CORNELL university libra: ^^(l^STENGINEERING ^^ imemory of K. Richtnyer 3 o' = Ol V c •?' ol^ = : or- !£ -j ■o T- < « in < O loo = 0) I CO =o |0 !■* iCM = 05 The original of tliis 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/cu31924004601989 DRILL-BOOK TRiaONOMBTRT GEORGE WILLIAM JONES, PROFESSOR OF MATHEMATICS IN CORNELL UNIVERSITY. FIRST EDITION. ITHACA, N. Y. GEOKGE AY. JONES. 189G. Copyright, 1896, by GEORGE WILLIAM JONES. Press of J. J. Little & Co. Astor Place, New York PREFACE. In 1881 a TEEATiSB 0]sr trigonometex was published under the joint authority of Professors Oliver, Wait, and Jokes. In 1889 this book was rewritten and reissued imder the same title and by the same authority. In all five editions have been printed. Professor Oliver died last March, and now that a new edi- tion of the book is called for and many changes are proposed, it seems better, perhaps fairer towards him, to issue it under my single name. It may be regarded, then, both as a new edi- tion of the older book, and as itself a new book. Among the more important changes are these : 1. The introduction, at the beginning, of a chapter on the EIGHT TRIANGLE, treating it as the pupil has been accustomed to think of it in plane geometry, and without the complex notions of directed lines and angles. In thi's chapter he learns, also, how to use tables of trigono- metric ratios and logarithms, and he gets some notion of the simpler applications of trigonometry to problems in surveying. 2. The second chapter, on the general properties of PLANE ANGLES, follows more closely the general lines of the old treatise, but it differs widely in details : in particular, it makes a much freer use of projections. 3. The third chapter, on plane triangles, shows a more radical departure. The habit of writers on trigonometry seems to have been to give broad and general definitions of trigonometric ratios, and to prove generally the propositions that relate to plane {ingles, and then, when they come to dis- cuss the properties of plane triangles, to fcrget all they had said before, and to fftU back on tlie ratios of positive acute angles. In the edition of 1889 I tried to make the definitions and the proofs general ; but the method then followed never satis- IV PREFACE. fied me, and I sought in vain for light in the many American and foreign text-books that I consulted. But now, through a happy suggestion of one of my assist- ants, j\[r. Fowler, I think I have overcome the difficulty. That suggestion was to use the exterior angles ; and by such use I have been able to make the proofs general and the formulse symmetric. So, in space trigouometry, I have been able to apply this suggestion to the discussion of the proper- ties of triedral angles and spherical triangles with the best results. ■i. Greater prominence has been given to the general tri- angles. 5. The proof of De Moivre's formula by aid of imaginaries has been left out : I propose to write a book, shortly, on HIGHER ALGEBRA, and it has seemed to me that there would be the best place to discuss the applications of imaginaries to trigonometry. 6. Most of the figures have been redrawn. On the other hand, many parts of the older book have been included without change, notably the discussion of derivatives and series, of directed areas, of astronomy, and of navigation ; and for the most part the examples have been taken bodily. As to the title of the book, it has seemed to me that the word treatise was too large for me ; and as I have meant my book primarily for class use, I have called it a drill-book. lu writing this book, I have been very fortunate in my as- sistants. To ilr. Charles S. Fowler and Dr. Virgil Snyder, instructors in mathematics in Cornell University, I am deeply indebted, both for their valuable suggestions, and for their unwearied labors in beating out the text and in preparing the questions and examples ; and, for its dress, I am no less indebted to my draughtsmen, Mr. John S. Reid and Mr. Hiram S. Gutsell, instructors in drawing, to my engravers, the American Bank Note Company, and to my printers, Messrs. J. J. Little & Co. George W. Jones. Ithaca, N. Y., January 1, 1896. SUGGESTIONS TO TEACHERS. V SUGGESTIONS TO TEACHEKS. There are many things in this book not meant for beginners. Below is a rough list of the chapters and parts of chapters that may be taken up at a first reading : the parts omitted are for advanced classes. And as to those parts which are included in the list, great caution must be taken lest too many examples, or too hard ones, be set ; for there are many of them, printed in a small space. No one can be expected to work them all, and the hardest of them should be reserved for the strongest pupils. But the profit comes to the pupil by hard thinking ; and the best part of the thinking is in answering the questions. Very often more than one figure is used to illustrate a prin- ciple : for the most part, the first figure is the simplest, and that one should be well understood before the others are looked at. Later the other figures may be taken up, and the generality of the principle will be felt only when they have all been studied. When the reasons are obvious, both theorems and corollaries are left without formal demonstration ; but students are ex- pected to state the proofs. In most cases theorems are given only in formula : it is best that these formulae be translated into words. In most cases answers to the examples are not given, and the student is left to test his own results : the testing is counted as not less important than the solution, and the habit of inde- pendent thought and self-reliance so cultivated as most valu- able of all. Only the main lines of the subject are developed in the text : collateral matters are outlined in the examples and left for the student to work out for himself. fOR A FIRST reading. I, all, pp. 1-31. II, §§ 1-9, 12, _ pp. 32-53, 58-60. III, §§ 1-4, ' pp. 62-75. IV, none. V, §§ 1-7, 9-15, pp. 104-130, 134-161. VI XEW SIGNS AXD WORDS. NEW SKtNS and words. Some of the less familiar sigus used in this book are these : >, largei- than. ; 3*-, not larger tlian : <, smaller tha)i ; 16, not sDiallrr ilinn ; >, not greater titan ; <, imf tcsn than : ■=t^, not equal to ; ■ ■ ■ . unit soon, rtieaning the contin- uance of a series of terms in tlie vin,y it has begun ; = , approarliex. meaning that the value of one expres- sion comes very close to that of another, without ab- solute equality ; = , . 3.8513 = .0071, log-1 5.8513 = 710000. ,4»v The mantissa not found in the table; Takeout the three-figure number of the tabular mantissa next less than the given mantissa, and to these three figures join the quotient of- the difference of these two mantissas by the tabular difference, .ff.y. V log 678 = 2.8312 and log 679 = 2.8319, . . log-i 2.8316 = 678f = 678.6, lor> 2.8317 = .067'e^ = .06787. The "use of trigonometric ratios and their logarithms is explained in works on trigonometry. LOGARITHMS OF NUMBERS. 1 (J 1 2 3 4 5 6 i » 9 i " 0000 0000 3010 4771 6021 6990 7782 8451 9081 9542 " 1 1 0000 0414 0793 1139 1461 1761 3041 8804 2663 2788- 2 3010 3222 3424 3617 3802 3979 4150 4314 4472 4034 3 4771 4914 6051 6185 6315 5441 5663 5682 6798 6911 4 6021 6128 6233 633S 6485 0632 6628 6721 6818 6902 5 6990 7076 7100 7243 7324 7404 7483 7659 7684 7709 6 7782 7853 7924 7993 8062 8139 8196 8261 8826 8388 7 8451 8513 8573 8633 8698 8751 8808 8865 8921 8976 ' ■ 8 9031 9086 9138 9191 9343 9394 9-345 9895 -9445 9494 9 9642 9690 9038 9685 9731 9777 9883 9868 9912 9956 10 0000 0043 0086 0138 0170 0212 0253 0894 0384 0374 11 0414 0-4 6-3 0492 0531 0569 0607 0645 0683 071.9 0765 13 0792 0828 0864 0899 0934 0960 1004 1038 ltf72 1106 13 llSO 1173 1200 1239 1371 1308 1335 1367 1S1)9 14 30 14 1401 1493 1523 1553 1684 1014 1044 1678 1708 1783 15 1761 1790 1818 1847 1875 1903 1981 1950 1987 2014 16 3041 2068 9(J»5 2133 2148 2175 8201 2227 226.'i 2270 17 2304 2330 TiSr.o 3380 2405 2430 2465 3480 2604 2529 18 2563 2577 2C01 8025 2648 2672 2695 3718 2742 2765 19 2788 2810 2833 2856 ,2878 2900 2023 2945 2907 2989 20 3010 3032 3064 3075 3096 3118 3189 3160 3181 3201 1 21 3222 3243 3208 3284 3304 3834 3345 3366 3386 3404 22 3124 8444 3464 3483 3502 3522 3541 3560 3679 3698 23 30 17 8636 3655 3674 3698 3711 3729 3 7 4 7. 3706 3784 24 3803 3820 8838 3856 3874 3892 3909 3112 7 3945 8962 26 3979 3997 401 1 4081 4048 4065 4088 4099 4116 4133 26 4150 4166 41N3 4200 4216 4238 4249 4365 4281 4298 27 4:!14 4330 4840 4363 4378 4393 4409 4425 4440 4456 28 4473 4487 4602 4618 4533 4648 4504 4679 4594 4600 29 4624 4039 4054 4669 4683 4698 4718 4738 4743 4767 30 4771 4786 4800 4814 4829 4843 4857 4871 48a6 4900 31 4914 4928 4942 4966 496!) 