°. 14. 15. ^ = 37° 60' 40". 37.84°. B = 127° 3'. 127.05°. C = 15° 6' 22". 15.11°. A = 40° 38' 22". 40.64°. B = 49° 21' 56". 49.36°. C = 89° 69' 48". 90°. A = 52° 20' 30". 52.34°. B = 107° 19' 6". 107.32°. = 20° 20' 26". 20.34°. A = 13° 12' 6". 13.2°. B = 30°2'44". 30.04°. C = 136° 45' 14". 136.76°. A = 46° 19' 52". 46.33°. B = 31° 17' 50". 31.3°. C = 102° 22' 18". 102.37°. A = 107° 55' 12. 107.92°. B = 35° 15' 34". 35.26°. = 36° 49' 18". 36.82°. 104° 28' 42". 104.48°. 16° 44' 6". 16.736°. 18. 14.8586. 14.86. 16. .53224. .5328. 17. .1188. 80. 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. E is due east of P. E 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° 7' 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° 67' 17". 28.96°. 20 ANSWERS 1. One solution. 2. Two solutions. 3. One solution. 4. No solution. 5. No solution. 6. One solution. 7. One solution, a right A. 8. No solution. 9. Two solutions. 10. JB = 32°36'33". Q = 109° 5' 27". c = 211.48. 11. 5 = 40° 40'. B' = 16° 44'. 0=78'-' 2'. 0' = 101° 58'. 6 = 15.737. 6' = 6.9753 15. 16. 17. 12. B = 42° 44' 23" A = 33° 1' 49". a = 92.942. 13. A = 18° 19' 28". C = 139° 18' 32". c = 1.3952. 14. 5 = 70° 47'. .8' = 14° 35'. G = 61° 54'. C = 118° 6'. b = 128.455. 6' = 34.2515. Exercise 39 A = 32° 55' 57". ^' = 147° 4' 3". G = 131° 33' 51". C' = 17°25'45". c = 1651.54. c' = 661.15. A = 43° 38'. B = 58° 3' 42". 6 = .32868. A - 90°. c = 25.64. 18. Bz C-- 6 = 19. A-. A'-. a ■■ a'-. B: B': 20. A: A' : G-. C'-- C - c' = 21. B-- B' -. C-. a -- b: 6': : 28° 16' 25". : 20° 25' 11". : .56045. : 103° 50' 22". : 13° 7' 8" = A = 15.354. : 3.589. : 44° 38' 23". = 135° 21' 37". : 35.91°. : 144.09°. : 111.72°. : 3.-54°. : 219.7. : 14.6. :55°. : 10.26°. : 67.63°. : 112.37°. : 20.11. : 4.372. other side = 129.1. 129.15. Other diagonal = j a-i' an Larger angle of parallelogram = | 173° 15' 8". 173.26°. 22. .4 = 25.22°. C = 49.51°. o = 135.46. 23. 4 = 20.79°. B = 132.99°. 6 = 136.7. 24. 4 = 16.25°. A' = 163.75°. G = 149.45°. C = 1.95°. c = 36.63. c' = 2.4518. 26. 5 = 122.81°. B' = 12.45°. C = 34.82°. C =145.18°. 6=441.6. 6' = 113.275. 26. A = 70.78°. G = 45.91°. a = 10.08. 27. A = 72.16°. A' = 9.22°. B = 58.53°. 5' = 121.47°. a = .19685. a' = .03313. 29. Two solutions : 1010.58 = 40. 1010.2 =40. 719.98 = 40'. 720.5 =40'. Smaller angle of parallelogram : [6° 44' [6.74°. 52". ANSWERS Exercise 40 1. 106.79. 4. 14290.6. 8. 1056.66. 106.8. 14290. 1056.25. 2. .30726. 5. 38983.64. 9. 1283.5. .30733. 38983.33. 10. 42148.46. 3. 125.223. 6. 113.55. 42130.77. 125.225. 7. .23404. .2341. 11. Area of parallelogram = cd sin A. 14. 106.798. 13. 600 V3 = 1039.2. Exercise 41 106.8. 21 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. 69.372. 5. 72.268. 69.37. 72.27. . 8968.5 ft. above the Colorado plain. 8958 ft. above the Colorado plain. 