4983 4597 5132 5011 5024 5038 32 5061 5065 5079 6092 5105 5119 5145 5169 6172 S3 5185 5198 5211 6224 5237 5250 5363 5376 5289 5302 34 6315 5328 6340 6353 6366 5378 5391 6403 5416 5488 35 5441 6453 6466 5478 5490' 6502 5614 6537 5539 5551 30 5563 5575 5587 5599 5611 5623 6635 5647 5658 5670 37 r,fiS3 6694 5705 5717 6729 5740 5752 5768 6775 6786 38 , 5798 5809 5821 6833 5843 6856 5866 5877 6888 5809 39 5911 5932 5933 5944 5956 6966 5977 5988 5999 6010 40 0021 6031 OU42 6063 6064 6075 6085 0096 6107 6117 ■ 41 6128 6138 8149 6160 6170 6180 6191 6801 6212 6223 42 6232 6248 6253 6203 6374 6284 6294 6804 0314 6336 43 335 6345 6366 6365 0375 6385 6395 6406 6415 6485 44 6435 6444 6454 6464 6474 0484 6498 6503 0518 6532 45 6532 6S42 6551 6561 0671 6680 6690 6599 6609 6618 46 6628 6037 0646 0656 Gil 6 6 0676 0684 6093 6702 6712 47 0721 6730 6739 6749 6758 0707 6776. 0785 6794 6803 48 6813 6821 6830 6839 0848 6857 6866 0876 0884 6893 49 6903 6911 6920 6928 6937 6946 6965 6964 6972 6981 50 1 •; 4 5 7 8 9 LOGARITHMS OF NUMBERS. ' 50 6 1 2 3 4 5 6 7 8 9 50 6990 6998 7007 7016 7024 7033 7048 7050 705^ 7067 51 7078 7084 7093 7101 7110 7118 7186 7135 7143 7152 52 7160 7168 7177 7185 7193 7208.7210 7218 7226 7235 53 7248 7251 7259 7267 7275 7884 7898 7300 7308 7316 54 7324 7332 7840 7348 7356 7864 7378 7380 7388 7306 65 7404 7il2 7419 7427 7435 7443 7451 7459 7466 7474 56 7482 7490 T497 7505 7613 7520 7628 7538 7643 7561 57 -7569 7566 7574 7682 7589 7597 7604 7612 7619 7637 58 7634 7642 7(r49 7657 7064 7072 7679 7086 7694 7701 59 7709 7716 7723 7731 7738 7745 7752 7760 7767 7774 60 7782 7789 7796 7803 7810 7818 7835 7888 7839 7846 61 7863 7860 7868 7875 7882 7889 7896 7903 7910 7917 63 7924 7931 7938 7945 795S- 7959 7966 T973 7980 7987 63 7993 8000 8007 8014 8021 8028 803B 8041 8048 8055 64 8062 8009 8076 8082 8089 80S6 8102 8109 8116 8183 65 8129 8136 8142 8149 8166 8162 8169 8176 8182 8189 66 8195 8S02 8209 8216 8222 8228 8235 8241 8248 8254 4^ 1-8261 8267 8274 8280 8287 8298 8299 8306 8312 -8319 8325 8331 8338 8344 8351 8367 8363 8370 8S7'6' 8382 69 8388 8395 8401 8407 8414 ' 8420 8436 8433 8439 8445 ^^P 8451*^457 8463 8470 8476 8483 '8488 8494 8600 8606 8613 8519 8625 8631 8537 8543 8549 8556 8561 8567 T8 8573 8679 8585 8691 8697 8603 8609 8615 8621 8627 78 8633 8639 8646 8651 8667 8663 8669 8675 8681 8686 T4 8698 8698 8704 8710 8716 8722 8727 8738 8739 8745 75 8751 8756 8762 8768 8774. 8779 8785 8791 8797 8802 76 8808 8814 8820 8886 8831 8837 8842 8848 8864 8859 77 8865 8871 8876 8882 8887 8893 8899 8904 8910 8915 78 8921 8927 8933 8938 8943 8949 8964 8960 8965 8971 70 8976 8983 8967 8993 8998 9004 9009 9015 9020 9025 80 9031 9036 9042 9047 9053 9068 9003 9069 9074 9079 81 9085 9090 9096 0101 9106 9112 9117 9122 9128 9133 82 9138 9143 9149' 9164 9159 9165 9170 9175 9180 9186 83 9191- 9196 9201 9800 9212 9817 9382 9287 9232 9238 84 9243 9248 9253 9268 9263 9869 9874. 9279 9284 9289 85 9294 9399 9304 9309 9316 9380 9385 9330 9335 9340 86' 9345 9350 9355 «360 9365 9370 9876 9380. 9385 939Q 87 9395 9400 9405 9410 9415 9420 9426 9430 9485 9440 88 9445 9450 9455 9460 9465 9-469 9474 9479 9484. '9489 89 9494 9499 9504 9509 9513 9618 9623 9538 9583 9638 90 9642 9547 9552 9557 9562 9566 9571 9676 9581 9586 91 9590 9595 9600 9605 9609 9614 9619 9624 9628 9633 92 9638 9643 9647 9652 9657 9661 9666 967i 9676 9680 93 9685 9689 9694 9699 970.3 9708 9718 9717 9722 9727 94 9731 9^36 9741 9746 9750 9754 9759 9763 9768 9773 95 9777 9782 9786 9791 9795 9800 9805 9809 9814 9818 96 9823 9827 9832 9830 9841 9845 9850. 9854 9869 9863 97 9868 9.872 9877 9881 9886 9890 9894 9899 9903 9908 98 9912 9917 9981 9926 9930 9934 9939 9943 9948 9958 99 9956 9961 9965 9969 9974 9978, 9983 9987 9991 9996 100 1 2 3 4 5 6 7 8 9 Xll TRIGONOMETRIC RATIOS. 1 ANGLE. SINES. COSINES. TANGENTS. COTANGENTS. ANGLE. Xat. Log. Nat. Log. Nat. Log. Log. Nat. 0»00' .0000 X 1.0000 0.0000 .0000 oo 00 CXI OO-OO' 10 .01129 7.463T I.OPOO 0000 .0029 7.4637 2.6363 343.77 50 20 .i:io.)8 ;(i48 1.0000 0000 .0058 7648 835S 171.89 40 30 .0087 9408 1.0000 0000 .0087 9409 0591 114.59 80 40 .0116 8.0658 .9999 0000 .0116 8.0658 1.9342 85.940 20 50 .0145 1627 .9999 0000 .0145 1627 8373 68.760 10 1«00' 0175 8.2419 .9998 9.9999 .0175 8.2419 1.7581 67.290 89°00' 10 .0204 3088 .9998 9999 .0204 3089 6911 49.104 50 20 .0233 3668 .9997 9999 .0233 3669 G831 42.964 40 80 .0262 4179 .9997 9999 .0262 4)81 6819 38.188 80 40 .0291 4637 .9996 9998 .029T 4688 5862 84.868 80 50 .0320 6050 .9995 9998 .0320 6063 4947 31.242 10 2'>00' .0349 8.5428 .9994 9.9997 .0349 8.5481 1.4569 28.636 88000' 10 .0378 5776 .9993 9997 .0378 5779 4221 26.432 6p 20 .0407 6097 .9992 9996 .0407 6101 3899 24.648 40 30 .0435 6397 .9990 9996 .0437 6401 3699 22.904 30 40 .0465 6677 .99S9 999S .0466 6682 3318 21.470 20 SO .0494 6940 .9988 9995 .0495 6945 3056 20.806 10 3''00' .0523 8.7188 .9986 9.9994 .0524 8.7194 1.2806 19.081 8 7 "GO-' 10 .0552 7423 .9985 9993 .0563 7429 2671 18.076 60 20 .0581 7045 .9983 9998 .0682 7662 2348 17.109 40 1 30 .0610 7857 .9981 9992 .0612 7865 2135 16.350 30 40 .0040 8059 .9980 9991 .0641 8067 1933 15.606 SO , 50 .0669 8251 .9978 9990 .0670 8261 1739 14.924 10 4°00' 0698 8.8436 .9976 9.9989 .0699 8.8446 1.1554 14.301 86»00' i '" .0727 8613 .9974 9989 .0729 8624 1876 13.727 60 20 .0756 8783 .9971 9988 .0758 8795 1205 13.197 40 30 .0785 8940 .9969 9987 .0787 8960 1040 12.706 80 40 .0814 9104 .9967 9986 .P816 9118 0888 18.851 20 50 .0843 9250 .9964 9985 .0846 9272 0728 11.826 10 S'OO' .0872 8.9403 .9962 9.9983 .0875 8.9420 1.0680 11.430 85°(0' 10 .0901 9545 .9959 9983 .0904 95i;S 0437 11.069 60 20 .0929 9682 .9957 9981 .0934 9701 0209 10.712 40 1 30 .0958 9816 .9954 9080 .0963 9836 0164 10.386 30 40 .0987 9945 .9951 9979 .0992 9966 0034 10.078 20 I 50 .1016 9.0070 .9948 9977 .1022 9.0093 0.9907 9.7882 10 6'nO' .10*-. 9.0192 .9945 9.9976 .1051 9.0216 0.9784 9.6144 84''00' 10 .1074 0311 .9942 9975 .1080 0336 9604 9.2553 60 20 .1103 0426 .9939 9973 .1110 0453 9547 9.0098 . 40 .30 .1132 0539 .9986 9978 .1139 0567 9433 8.7769 80 40' .11 in 0648 .9982 -9971 .1169 . 0678 9322 8.5555 20 : 50 .1190 0755 .9929 9969 .1198 0786 E214 8.3450 10 7°00' .1210 9.0859 .9925 9.9968 .1228 9.0891 0.9109 8.1443 83°00' 10 .131S 0961 .9922 9960 .1267 0995 9005 7.9630 • 50 20 .1276 1060 .9918 9964 .1287 1096 8904 7.7704 40 30 .13-05 1157 .9914 9903 .1317 1194 8806 7.6958 80 40 .1334 1252 .9911 9961 .1846 1291 8709 7.4287 20 50 .1363 1345 .9907 9959 .1376 1385 8615 7.2687 10 S-OO' .1392 9.1436 .9903 9.9968 .1406 9.1478 0.8588 7.1154 82''00' 10 .1421 1525 .9899 9956 .1435 1669 8431 6.9682 60 20 .1449 1612 .9894 9954 .1465 1658 8342 6.8269 40 30 .1478 1697 .9890 9953 .1495 1746 8255 0.6912 30 40 .1507 1781 .9886 9950 .1624 1831 8169 6.5606 20 50 .1536 1863 .9881 9948 .1564 1915 8085 6.4348 10 9»00' .1564 9.1943 .9877 9.9946 .1584 9.1997 0.8003 6.3138 81°00' Nat. Log. Nat. Ln;. Nat. Log. Log. Nat. an<;le. rOHNES. 8TNE- roTANOENTS. TANGENTS. AN(J1.K. TRIGOXOilETRIC RATIOS. ANULE. SIKES. COSINES. TANGENTS. COTANGENTS. ANGLE, Nat. Log. Nat. Log. Nat. . Log. Log. Nat. 0°00' .1564 9.1943 .9877 9.9946 .1584 9.1997 0.8003 6.3138 81°0'0' 10 .1593 2022 .9872 9944 .1614 2078 7922 6.1970 50 20 .1622 2100 .9868 9948 .1644 2158 7848 6.0844 40 30 .1650 2176 .9863 9940 .1673 2236 7764 5.9758 30 40 .1679 2251 .9858 9938 .1708 8313 7687 5.8708 20 50 .1708 2324 .9853 9936 .1733 2389 7611 5.7694 10 ICOO'^ .1736 9.2397 .9848 9.9934 .1763 9.2463 0.7537 5.6713 80°00' 10 .1765 2468 .9843 9981 .1793" 2586 7464 5.5764 60 30 .1794 2538 .9888 9929 .1823 2609 -7891 5.4845 40 80 .1822 2606 .9833 9937 .1858 2680 7820 5.3955 30 40 .1851 2674 .9827 9984 .1888 2750 7850 5.3093 20 50 .1880 2740 .9882 9982 .1014 2819 7181 5.2257 10 11°00' .1908 9.2806 .9816 9.9919 .1944 9.2887 0.7113 5.1446 79°00' 10 ■11937 2870 .9811 9917 .1974 2953 7047 5.0658 50 20 .1965 2934 .9805 9914 .2004 3020 6980 4.9894 40 30 .1994 2997 .9799 9912 .2035 3085 6915 4.9152 30 40 .2022 3058 .9798 9909 .2065 3149 6851 4.8430 20 50 .2051 3119 .9787 9907 .3095 3213 6788 4.7729 10 12">00' .2079 9.8179 .9781 9.9904 .2126 9.3275 0.6725 4.7046 78°00' 10 .3108 3238 .9775 9901 .8150 3336 6664 4.6382 50 SO .2136 3296 .9769 9899 .2186 3397 6003 4.5736 40 30 .2164 3353 .9768 9896 .2217 3458 0542 4.5107 30 40 .2193 3410 .9757 9893' .2247 3517 6483 4.4494 20 50 .2221 3466 .9750 9890 .2878 3576 6434 4.3897 10 13° 00' .2250 9.3521 .9744 9.9887 .2309 9.3634 0.6866 4.3316 77°00' 10 .2278 3575 .9737 9884 .2339 3691 6309 4.2747 60 20 .2300 3629 .9730 9881 .2370 8748 6252 4.2193 40 30 .2334 3682 .9724 9878 .2401 3804 6196 4.1863 30 40 .2363 3734 .9717 9875 .2483 3859 6141 4.1126 20 50 .8391 3786 .9710 9872 .2468 3914 6086 4.0611 10 14°00' :2419 9.3837 .9703 9.9869 .2493 9i3968 0.6082 4.0108 76°00' 10 .2447 8887 .9^96 9866 .3534 4021 5979 3.9617 50 20 .2476 3937 .9689 9863 .2555 4074 5926 3.9136 40 30- .2504 3986 .9681 9859 .2586 4127 5878 3.8667 30 40 .2532 4035 .9674 9856 .2617 4178 6822 3.8208 20 50 ,2560 4083 .9667 9853 .2648 4230 5770 3.7760 10 15° 00' .2588 9.4130 .9659 9.9849 .2679 9.4281 0.5719 3.7321 75°00' 10 .2616 4177 .9652 9846 .2711 4381 5669 8,6891 60 20 .2644 4223 .9644 9848 -.2742 4881 5619 3.6470 40 30 .2678 4289 .9636 9889 .2773- 4430 5570 3.6058 SO 40 .27d0 4314 .9688 9836 .2805 4479 5521 3.5656 20 50 .2728 4359 .9621 9832 .2836 4527 5473 3,5261 10 16°o0' .2756 9.4403 .9618 9.9828 .8867 9.4575 0.5425 3.4874 74°00' 10 .2784 4447 .9605 9825 .2899 4622 5378 3.4496 60 , 80 .2812 4491 .9596 9821 .2981 4669 5331 3.4124 40 30 .2840 4583 .9588 9817 .2962 4716 5284 3.3759 ,30 40 .2868 4576 .9580 9814 .8994 4762 5238 3.3402 80 50 .2896 4618 .957.2 9810 .8086 4808 5198 3.3052 10 IVOO' .8924 9.4659 .9563 9.9806 .3057 9.4853 0.5147 3.2709 73°00' 10 .8952 4700 .9555 9802 .8089 4898 5103 3.3371 60. 20 .8979 4741 .9546 9798 .3131 4943 5057 3.3041 40 30 .3007 4781 .9537 9794 .3153 4987 5013 8.1716 30 40 .3035 4881 .9528 . 9790 .8185 5031 4969 3.1397 20 50 .3062 4861 .9520 9786 .3817 5075 4925 3.1084 10 ISoOO' .8090 9.4900 .9511 9.9782 .3349 9.5118 0.4882 3.0777 72°00' Nat. Log. Nat. Log. Nat. Log. Log. Nat. ANGLE. COSINES. SINES. COTANGENTS. TANGENTS. ANGLE. TUIGOXOMETliir RATIOS. 1 ANGLE. SINKS. COSINES. TANGENTS. COTANGENTS. AN(.l,n, Nat. Log. Nat. Log. Nat. Log. Log. Not. 18°00' .3090 0.4900 .9511 9.9782 .3249 9.5118 0.4882 3.077 7 72°00' 1 10 .3118 4939 .9502 9778 .3281 6161 4839 3,0 4 75 50 1 20 .3145 4977 .9492 9774 .3314 5203 4797 8.0178 40 i 30 .3173 5015 .9483 9770 .3346 5245 4765 2.9887 30 i 40 .3201 5062 .9474 9765 .3378 5287 4713 2.9600 20 50 .3228 5090 .9465 9761 .3411 5329 4871 8.9319 ■ 10 19°00' .3256 9.51Sf, .9455 9.9767 .3443 9.5370 0.4630 2.9042 71<'00' 10 .3283 5163 .9446 9752 .3476 5411 4589 2.8770 50 20 .3311 5199 .9436 9743 .3508 5451 4549 2.8502 40 30 .3338 5235 .9426 9743 .3641- 5491 4509 2.8239 30 40 .3366 5270 .9417 9739 .3574 5531 4169 2.7980 SO 60 .3393 5306 .9407 9734 .3607 5571 4429 2.773 5 10 20-00' •3480 9.5341 .9397 9.9730 .3640 9.5611 0.4389 2.7475 70°00' 10 .3448 5375 .9387 9725 .3073 5060 4350 2.7228 50 20 .3475 .1409 .9377 9721 .3706 6680 4311 2.0985 40 30 .3502 5443 .9867 9716 .8739 5727 4273 2.6746 80 40 .3520 5477 .9356 9711 .3772 5766 4234 2.6511 20 50 .3557 5510 .9346 9706 .3805 5804 4196 2.6279 10 21''00' .8584 9.5543 .9336 9.9702 .3839 9,5842 0.4158 2.6051 69»00' 10 .3611 5576 .9325 9697 .3872 5879 4121 2,5820 50 20 .3638 5609 .9315 9692 .3906 5617 4083 2.5605 40 30 .3665 5641 .9304 9887 .3939 5934 4046 2.5388 30 40 .3692 5673 .9293 9682 .3978 6991 4009 2,5172 20 > 50 .3719 5704 .9283 9677 .4006 6028 3972 2,4960 10 22°00' .3746 9.6730 .9272 9.9672 .4040 9.6064 0.3936 2.4751 68°00' 10 .3773 5767 .9261 9667 .4074 0100 3900 2.4545 50 20 .3800 5798 .9250 9661 .4108 6136 3864 2.4342 40 30 .3827 5828 .9239 9656 .4142 6172 3828 2.4142 80 40 .3854 5859 .9228 9651 .4176 6208 3193 2.3945 20 50 .3881 5889 .9210 9646 .4210 6243 3757 2.3750 10 23'>00' .3907 9 5919 .9205 9.9640 .4245 9.6279 0.3721 2.8659 67°00' 10 .3934 5948 .9194 9636 .4279 0314 3086 2.3869 50 20 .3961 5978 .9182 9629 .4314 6348 3652 2.3188 40 30 .3987 6007 .9171 9624 .4348 6383 3617 2.2998 30 40 .4014 6036 .9159 9618 .4383 6417 3583 2.2817 20 50 .4041 6065 .9147 9613 .4417 6452 3648 2.2687 10 2"4°00' .4067 9.6093 .9135 9.9607 .4452 9.6486 0.3614 2.2460 66»00' 1 10 .4094 6121 .9124 9S02 .4487 6520 3480 2.2286 50 20 .4120 6149 .9112 9596 .4522 6553 3447 2.2113 40 30 .4147 6177 .9100 9590 .4557 6687 3413 2.1943 30 j 40 .4173 6205 .9088 9684 .4592 6620 3380 2. 1775 20 ! 50 .4200 6232 .9075 9579 .4628 6654 3340 2.1609 10 25»00' .4226 9.0259 .9063 9.9573 .4663 9.6687 0.8313 2.144 5 05''00' 10 .4253 6280 .9051 9567 .4699 6720 3280 2.1 am; 50 ' 20 .4279 6:il3 .9038 9561 .4734 6752 3248 2.1 123 40 30 ..1305 6340 .9026 9556 .4770 6786 3215 2.0905 30 40 .4331 0306 .9013 9549 .4800 6817 3183 2.0809 20 50 .4.J58 6392 .9001 9543 .4841 6850 3150 2.0055 10 26°00' .4384 9.6418 .8988 9.9537 .4877 9.6882 0.3118 2.0603 (i4»00' 10 .4410 6444 .8976 9530 .4913 6914 3086 2.0363 50 20 .4436 6470 .8902 9524 .4950 6946 3054 2.0204 40 30 .4462 6495 .8949 9518 .4980 6977 3023 2,0057 30 40 .4483 6521 .8936 9512 .6022 7009 2991 1.9912 20 50 .4614 6546 .8923 9505 .5059 7040 2960 1.9768 10 ■ZT'OO' .4540 9.6570 .8910 9.9499 .6095 9.7072 0.2928 1,9626 es'-oo' Nat. Log. Nat. Log. Nat. Log. Log. Nat. 3ENT8. ANGLE. COSINES. BINES. COTANGENTS. TAN ANGLE. TRIGONOMETRIC RATIOS. JlSBJJS. SINES. coarsEf. TANGENTS. COTANGENTS. ANGLE. Nat. Log. Nat. Log. Nat. Log. Log. Nat. sr^oo' .4540 9.6570 .8910 9.9499 .5096 9.7072 0.2938 1.9626 6S°00' 10 .4566 65&5 .8897 9492 .5132 7108 2897 1.9486 50 20 .4592 6620 .8684 9486 .5169 7184 3866 1.9847 40 30 .4617 6644 .8870 9479 .6306 7166 2835 1.9210 30 40 .4643 6668 .8857 9473 .6343 7196 2804 1.9074 20 60 .4669 6698 .8843 9466 .5280 7836 3774 1.8940 10 SS-OO' .4695 9.6716 .8829 9.9459 .5817 9.7867 0.8748 1.8807 62-00' 10 .4720 6740 .8816 9453 .6864 7887 3718 1.8676 60 20 .4746 6763 .8802 9446 .6392 7817 2683 1.8646 40 30 .4772 6787 .8788 9439 .5430 7348 2652 1.8418 30 40 .4797 6810 .8774 9432 .5467 7878 2622 1.8291 20 50 .4883 6833 .8760 9485 .5605 7408 2693 1.8165 10- S9°00' .4848 9.6856 .8746 9.9418 .6543 9.7488 0.2663 1.8040 61°00' 10 .4874 6878 .8738 9411 .5581 7467 2538 1.7917 60 80 .4899 6901 .8718 9404 .5619 7497 2508 1.7796 40 30 .4984 6923 .8704 9397 .5658 7626 2474 1.7675 30 40 .4950 6946 .8689 9390 .5696 7656 8444 1.7566 80 50 .4975 6968 .8675 9383 .6736 7585 2415 1.7437 10 80»00' .6000 9.6990 .8660 9.9376 .8^46 9368 .5774 9.7614 0.2386 1.7321 60°00' 10 .5025" 7018 .5818 7644 3866 1.7205 50 20 .5050- 7033 .8631 9361 .5861' 7078 2327 1.7090 40 30 .5075 7055 .8616 9363 .5890 7701 2299 1.6977 30 40 .5100 7076 .8601 9346 .5930 7780 2270 1.6864 80 50 .5125 7097 .8587 9388 .5969 7759 2841 1.6763 10 31° 00' .5160 9.7118 .8572 9.9331 .6009 9.7788 0.2212 1.6643 59°00' 10 .5175 7139 .8557 9823 .6048 7816 2184 1.6534 50 20 .5200 7160 .8542 9316 .6088 7846 3155 1.6486 40 30 .5386 7181 .8526 9808 .6138 7873 2127 1.6319 30 40 .5860 7201 .8.511 9800 .6168 7903 3098 1.6218 20 50 .5276 7222 .8496 9292 .6308 7980 3070 1.8107 10 82°00' .5399 9.7242 .8480 9.9284 .6249 9.7958 0.2042 1.6008 58°00' 10 .6324 7262 .8465 9376 .6289 7986 2014 1.5900 50 20 .5348 7288 .8450 9268 .6380 8014 1986 1.5798 40 30 .6373 7302 .8434 9360 .6371 8042 1968 1.5697 80 40 .6398 7832 .8418 9262 .6418 8070 1930 1.6597 20 50 .5422 7342 ..8403 9244 .6453 8097 1903 1.5497 10 33°00' .5446 9.7361 .8387 9.9236 . .6494 9.8125 0.1875 1.6899 57''00' 10 .5471 7380 .8871 9228 .6536 8153 1847 1.6301 BO 20 .5495 7400 .8355 9219 .6577 8180 1820 1.5804 40 30 .6519 7419 .8339 9811 .6619 8208 1792 1.5108 30 40 .5644 7438 .8323- 9208 .6661 8235 1765 1.5013 20 50 .6668 7457 .8307 9194 .6703 8263 1737 1.4919 10 34°00' .5592 9.7476 .8290 9.9186 .6745 9.8390 0.1710 1.4826 66<'00' 10 .5616 7494 .8274 9177 .6787 8817 1683 1.4733 60 30 .5640 7613 .8258 9169 .6830 8844 1666 1.4641 40 30 .S664 7631 .8241 9160 .6873 8371 1639 1.4650 30 40 .5688 7550 .8825 9151 .6916 83fl8 .1602 1.4460 20 50 .6712 7668 .8208 9143 .6959 8485 1575 1.4370 10 35'>00' .6736 9.7586 .8192 9.9184 .7008 9.8458 0.1548 1.4281 55°00' 10 .5760 7604 .8175 9185 .7046 8479 1521 1.4193 60 20 .5783 7628 .8158 9116 .7089 8606 1494 1.4106 40 30 .5807 7640 .8141 9107 .7133 8533 1467 1.4019 80 40 .6881 7657 .8124 9098 .7177 8559 1441 1.3934 20 50 .5854 7675 .8107 9089 .7881 8586 1414 1.3848 10 36°00' .5873 9.7692 .8090 9.9080 .7265 9.8613 0.1387 1.3764 64°00' Nat. Log. Nat. Log; Nat. Log. Log. Nat. ANGLE. COSINES. SIN^B. COTANGENTS. TANGENTS. AltGLE. XTl TRIGONOMETRIC RATIOS. ANGLE. SINES. COSINES. TANGENTS. COTANGENTS. ANGLE. Nat. Log. Nat. Log. Nat. Log. Log. Nat. 36°00' .5878 9.7692 .8090 9.9080 .7266 9.8613 0.1387 1.3764 64-00' 10 .5901 ■ 7710 .8073 9070 .7310 8639 1361 1.3680 60 ao .6925 7727 .8056 9061 .7355 8606 1334 1.8597 40 30 .5948 7744 .8039 9062 .7400 8692 1308 1.3614 80 40 .5972 7761 .8031 9042 .7445 8718 1282 1.8432- 20 50 .6996 7778 .8004 9033 .7490 8745 1266 1.S35I 10 37">0O' .6018 9.7796 .7986 9.9023 .7536 9.8771 0.1229 1.3270 53-00' 10 .6041 7811 .7969 9014 .7581 8797 1203 1.3190 50 20 .6006 7928 .7961 9004 .7627 8824 1176 1.3111 40 30 .6088 7844 .7934 8995 .7673 8850 1150 1.3032 30 40 .6111 7861 .7916 8985 .7720 8876 1124 1.3954 30 50 .6134 7877 .7898 8975 .7766 8902 1098 1.2876 10 38°00' .6157 9.7893 .7880 9.8965 .7818 9.8928 0.1072 1.2799 52-00' 10 .6180 7910 .7862 8955 .7860 8964 1046 1.2723 60 20 .6202 7926 ,7844 8945 .7907 8980 1020 1.2647 40 30 .6225 7941 .7826 8936 .7964 9006 0994 1.2572 80 40 .6248 7957 .7808 8926 .8002 9032 0968 1.2497 20 50 .6271 7973 .7790 8015 .8060 9058 0942 1.2423 10 sg-oo' .6293 9.7989 .7771 9.8905 .8098 9.9084 0.0916 1.2349 51-00' 1 10 .6316 8004 .7763 8895 .8146 9110 0890 1.2276 50 ao .6338 8020 .7735 8884 .8196 9185 0865 1.2203 40 30 .6361 8085 .7710 8874 .8243 9161 0889 1.2181 30 40 .6383 8050 .7698 8864 .8292 9187 0818 1.2059 20 50 .0406 8066 .7679 8863 .8342 9312 0788 1.1988 10 40''00' .6428 9.8081 .7660 9.8843 .8391 9.9238 0.0762 1.1918 50-00' 10 .6450 8096 .7642 8832 .8441 9264 0736 1.1847 50 30 .Ijl72 8111 .7623 8821 .8491 9289 0711 1.1778 40 ! 