14144.5 ft. above sea level. 14134 ft. above sea level. 30. 29. 30. 32. 33. Height = 97.083. Height = 97.08. Distance = 71.787. Distance = 71.78. 10.274. 10.273. 6.6101. 6.61. = 43.452, 43.43. 58.342 ^58.346. 146° 52' 47' 146.88''. 33° V 13". 33. 12°. Difference of latitude = difference of departure New latitude = 34° 23' North. New longitude = 38° 24' W. 7. 373.33. 11..: 8. 69.837. 69.85. 9. 167.05. 167.03. 12. 10. 1016.6. 1016.8. .1 13. 16.83. ( 14. Other side Other diagonal 15. Height =42.93. Height = 42.92 ft. Distance = 104.66. Distance = 104.675 ft. 16. 11.36. 18. 4.2818. 5.573. 4.283. 17. .1189. 19. 1496.517. 1496.66. First answer = 4.487 mi., 4.488 mi. Second answer = 9.16 mi. =(! 21. 996.94. 997.25. 22. 401.52. 401.54. 443.54. 443.5. 974185. 973.72. 25. 220.7. 26. 16.58. 23. 24. 27. 28. 6739.33 m. 6738.6 m. 9° 6' 7". : 247.49 mi. 152.69 ft. 152.7 ft. 85.854 ft. 85.89 ft. 38.566 ft. i38.56 ft. 2049 = resultant. 61° 35' 1" 61.59° 31. 114.85 ft. = distance between observers. = distance of first observer from the rock. = angle the resultant makes with OX. 22 ANSWERS 39 367.63 ft. 1 _ ^j^g opposite tower. 367.9 ft. J 90.032 ft. 90.04 ft. and 379.125 ft. 379.1 ft. _ the other two sides ~ respectively. 36. 167.7. 36. 1813.6 ft., 1813 ft. 37. 162.08 ft., 162 ft. 38. 97" 6' 25" 97.11° and 14° 53' 35" 14.89° respectively. 40. 48 ft. and 108 ft. respectively. 41. 40° 0' 16" 1 _ g^ .g jj^g gi g makes with the embankment. 40° J ^ 29.448 ft., 29.45 ft. = width of base. 42. 161.32, 161.3. 43. 22° 49' 46". 44. 86 mi. 22.83°. 85.8 mi. 135° = — . 60° = - 210° = i^ 6 270° = ^^ 2 225° = — 72° = f 315° = ^ 4 9. Complement of Complement of Complement of Complement' of Complement of Exercise 42 - = 30°. 6 3. 21 = 45°. 4 1 = 60°. 3 4. ^ = 120°. 3 5. i2: = i44°. 5 6. — = 108°. 6 7. — = 252°. 5 8. 15 2: =21, 60' 6 3' • 1° = .01745 radian. 16" = .0000775 radian. 2' 15" = .0006545 radian. 5° 14' = .0913374 radian. 2 radians =114° 35' 30". 3.2 radians = 183° 20' 48". .003 radian = 0° 10' 18.8". Arc 21 in. long = f radian. Arc 7 in. long = \ radian. 6. JJ = 28 in. Eadians = 1.118. Angle = 64° 3' 22.5". Angles = 70°; 40°. = 1.2215 radians; radian. ' ; supplement = — , 150°. ' ; supplement = ^ , 120°. 1 = 11,30° 3 6' - , 45° = 1 , 46° ; supplement = — , 135°, 4l i 4 21 = l£, 70°; supplement =-2: 160°. 9 18 9 ' 5^ = 2£ _ 4QO . supplement = IStt ^ ^^f^^^ 18 9 18 ANSWERS 10. sinr = l. cos = i Va 2 sin3z = lV2. 4 2 cos = - i V2. 2 tan = }V3. cot = Vs. tan = cot = — 1. sec = 1 Vs. CSC = 2. sec = — V2. CSC = V2. sinJ = lV3. 3 2 cos = |. Bin!z=_i. cos = -|V3. tan = VS. cot = iV3. tan = J\/3. cot = Vs. sec = 2. csc = fV3. sec = — 1 Vs. CSC =—2. 23 sin^ = cos^=iV2. sinI^ = -lV2. oos = iV2. 4 4 2 4 2 2 tan = cot = — 1. tan - = cot - = 1. ,- ^ 4 4 sec = V2. esc = — V2. seo- = osc-= V2. 