30 .6494 8136 .7604 8810 .8541 9315 0685 1.1708 30 40 .6517 8140 .7585 8800 .8591 9341 0669 1.1640 20 50 .6539 8156 .7566 8789 .8642 9366 0634 1.1671 10 41«00' .6561 9.8169 .7547 9.8778 .8693 9.9392 0.0608 1.1504 49-00' 10 .6583 8184 .7528 8767 .8744 9417 0588 1.1436 50 20 .6604 8I9« .7509 8766 .8790 9443 0557 1.1369 40 30 .6626 8213 .7490 8745 .8847 9468 0632 1.1808 80 40 .6648 8227 .7470 8783 .8899 9494 0506 1.1237 20 50 .6670 8241 .7451 8722 .8963 9519 0481 1.1171 10 43-00' .6691 9.8256 .7431 9.8711 .9004 9.9544 0.0466 1.1106 48-00' ■ 10 .6713 8269 .7412 8699 .9057 9570 0480 1.1041 50 20 .6734 8283 .7392 8688 .9110 9695 0405 1.0977 40 30 .6756 8297 .7373 8676 .9163 9621 0379 1.0913 30 40 .6777 8311 .7358 8666 .9317 9646 0354 1.0850 20 50 .6799 8324 .7333 8653 .9371 9671 0339 1.0786 10 43°00' .6820 9.8338 .7314 9.8641 .9325 9.9697 0.0303 1.0724 47''00' 10 .6841 8351 .7294 8699 .9380 9732 0278 1.0661 50 20 .6862 8365 .7274 8618 .9435 9747 0258 1.0699 40 30 .6884 8378 .7254 8606 .9490 9772 0228 1.06S8 30 40 .6905 8891 .7234 8594 .9545 9798 0202 1.0477 20 60 .6926 8405 .7314 8582 .9601 9828 0177 1.0416 10 44°00' .6947 9.8418 .7193 9.8569 .9667 9.9848 0.0152 1.0356 46-00' 10 .6967 8431 .7173 8557 .9718 9874 0126 1.0296 50 20 .6988 8444 .7163 8546 .9770 9899 0101 1.0235 40 30 .7009 8457 .7133 8532 .9827 9924 0076 1.0176 80 40 .7030 8469 .7112 8620 .9884 9949 0051 1.0117 20 50 .7050 8482 .7092 8607 .9942 9975 0025 1.0068 10 45''00' .7071 9.8495 .7071 9.8495 I.OOOO 0.0000 0.0000 1.0000 46-00' Nat. Log. Nat. Log. Nat. Log. Log. Nat. ANGLE. COSINES. SINES. COTANGENTS. TANGENTS. ANGLE. . 1 TRIGONOMETRY. Teigonometry is that branch of mathematics which treats of the numerical relations of angles and triangles. It is essentially algebraic in character, but is founded on geometry. I. THE EIGHT TEIANGLB. §1. TRIGONOMETRIC RATIOS. Theoe. 1. If from a jjoini in one side of an acute angle a perpendicular fall on the other side, then, in the right trian- gle so formed, the ratio of the side opposite the angle to the hypotenuse is the same, ivhatever point he tahen / and so for that of the adjacent side to the hypotenuse, for that of the opposite side to the adjacent side, and for the reciprocals of these three ratios. For let A be any acute angle ; b, b' • • • points on either bound- ing line ; a, a' • ■ • perpendiculars from b, b' • • • to the other line at C, c' ■ • ■ ; 5, 5' ■ • • the lines AC, Ac' • • • ; and c, c' • • • the lines ab, ab' • • • ; then'.'the right triangles ABC, ab'c'- • • are similar, .•. the ratios a/c, a'/c' ■ • ■ are all equal ; and so for the other ratios l/c, I'/c' • ■ • , a/h, a'/h'- b/a, b'/a'---, c/b, c' /V ■ ■ ■ , c/a, c'/a'---. THE BIGHT TRIANGLE. [I. But if an angle be taken greater or less than A, the tri- angles so formed are not similar to these, and the ratios are diiierent from those for the angle a. For this reason an acute angle has its six ratios distinct from the ratios of every other angle, and if one of the ratios be given the angle can be constructed. These ratios are the six principal trigonometric functions of an angle, and they are named as follows : opposite side to hypotenuse, the sine of the angle, adjacent side to hypotenuse, the cosine, opposite side to adjacent side, the tangent, adjacent side to opposite side, the cotangent, hypotenuse to adjacent side, the secant, hypotenuse to opposite «ide, the cosecant. "When written before the name of the angle, the words sine, cosine, tangent, cotangent, secant, cosecant may be abbreviated to sin, cos, tan, cot, sec, esc. Standing alone, the abbreviations have no meaning. If ABC be any right triangle with c the right angle, a the side opposite the acute angle a, J the side opposite the acute angle B, and c the hypotenuse, then the six ratios of each of the acute angles may be expressed in terms of the three sides of the triangle, as below. sin a — a/c, esc a = c/a, and sin b = h/c, esc b = c/h, cosA='d/c, secA = c/J, cos B = a/c, sec B = c/a, tan A = a/5, cot x = b/a, tanB = J/a, cot B = a/5. Note. In the discussion of the right triangle that follows, the triangle is always lettered as in the figures above ; i.e., §1..] TRIGONOMETRIC RATIOS. 3 with c for the right angle, c for< the hypotenuse, A, B for the acute angles, a, b for the sides opposite a, b. The expression sin~^a/c nieans the angle whose sine is a/c ; COS"' J/c, the angle whose cosine is 6/c ; tan~' a/h, the angle whose tangent is a/b, and so for the other ratios. U'liey are read : the anti-sine of a/c, the anti-cosine otb/c, the anti- tangent of a/b, and so on. E.g. if sin A = -^, ifcosB = |, iftanF=6, ifcotx = V3, then A = sin-'-|, B = cos~'|, p = tan-'6, x = cot-'V3. The index ~' is to be carefully distinguished from the nega- tive exponent ; it is analogous to that in the expression log"' 2, which is read the anti-logarithm of 3 and means the number whose logarithm is %. The positive powers of the trigonometric ratios are com- monly written in the form sinV, cos'b, instead of (sin a)", (cos b)" ; but their reciprocals are written in the form of fractions, or with the exponent without the bracket. E.g. 1/sinA, or (sin a)"', not siu-'A ; I/cos^'b, or (cosb)"''. QUESTIONS. 1. If the sides a, b, c, of a right triangle Abc be 3 feet, 4 feet, 5 feet, what are the six ratios of A and of B ? if the sides be 3 yards, 4 yards, 5 yards ? if 3 miles, 4 miles, 5 miles ? 3. In a right triangle abc the sides a, c are 13 yards and 13 yards : find 6- and the six ratios of a and of B. So, if a, b be 13 feet and 5 feet, find c and the six ratios. 3. Construct the right triangle abc with the hypotenuse c 5 feet, and a side a 3 feet. What is the sine of the angle A.? From this construct a right triangle ABC if sin A = |.- So,ifcosA = f, iftanA = |, ifcotA = f, ifsecA = |. .#*Mi, Construct sin-' "ii cos-'|, tan-'|, cot"' f , sec"' f . 5. Find the six ratios of one of the acute angles of a right isosceles triangle. 6. Draw a perpendicular from the vertex to the base of an equilateral triangle, and find the six ratios of the acute angles of the right triangles so formed. 4 THE RIGHT TRIANGLE. [I, THS. 7. In a right triangle ABC, let the hypotenuse c be 12 feet and the angle A be 30°: find the sides a, b, given sin 30° = .5, cos 30° = .866, nearly. 8. In a right triangle abg, let the side a be 13 yards and the angle a be 35° : find the sides b, c, given sin 35° = .574, tan 35° = .7. 9. In a right triangle abc, let the side b be 12 miles and the angle Abe 40°: find the sides c, a, given cos 40° = . 766, cot 40° = 1.192. 10. In a right triangle abc, let the hypotenuse c be 13 feet and the side a be 8.484 feet : find the side b and the angle A, given sin 45° = .707. y 11. In a right triangle abc, let the side a be 12 yards and the side b be 9.948 yards : find the side c and the angle A, given sin 50° = .766, tan 50° = 1.193. 12. In a right triangle abc, let the side b be 13 miles and the hypotenuse c be 30.9 miles : find the side a and the angles, given cos 55° = .574, tan 55° = 1.438. 13. In a right triangle abc, let the side a be 13 metres and the hypotenuse c be 33^ metres : find the side b and the angles, given sin 21° 6' = .36, cos 21° 6' = .933. -. Verify the work by showing that a' + b'' — c'. 14. Draw two right triangles abc, a'b'c', having A larger tliuii a', and show which of the ratios of A are larger, and which are smaller, than the like-named ratios of a'. 15. Draw a right triangle having an acute angle less than half a right angle, and show which of the ratios of that angle are larger than unity, and which are smaller, j 16. Draw a right triangle having one acute angle very small, and show which of the ratios of this angle are very small, which are very large, and which are near unity. As the angle is made smaller and smaller, approaching zero, to what do these ratios approach ? So, what are the ratios of the other acute angle, which is very near a right angle ? 3-4, §1.} TEIGONOMETEIC RATIOS. Theoe. 2. If A.le any acute angle, then : sinA • cscjl^ 1, cos A -jm^ 1, -.JmLLSQtA = 1- For, from any point B of eit|ier side of the angle, let fall a perpendicular Bc up(3n the other side, as in theor. 1; then'. "in the right triangle abcso formed, siia.A = a/c, gsga = c/a, [df. .•. sin A- osc A = a/c-c/a = 1. q.e.d. .• cos A = 5/c, sec A = c/l, So, So, . cos a'- sec a =:: b/c • c/5 = 1 . Q.E.D. [df. [df. tan A = a/h, cot A = I /a, .•. tanA-cot A = ffl/&-57« = l. q.e.d. Theoe. 3. If Abe any acute angle, then': For •.• in the right triangle abc, formed as in theor. 1, sinA=a/c, cosA = 5/c, tanA = ff/5, cotA = 5/fl!, [df .•. sin a/cos a = a/c : b/c = a/b = tan A, and cos A/sin A = 6/c : ffl/c = 5/« = cot A. q.e.d. Theoe. 4. If a be any acute angle, then : si7i^A + cos' A = 1, 1 + tan' A = S^C^A, l-\-COt'A = CSC^A. For ■ and • : in the right triangle abc, -.d'/c' + br/c-'^l; . • sin A = a/c, cos A = l/c, \ sin''A + cos''A=l. a' + b'^. •3 [diy. by c". [df. Q.E.D. So, and • .• tan A = a/b, sec A ^'c/b, : l+tan°A = sec'A. [div. by b\ [df. Q.E.D. So, and •, l + bya' = cya'; , ■ cot A = b/a, esc A = c/a, : 1 + cofA = CSC^A. . [diy. by a'. Q.E.D. 6 THE RIGHT TRIANGLE. [I, TH. /^ Theor. 5. If A. he any acute angle and B the complement of X, then: sinA = cosB, tanA = cotB, !^a c\ B There are four cases : 1. Given c, a, the hypotenuse and an acute angle : then B = 90° — A, (7 = c-sinA, 5 = c-cosa. Checks : tan B = i/«, ¥={0 + 0) {c-a), n'' = {c+b) (c-h). 2. Given b, a, a side and an acute angle : then B = 90°-A, c=J/^osa, a = 5-tanA. Checks : cos B = ff/c, b' = {c + a){c-a), a'=(c + 5) (c-5). 3. Given c, b, the hypotenuse and a side : then cos A = b/n, b = 90°-a, rt = &-tanA. Checks: cos B = a/c, b''={c + a) (c-a), a'-{c + b) {c-b). 4. Given a, b, the two sides about the right angle: then tanA = a/Z>, B=;90° — a, c = Zi/cosa. Checks : cos B = a7c, b' = {c+a) {c-a), a'={c + b) (c-b). §3.] THE SOLUTION OF RIGHT TRIANGLES. 11 E.g. let c = 125, A = 40° ; then b = 90° - 40° = 50° j and, with natural functions, the work may take this form : sin 40° = .6428 cos 40° = .7660 , xl35 xl'SS a = 80.35. chech : i;an b = b /a, 80,35)95.75(1.1917 C- & = 95.75. (c + J)(c-S) = a' = 125 tan 50° = 1.1918 I: c + b- = 95.75 = 220.75 « = 80.35 C-1-. = 29.25 X X 80.35 -J' = 6456.9375 a'' = 6456.1225 So, with logarithmic functions, the work may take this form : log-sin 40° = 9.808 1 log-cos 40° = 9.8843 log 135 = 2.0969 + log 125 = 2.0969 + log a = 1.9050, rt = 80.35 log 5 = 1.9812, 5 = 95.76 check: c=126 1= 95.76 c + 5 = 220.76 log, 2.3439 log « = 1.9050 c-l- 29.24 1 .4660 x2 log (c"-i') = 3.8099. log a" = 3.8100. Note. The two solutions do not quite agree, and the checks are not perfect ; the defects arise from the use of the small tables. More exact results come from larger tables, that give the ratios correct to five, six, or seven figures. QUESTIONS. Solve these right triangles, using natural functions, given : 1. c,.40 yds.; A, 30°. 2. c, 12..5 ft.; B, 68° 10'. 3. I, 187 metres ; A, 55° 20', 4. a, 7.57 in.; b, 9° 30',- 5. 5, 18.5 ft.; c, 125 ft. 6. c, 37 mi.; a, 25.2 mi. 7. a, 59.3 yds.; 5, 45.7 yds. 8. a, 4 ft. 6 in.; J, 12ft. 9. in. Solve thgse right triapdes, using logarithmic functions, given : 9.^c, m ft^ Af 60°. 10. c, 18.7 yds.; B, 76° 15'. 11. . 5, -45.9 yds. ; A, 59° 15'. J^. a, 18.3 chs. ; B, 55° 12'. 13. 5, 597 m. ; c, 676 ra. 14. a, 1278 yds. ; c, 1355 yds. 15. a, 27.85 in,; S, 5519 in. 18^ a, 8539 ft.; J, 2815 ft. 12 THE lUGHT TRIANGLE. [I. g 4. ISOSCELES AND OBLIQUE TRIANGLES. In an isosceles triangle, the perpendicular from the vertex to the base divides the triangle into two equal right triangles ; and if two parts of one of these triangles be given, this tri- angle may be solved, and so the whole triangle is solved. If three parts of an oblique triangle be given, always includ- ing a side, a perpendicular may fall from a vertex to the opposite side and so divide the given triangle into two right triangles, and by their solution the triangle is solved. Let ABC be any oblique triangle, a, h, c the sides opposite the angles a, b, c ; ji the perpendicular ad from a to a ; x, y the segments cd, bd, of a. The statements below apply directly to the second of the three figures ; but with slight modifications suggested by the figures themselves, they apply to the other figures as well. There are four cases : 1. Given a, t, c, the three sides : fhQn:-p' + x^=V, / + ?/ = f', .■.x'-y-' = ¥~c-; and ■.■x + y = a, .-. x — y={V — c')/a, .:x=^[a + (F-c')/a] = {a' + h'-c')/2a, and y = i [_a-{¥-?^/a] = {a'-b'' + c')/2a; and two parts of each right triangle are known. 2. Given b, B, c, a side and two angles : then, in the right triangle A CD, b and c are known, and^ and x may be computed ; and, in the right triangle ABT>,p and_B are known, and c and y may be computed. a = x + y, a = 180°-(b-1-c). §4.j ISOSCELES and' OBLIQUE TEI ANGLES. 13 3. Given c, a, b, two sides and the included angle : then, in the right triangle abd, c and b are known, and ^ and ^ y may be computed ; and, in the right triangle acd, p is known, x = a — y, and I and C may be computed. A = 180°-{B + c). 4. Given 5,'C, b, two sides and an opposite angle : then, in the right triangle abd, c and B are known, and^ and y may be computed ; and, in the right triangle acd, h and p are known, and x and c may be computed. a = y±x, a=180°-(b + c). QUESTION'S. Solve these isosceles triangles, given : 1. The sides 10 yards, and the base 16 yards. 2. The vertical angle 90°, and the base^lO yards. 3. The base 10 yards, and the base angles 70°. 4. The vertical angle 70°, and a side 12 yards. 5. The base 18 yards, and a side 13 yards. G. If two sides and an angle opposite one of them be given, h, c, B, the side c is given in length and position both, a in po- sition but not in length, b in length but not in position, and I) finds its position only as it swings about A as a hinge till its lower end rests on the line of the base : if then the angle B be acute, and if the swinging side i be shorter than the perpen- dicular^, is a triangle possible ? is there a triangle if b be just as long asp? ot what kind is it ? is there one triangle or two if b be longer than p, but shorter than c? if 5 be just as long as c ? if 5 be longer than c ? Draw figures to illustrate. So, if B be right or obtuse ? Solve these oblique triangles,, given : 7. fl, 13 ; 5, 15 ; c, 17. 8. a, 357 ; b, 537 ; c, 735. 9. c, 5 ; a, 7 ; Bi 65°. 10.' a, 537 ; b, 753 ; c, 119° 15'. 11. b, 30 ; B, 55°; c, 48° 25'. 12. a, 7.5 ; a, 84° ; B, 42° 37'. , 13. b, 5, 10, 15, 20, 25 in turn ; c, 20 ; b, 30°. 1-i THE RIGHT TRIANGLE. [I, §5. HEIGHTS AND DISTANCES. The plane of the horizon at any point on the earth's surface is the plane that is tangent to the surface, i.e. to the surface of still water, at that point ; it would therefore be perpen- dicular to the radius of the earth, if the earth were a perfect sphere. The direction perpendicular to the horizou-plane is determined by a plumb line ; it is a vertical line, and any plane containing this line is a vertical plane. Any plane parallel to the horizon-plane is a horizontal plane, and such a plane may be determined by a spirit level. An angle lying in a horizontal plane is a horizontal angle, and an angle lying in a vertical plane is a vertical angle. The vertical angle made with the horizontal plane by the line of sight from the observer to any object is its angle of elevation if the object be above the observer, and its angle of depression if it be below him. Ordinary field instruments measure horizontal and vertical angles only. By distance is meant the horizontal distance, unless otherwise named ; and by height is meant the vertical distance of a point above or below the plane of observation. A surveyor's chain is four rods long and it is divided into a hundred links. Ten square chains make an acre. To find the height above its base of a vertical column, ap, whose base is accessible. 1. If the column ap stand on a horizontal plane : p O A Prom the base a measure any convenient distance AO, and the angle aop ; and solve the right triande AOP. for AP. i.5.] HEIGHTS AND DISTANCES. 2. If the column pq stand on an inclined plane ; 15 Let p be the top of the colnmn, q the point at the base of the column below p, and A a point below p in the hori- zontal plane through the point of observation, o ; measure any convenient distance Qo along the plane, and the angles of elevation, or deptessiou aop, acq ; solve the right triangles aop, acq : theu pq = ap±aq. To find the distance from the observer, and the height above its base, of an inaccessible but visible vertical column. Let p be the top of the column, Q the base, b the position of the observer, a the point vertically below p in the horizontal plane through b ; take any other convenient point of observation c, and measure the horizontal line bc, the horizontal angles ABC, ACB, and the vertical angles abp, abq ; solve the horizontal oblique triangle abo for ab, and the vertical right triangles abp, abq for ap, aq : then PQ=:AP±AQ. IG THE EIGHT TRIANGLE. [I. If the observer be in the same horizontal plane as the base, the line bq coincides with ba, and bap is the only vertical triangle to be computed. To find the distance apart of ttvo objects that arc separated by iiH impassable harrier. 1. If both objects be accessible : J&---^ Let E, F be the two objects, and G the point of observation ; measure the horizontal lines ge, qf and the horizontal angle EGF, and compute ef. 2. If both objects be inaccessible : Let C, D be the two objects ; measure any convenient line AB and the horizontal angles abc, abd, bac, bad ; in triangle abd compute bd ; in abc compute bc ; in bod compute CD. This is the method of triangulation j ab is the base line. QUESTIONS. 1. At 120 feet distance, and on a level with the foot of a steeple, the angle of elevation of the top is 62° 27' : find the height. [230.03 feet. 2. From the top of a rock 326 feet above the sea, the angle of depression of a ship's hull is 25° 42' : find the distance of the ship. [677.38 feet. 3. A ladder 29^- feet long standing in the street just reaches a window 25 feet high on one side of the street, and 23 feet high on the other side : how wide is the street ? [34.13 feet. §5.] HEIGHTS AND DISTANCES. 