11- li radians = 68" 45' 18". 4 4 sin^ = l. oot^ = 0. 18- B=4in. 2 2 :4 = 143° 14' 22.5". cos- = 0. sec- = 00. 14. a = 12.5 in. 2 2 ^ = 14° 19' 26i". taA^ = = 93° 1' 20" (93.02°). c = 102 87 c = 295.26. ^ ^Qg^gg^ 57. 0=79° 42' 48" (79.71°). 62. 17° 41' 20.8" (17.695°). a = 2506.6. 162° 18' 39.2" (162.303°). = 2506.1. [305.21. c = 2512.4. 1 305.3. = 2511.8. J43.602. 58. A right triangle. 1 43.59. A = 50° 46' 12" (50.76°). 63. 876.34. 5 = 39° 13' 48" (39.24°). 876.4. 64. .30°, 150°, 210°, 330°. 65. 30°, 60°, 120°, 150°, 210°, 240°, 300°, 330°. ANSWERS 27 66. 90°, 270°; 60°, 120°. 67. 0°, 180°; 45°, 135°, 225°, 315°. 68. 180°; 30°, 150°, 270°. 69. 45°, 135°, 225°, 315°; 15°, 165°, 195°, 345°. 70. 240°, .300°. 71. (64)- n 180° ± 30°. (68) (2 n + 1) 180°. (65) re 90° ±30°. |(45° + ji 180"). (66) 90° + n 180°, (in + 1) 90° ± 30°. (69) n 180° ±45°, n 180° ± 15°. (67) n 180°, n 180° ± 45°. (70) (4 n + 3) 90° ± 30°. 72. J. 78. 1.087 +, .9127+. 73. 60°. , 80. 4|. 74. 90°, 270°; 120°, 240°; 30°, 150°. ^^ 60°, 120°, 240°, 300°. 75. 0, V3, -Vs. 76. lift. 82. ^ + ^^^"\ 126.06°. '^"'^ - 1 77. 2.2416 radians. 83. 1 ft. 128.44°. 84. 30°, 150°, etc. ; 45°, 315°, etc. ; 270°, etc. ; 150°, 210°, etc. 85. 2. 99. 29.698 ft. 8Y. ^ 4870.5 sq. ft. 5 _ 7 _ 4870 sq. ft. 88. 225°, 315°; ^, LJL. ' 4 4 100. 30°. 90. x = hy = l. 101- 0° 3' 26" (.05729°). 91. 130.39 ft. 102- 1910.1ft. 130.41 ft. 1910-3 ft. 92. 9° 24' 28" (9.41°). 103. 17.262. 93. 30°. ^''■^^■ 94. 88.328. 104- - h t- 1^5. - 1. -f- 88.32. ' 105. 135°, 315°. 45° 37' 15" (45.62°). j„^_ 5_ ^^Vl. 95. 88.09 ft. 88.1ft. 108. (1) (4re + l)|±|- 96. 92,742,500 mi. ._ ^ 92,750,000 ml. (2) (*« + 3)| ± |- 97. 21.835. V„, ,„ ,3n 21.805. (3)2«.+_. 98. 529.49 yd. 529.4 yd. 109. Cos 2 a; = 1 - 2 sin^o; = 2 cos^a; -1 = 1 ~ = — 1 csc^K seo^x = ^ 1 = 1 2_. 1 + tan^a; 1 + oot^x 28 ANSWERS 110. cos i{A + B) + cos I (^ - B) . 140. 2n7r-5:. 6 111. 1 -h^. 142. 1+A;2 15 ft. 112. ^, 5!; 120°, 60°- 143. 20(3 + ^3). 3 ' 3' 145. — 4. 114. 4i- 147. No. 116. il- 148. 237,144.4 sq. ft. 117. Yes; no; no; no. 237,105.3 sq. ft. ±Vs. 149. 39,190,000 mi. 120. 121. cos 8 ,4 — cos 10 A. 151. iVlB. 152. _}_ 122. 30°. ) 5 T^' • 123. f, -t; ±f, ±i 153. ±1, ±h 124. 2 A. 154. 23° 22' 25" (23.37°). 155. Possible when a = b. 125. 122.44 ft. 122.46 ft. Impossible if a > or < 6. 157. 0°, 90°, 210°. 127. 30° 158. 75°. 128. - II, If, - II- 160. Two cases. 14.44 mi. 129. 3° 35' (3.577°). 66.361 mi. 7° 7' 30" (7.125"). 66.37 mi. 14° 2' 15" (14.87°). 165. 53° 7' 49". 130. 1. 167. 1 : V2. 131. 1 +V2 169. 81.963 ft. V4+2V2 170. 85 yd. 132. 30°, 150°, 210°. 330°, 390°. 26 yd. 135. 7912.8 mi. 172. 25,240,000,000,000 mi. approx. 7914.6 mi. 173. 1084.82. 1084.11. 136. No. ■ 0—6 1 + ab 137. 3.75. 174. 1 + aft' b — a 138. ±}V3. 175. h -h 139. a 176. 1492.45 ft. VI -a^ 1492.3 ft.