17 4. From the top of a hill I observe two successive mile- stones in the plain below, and in >a straight line before me, and find their angles of depression to be 5° 30', 14° 20' : what is the height of the hill ? [815.85 feet. 5. Two observers on the same side of a balloon, and in the same vertical plane with it, are a mile apart, and "find the angles of elevation to be 17° and 68° 35' respectively : what - is its height ? [1§36 feet. / 6. From the top of a mountain 1^ miles high, the dip of the sea-horizon (angle of depression of sky-and- water line) is 1° 34' 40" : find the earth's diameter, and the distance of the sea-horizon. f 7. What is the distance and the dip of the sea-horizon from the top of a mountain 2| miles high, the earth's mean radius being 3956 miles ? [3° 8' 8". !>4^ 8. If the dip of the sea-horizon be 1°, find the height of the mountain, and the distance of the sea-horizon. -, 9. How far should a coin an inch in diameter be held from 'the eye to subtend an angle of 1° ? 10. Given the earth's equatorial radius, 3963.76 miles, and tlie angle this radius subtends at the sun, 8". 81 : find the dis- tance of the earth from the sun. [93 780 000 miles nearly. 11. Find the distance across a river, if the base ab be 475 feet ; the angle A, 90° ; the "angle B, 57° 13' 30". [737.68 feet. 13. G-iven CA, 131 feet 5 inches ; BC, 109 feet 3 inches ; the angle c, 98° 34' : what is the distance ab ? [183 feet. 13. Two ships lying half a mile ,apart, each observes the angle subtended by the other ship and a fort ; the angles are found to be 56° 19' and 63° 14' : find the distances of the ships from the fort. . [3535, 3710 feet. 14. Given the base ab, 131^ yards ; the angle bad, 50° ; the angip BAG, 85° 15' ; the angle dbc, 38° 43'; the angle DBA, 94° 13' : what is the distance CD ? Check the work by making two distinct computations from the data. [139.99 yards. 3 18 THE EIGHT TRIANGLE. §6. COMPASS SURVEYING. [I, In compass surveying, the leaving of a point is the hori- zontal angle which the line of sight from the observer to the point makes with the north-and-south line through the point .of observation. This angle is found by aid of the compass. The latitude of a point is its distance north or south of a given point. The latitude of a line is the length of its pro- jection on a north-and-south line ; and its departure is the length of its projection on an east-and-west line. E.g. in the figure below, representing a field, the starting point is A, the bearings of the lines ab, bo- ■ •, taken in order, are : s. 70° 30' e. (70° 20' east from south), s. 10° 15' E., N. 55°35'e., k. 18°45'w, s. 40°55'w., s. 37° 15' w. ; and the lengths of these lines, in chains, are : 6,37, 4.28, 13.36, 1496, 11.15, 8.00. k' ■N 1. i. E f' / \ P" \ 1 d' / V A *^*^^ ^ ^ b' c' S C"C Through all the points A, B • • • , are drawn north-and- south lines, marked on the figure with arrows, and east-and- west lines perpendicular to them. The north-and-south line through the starting point a is distinguished as the meridian. The latitude of ab is the length of ab', the projectioji of ab on the meridian, and it is computed by multiplying 6.37, the length of ab, by the cosine of 70° 30', the bearing of ab. §6.J COMPASS sueteyi:n'G. 19 So, the departure of ab is the length of b'b, i.e. the prod- uct of 6.37 by the sine of 70° 20'. The latitude of the line BC is the length of Bc'', i.e. the product of 4.28 by the cosine of 10° 15', and the departure of BC is the length of c"o, i.e. the product of 4.28 by the sine of 10° 15'; and so for the latitudes and departures of the other lines, as shown in the table below. The meridian distance of a point is the distance of the i:)oint east or west from the meridian, and the double meridian distance of a line is the sum of the meridian distances of its ends. E.g. the meridian distance of the point B is b'b, and that of c is c'c, which is equal to b'b-|-c"c. So, the double meridiati distance of the line ab is + b'b, and that of BC is b'b + c'c. When a surveyor has run round a field, e.g. that which is described above, and has found and set down the lengths and bearings of the sides, he. has next to compute the latitudes and departures of the sides, the meridian distances of the cor- ners, and the double meridian distances of the sides as shown above. He is then ready to compute the areas of certain ti-apezoids and right triangles, and finally the area of the field ; and he takes care to set down his work in such form that it can be easily understood and reviewed, generally in the form of a table as below. EEAKING. DIS- TANCE. DEP. M.D. D.M.D. LAT. DOUBLE AREA. + — AB S. 70° 30' E. 6.37 5.998 5.998 5.998 -3.144 -12.860 BC s. 10° 15' b; 4.38 .761 6.759 12.757 -4.213 -53.733 CD fj.55°35'B. 12.36 10.196 16.955 23.714 6.985 165.642 DE N. 18° 45.' w. 14.96 -4.809 13.146 29.101 14.160 412.245 EF s. 40° 55' w. 11.15 -7.303 4.843 16.989 -8.436 -143.149 PA s. 37° 15' V. 8.00 -4.843 0. 4.843 -6.369 -30.845 +577.887 -340.586 337.801/3=168.651 square£hains=±16.865 acres. 337.301 -240.586 20 THE RIGHT TRIANGLE. [I. In this figure there are two right triangles ab'b, Ff'a and four trapezoids bb'c'c, cc'd'd, dd'e'e, ee'f'f, so related that the area of the polygon abcuef is the excess of the sum of the two trapezoids cc'd'd, dd'e'e over the sum of the two tri- angles and the other two trapezoids. = i [ - ab' • b'b - b'c' • (b'b + c'c) + c'd' • (c'c + d'd) + d'e' • (d'd + e'e) — e'f' • (e'e + f'f) - f'a • f'f] and it remains only to compute the lines ab', b'b- ■ ■, and to add, subtract, and multiply as shown below. In detail the work may take this form : 1. To compute the latitudes and departures s. 70° 20' E. cosine sine .3365 .9417 6.37 AB 6.37 -2.144 5.998 N". 18°45' w. cosine sine .9469^ .3214^ 14.96 de 14.96 s. 10° 15' E. cosine sine .9840 .1779 4.28 BC 4.28 -4.212 .761 s. 40° 55' w. cosine sine .7556^ .6550 11.15 EF 11.15 of the sides : N. 55°35'e. cosine sine .5652 .8249 12.36 CD 12.3 6 6.986 107196 8. 37° 15' w. cosine sine .7960 .6053 8.00 FA 8.00 § 6.] COMPASS SURVEYING. 31 North latitudes, nortliin.gs, are called positive ; south lati- tudes, southings, negative. East departures, eastings, are called positive ; west departures, westings, negative. 3. To coinpute the meridian distances : B C D K F A 5.998 5.998 6.759 16.955 12.146 4.843 + .761 + 10.196 - 4.809 - 7.303 -4.843 6.759 16.965 13.146 4.843 3. To compute the double meridian distances : AB BO CD DE EE EA 5.998 5.998 6.759 16.955 13.146 4.843 + 6.759 + 16.955 + 12.146 + 4.843 12.757 23.714 29.101 10.989 4. To compute the douhle areas : abb' bb'g'c cc'd'd dd'e'e ee'f'e ee'a, 5.998 13.757 23.714 39.101 16.989 4.843 X -2.144 X -4.212 x +6.985 x +14.166 x -8.426 x -6.36 9 12.860 -53.732 +165.642 +413.245 "143.149 -30845 QUESTIONS. 1. A surveyor, starting from A, runs n. 22° 37' b. 3.37 chains to b; thence N. 80" 24' E. 3.81 chains to c ; tjience s. 41° 12' E. 5.29 chains to n ; thence s. 62° 45' w. 6. 22 J chains to E : find the latitude and meridian distance of b, o, d, e from A ; find the bearing and distance of A from e ; find the area of the field abode. 2. Starting at A and chaining along the .surface of the ground, a surveyor runs N. 81° 10' E. 48 chains to B, at an elevation of 4° 15' ; thence N. 30° 25' w. 126 chains to c, at an elevation of 3° 40' ; thence s. 73° 50' w. 45 chains to d, at an elevation of 2° 40' ; thence s. 60° E. 85 chains to E, at a depression of 4° 15' : find the horizontal distances AB, BC, OD, DE, and the heights of b, g, d, e above A ; find the bearing, distance, and angle of depression; of a from e ; find the area of the field abode. a3 GENERAL 'PKOPERTIES OF PLANE ANGLES. [II, II. GENEEAL PEOPERTIES OP PLANE ANGLES. Hitherto the lengths of the sides of a triangle and the magnitudes of the angles have been mainly considered, and little attention has been paid to their directions ; but greater generality, as well as greater definiteness, is given to the definitions and theorems of trigonometry if the Mnes and angles be thought of as directed as well as measured. Nor is this a new thing : in geography and navigation longitudes are distinguished by the words east and west, and latitudes by north and south ; a surveyor speaks of liis northings and southings and of his eastings and westings, and he writes down the bearings of his lines with the sig- nificant letters N, s, e, \t ; in physics the directions and inten- sities of forces are represented By the directions and lengths of lines. Even the language is not new : the mathematician merely makes use of the familiar algebraic words positive and nega- tive as more convenient to him than the commoner words north, south, east, west, up, down, right, left, forward, backward. §1. DIRECTED LINES. Hereafter every straight line will be regarded as having not only position but direction also, meaning thereby that a point moving along the line one way will be regarded as moving forward, and a point moving along the line the other way as moving backward. The direction of the line is assumed to be that of forward motion. If a line represent a force or an actual motion, like tiiat of the winds and the tides, it has a natural direction ; otherwise its direction may be assumed at will. E.g. with a double-track east-and-west railway, the south track may be used habitually by east-bound trains, and the north track by west-bound trains. On the south track a train moves forward when goiug east, and it goes west only §!•] DIEECTED LINES. 33 when backing. On the north track forward motion is west- ward motion. The two tracks may be regarded as two par- allel lines lying close together and having opposite directions. A segment of a line is a limited portion of the line that reaches from one point, the initial point of the segment, to another point, the terminal point. A segment is a 2^ositive segment if it reach forward, in the direction of the line, and a negative segment if it reach backward. It is convenient also to speak of the positive and negative ends of a line, meaning by the positive end that end which is reached by going forward along the line, from any start- ing point upon it, and by the negative end that end which is reached by going backward. E.g. if a north-and-south line be directed from south to north, then the north end is tlie positive end and the south end is the negative end of the line; segments of this line reaching northward are positive segments and segments reach- ing southward are negative segments. The direction of a line is indicated by an arrow, or by naming two of its points, the direction being from the point first named towards the other. The direction of a segment is shown by the order of the letters at its extremities, the initial point being najmed first and tlie terminal point last. U.g. the indefinite line op has its positive direction from o to P, and the segment ab of the line op is the segment that reaches from the point A to the point B. If two segments, not necessarily upon the same line, have the same length and be both positive or both negative, they are equal segments ; if they have the same length, and be one positive and the other negative, they are opposite segments. ADDITIOK OF SEGMENTS OF A STRAIGHT LINE. Two or more segments of a straight line are added by placing the initial point of the second segment upon the terminal of the first, the initial point of the third segment 2i GENERAL PKOPERTIES OF PLAKE AKGLES. [II, upon the terminal of the second, and so on ; and the sum of all the segments so added is the segment that reaches from the first initial to the last terminal point. When a positive segment is added,' the terminal point slides forward ; when a negative segment is added, it slides backward. E.g. in the figures below, AB + BA = 0, AB-|-BC = AC, AB + BO + CA = .0, AB + BC + CD = AD, AB + BC + CD + DA = 0. CD A H 1 D This addition is analogous to the addition of like numbers, positive and negative, in algebra. One segment is subtracted from another by adding the op- posite of the subtrahend to the minuend, or by placing the initial point of the subtrahend upon that of the minuend ; the remainder is then the segment that reaches from the ter- minal point of the subtrahend to that of the minuend. QUESTIONS. 1. If from a given starting point one man walk east and another west, each a hundred yards, how far apart are the two men ? how far, and in what direction, is the first man from the second ? the second man from the first ? 3. If the river run five miles an hour, how fast does a boat go, with the current, if tlie crew can row four miles an hour in still water ? against the current ? 3. If longitudes alone be under consideration, and west longitudes be marked +, how may east longitudes be marked ? how may north and south latitudes be then distinguished ? 4. If a traveller go east 50 miles, then west 30 miles, then west 60 miles, then east 30 miles, how far has he gone ? and how far, and in what direction, is he from the starting point ? § 2. ] DIRECTED PLANES AND ANGLES. 25 §3. DIRECTED PLANES AND ANGLES. Hereafter every plane will be regarded as having direction, meaning thereby that a line swinging about a point in the plane one way will be regarded as swinging forward, and a line swinging the other way as swinging backward. The direc- tion of the plane is that of the forward motion of the line. If the swinging line has a natural motion like that of the hands of a clock, or a spoke of the fly-wheel of an engine, or an equatorial radius of the earth, then the direction of the plane is determined by this motion ; otherwise its direction may be assumed at will. ; This swinging motion, as viewed from one side of the plane, is clockwise, i.e. left-over-right, and counter-clockwise, i.e. right-over-left, as viewed from the other side. E.g. the apparent daily motion of the sun, as seen by an ob- server in the northern hemisphere, is clockwise, and as seen by one in the southern hemisphere it is counter- clockwise ; but to both of them it is the same east-to-west motion, and the plane of the sun's apparent path is an east-to-west plane. So, the real motion of an equatorial radius of the earth is counter-clockwise if viewed from a point in the northern hemisphere, and clockwise if from a point in the southern hemisphere ; but it is the same west-to-east motion, and the plane of the equator is a west-to-east plane, whose direction is opposite to that of the sun's apparent path. An observer to whom forward motion appears counter-clock- wise is in front of the plane, and looks at its face ; one to whom forward motion appears clockwise is back of the. plane. E.g. the plane of the equator faces northward, and points in the northern hemisphere are in front of it ; but the plane of the sun's apparent path faces southward. in plane trigonometry the reader always looks at the face of his plane, and to him, therefore, forward motion is always counter-clockwise motion. 26 GENERAL PROPERTIES OF PLANE ANGLES. [II, DIRECTED ANGLES. A plane angle lias been variously defined as " tlie opening between two lines," as "the inclination of one line to another," as "the difference of direction of two lines," and as "the por- tion of the plane between the two lines." The words " incli- nation " and "difference of direction" appear to define the magnitude of the angle rather than the angle itself ; but whichever of these definitions be used, it is manifest that an angle may be generated by swinging a line, in the plane of the angle, about the vertex, from one of its bounding lines to the other. The first position of the swinging line is the ini- tial line, and the last position is the terminal line, of the angle. E.g. the minute-hand of a clock generates a right angle every fifteen minutes, and four right angles in an hour. If the generating line swing forward, in the direction of the plane, it generates SbjMsitive angle; if it swing backward, it generates a negative angle. Since, in plane trigonometry, the reader always looks at the face of his plane, it follows that positive angles, are counter- clockwise angles, and negative angles are clockwise angles. T7ie angle of two lines is the smaller of the two angles which lie between their positive ends and reaches from tlie posi- tive end of the line first named to the positive end of the other. E.g. if the two lines a'a, b'b cross at 0, the angle of the two lines a'a, b'b is aob, and the angle of the two lines b'b, a'a is boa. The two bounding lines may be designated by single letters, the initial line being named first. E.g. if I, m stand for the two lines a'a, b'b, then Im stands for the angle aob and ml for the angle boa. §2.] DIEECTED PLANES AND ANGLES. 27 NORMALS. One line is normal to another if the first line make a posi- tive right angle with the other. E.g. in the figures below, ob is normal to oa, but not oa to ob. EQUAL AND CONGRUENT ANGLES. If two angles differ by one or more complete revolutions, they are congruent ; if, when placed one on the other, their initial lines coincide and their terminal lines coincide, they are equal or congruent. X E.g. in the figures above all the angles aob, whether positive or negative, are congruent, and the angles aob, a'ob' are equal, but not aob, b'oa'. 28 GENERAL PROPERTIES OF PLANE ANGLES. [II, The smallest aiigle, positive or negative, of a series of con- gruent angles is the primary angle ; and the primary angle is always meant if no other be indicated. It is always smaller than two right angles. QUESTIONS. 1. If a surveyor by mistake write n. 30° e. 12 chains, instead of N. 30° w. 12 chains, what is his error ? and what is the effect, in his map, on the position of every subsequent line and point ? 2. If the line a be normal to the line h, what angle does 1) make with a ? 3. Through what angle has the hour-hand of a clock swept from VI midnight to 12 noon ? the minute-hand ? the second- hand ? 4. If the moon revolve about the earth once in four weeks, what is its angular motion in a year ? in a day ? 5. How great is the angular motion of the earth upon its own axis in a day ? in an hour ? in a year ? So, how great is its angular motion in its orbit about the sun in a year ? in a day ? in a century ? 6. "What is the angle between a north wind and a north- east wind. ? a north wind and a southwest wind ? 7. If the current carry a chip due south, and the wind carry a feather due east, what is the angle between the arrows that show the directions of the motions of the chip and the feather ? 8. If two forces act upon a body, the one vertical and the other horizontal, what is the angle between them ? its sign ? 9. If three equal forces acting upon a body be parallel to the three sides of an equilateral triangle, what are the angles between them ? Discuss the eight possible cases. 10. If the two hands of a clock start together at noon, what is the angle between them at one o'clock ? at two ? at three ? at six ? at nine ? at twelve ? §2-] DIRECTED PLANES AKD ANGLES. 29 ADDITION OF CO-PLANAK ANGLES. Two or more co-planar angles are added by placing the ini- tial line of the second angle upon the terminal of the first, the initial line of the third angle upon the terminal of tlie second, and so on ; and the sum of all the angles so added is one of the congruent angles reaching from the first initial to the last "terminal line. This definition applies whether the vertices of the angles be at the same point or at different points. When a positive angle is added, the terminal line swings forward ; when a negative angle is added, it swings backward. E.g. in the figures below, ah + ia = (or one of the congruents of 0), ai + bc = ac, ab + hc + ca = 0, ab + bc + cd=ad, ab + bc + cd + da = 0. One plane angle is subtracted from another by adding the opposite of the first angle to the other, or by placing the initial line of the first angle upon that of the second ; the remainder is then the angle that reaches from the terminal line of the first angle to that of the other. If the sum of two angles be a positive right angle, either angle is the complement of the other ; and if their sum be two right angles, either angle is the supplement of the other. 30 GENERAL PROPERTIES OF PLANE ANGLES. [II, TH, QUESTIONS. 1. Show what angles must be added to the angles below to make the sums positive right angles, and so construct their complements. / I — \ (3^ Qj-*- C^T-^ ' Kj ' /7~^ r^! . fe TT'-K" 2. Show what angles must be added to these angles to make the sums two positive right angles, and so construct their supplements. 3. Show that the angle of two lines equals the angle of any normals to them. ^ 4. If a surveyor, in running round a field, turn at the cor- ners always to the left, what is the sum of the exterior angles of the field ? if he turn always to the right ? if he turn some- times to the right and sometimes to the left ? 5. If the wind shift from north to northeast, and then from northeast to southeast, through what angle has it shifted ? 1, §3. J PROJECTIONS. 31 § 3. PROJECTIONS. The orthogonal projection of a point upon a line is the foot of the perpendicular from the point to the line ; and, in this book, by projection is always meant orthogonal projection. The line on which the projection is' made is the line of pro- jection, and the perpendicular is the projectmg line. The projection of a segment of one directed line upon another directed line is the segment of the second line that reaches from the projection of the initial point of the given segment to the projection of its terminal point. The projec- tion is positive if it reach forward in the direction of the line of projection, and negative if it reach backward ; its sign may.be like or unlike that of the projected segment. E.g. the shadow of a post on a plane perpendicular to the sun's rays is an orthogonal projection of the post. Projections upon the same line are like projections. Theok. 1. Jf segments of one directed line le projected iipon another such line, the ratios of the projections to the segments are equal. Let I, m, be any two directed lines ; take ab, cd, ef- • • segments of m, and let a'b', c'd', e'f'- • • be their projections on I ; then will\A'B'/AB = C'D7CD = E'F7EF- • • For •.• the projecting lines aa', bb'- • • are ail parallel, and contrary segments of the same line have contrary pro- jections on another line, .•. the segments and their projections are proportional ; i.e. a'b'/ab =c'd7cd = e'f7ef=oa'/oa=:of7of- • •, both in magnitude and sign. q.e.d. 32 GENErtAL PROPERTIES OF PLANE ANGLES. [II, TH. In tlie first figure the segments AB, ef are positive, and so are their projections a'b', e'f', but CD, c'd' are negative, and all the ratios are positive. In the other figure ab, c'd', bf are positive, but a'b', 6p, e'f' are negative, and all the ratios are negative ; i.e. the ratios are positive if the primary angle of the two lines be acute, positive or negg^ive ; they are nega- tive if the primary angle be obtuse. " Cor. 1. Equal serjinents of one line have equal projections on another line, and ojJiiosite segments have opposite projections. Cor. 2. If on each of tioo directed lines equal seffmenfs of the other line be projected, the projections are equal. E.g. let I, m be any two lines, ab a segment of I, and CD, EF segments of m equal to ab, let a'b' be the projection of ab on tn and c'd', e'f' the projec- tions of CD, EF on I, then a'b', c'd', e'f' are equal in magnitude and sign. In the first figure the segments and their projections are all positive ; in the other figure the segments are negative, and their projections are positive. Cor. 3. // there be two equal angles, and if equal segments of the bounding lines be projected, each upon the other bound- ing line of its angle, these projections are equal. For tlie two figures may be placed one upon the other, and then cor. 3 becomes a case of cor. 2. Ij §3. J PKOJECTIONS. 33 QUESTIONS. 1. A line is 5 feet long and its projection on another line, a, is 4 feet long : how long is its projection on a normal to a ? Can the signs of the projections be found from the data ? 3. If a, h be two directed lines at right angles to each other, how long is that segment, whose projections on a, b are 5 feet and 12 feet ? -5 feet and -13 feet ? Can the sign of the segment be found from the data ? 3. Construct lines so that segments of one being projected on the other, the ratios of the projections to the segments shall be 1, |, |, i, i, ^, O ; -|, —J, -f, -1, in turn. 4. A pole ten feet long points northward and makes an angle of 45° with the level ground : how long is its shadow, if the sun be directly overhead ? So, how long is its shadow on a north-and-south wall, at sunrise, if the sun rise due east ? Of these two shadows, which is the longer ? So, which is the longer if the inclination be 60° ? From what point of view would the pole appear to be ver- tical ? from what point horizontal ? 5. Describe an isosceles triangle by walking due east 100 yards, then northwest 70.7 yards, then southwest 70.7 yards, thus giving direction to the sides. Project the two sides of this triangle upon the base : what relation have these two projections ? So, project these two sides upon the bisector of the vertical angle : what relation have the two projections now ? '' 6. In an equilateral triangle, whose sides are directed by walking about it and turning to the left at the vertices, how do the projections of the sides upon the base compare in length ? in sign ? So> the projections upon a normal to the base ? 7. Can a segment of a line be so projected upon another "lirie, that the projection is longer than the segment itself ? 8. Taking note of signs, what is the range of magnitude for the ratio projection/segment ? segment/projection ? 3 34 GENERAL PROPERTIES OF PLANE ANGLES. [", § 4. TEIGQN0METRTC RATIOS. If a segment of the terminal line of anjangle be projected upon the initial line and upon a normal to the initial line, the first project^n* may be called the major projection of the segment, tne other the hithiov projection, and the ratios of these two projections to ihi, segment and to each other are Burned as below : ^ 'minor projeoVon/segraerft, the sine of the angle, major projection/segmeiit, the cosine, minor projection/major projection, the tangent, major projection/minor projection, the cotangent, segment/major projection, the secant, segment/minor projection, the cosecant. These definitions apply to all angles whatevej»their magni- tudes or signs, and they include as a special case the defini- tions given on page 3. a«.'' X X Y, X ^ 1 \r X y ^>v ■n > E.g. in the figures above, let xop be any angle a ; let ot be normal to the initial line ox, r any segment of the ter- minal line OP, X, y the major and minor projections of r ; then sin a — y/r, cos a = x/r, tan a = y/x, esc a — r/y, sec a — r/x, cot a = x/y. §4. J TRIGONOMETRIC RATIOS. 35 The segment r may be taken positive or negative ; for if the segment be reversed both projections are reversed, and the ratios are unchanged. Two other functions in common use are the versed sine and coversed sine ; they are defined by the equations vers or == 1 — cos a, covers « = 1 — sin a. QUESTIONS. 1. How do the major and minor projections of the seg- ment of a line compare in length with the segment itself ? how with each other ? 2. Can the sine of an angle be larger than 1 ? as large as 1 ? smaller than 1 ? can the sine be naught ? the cosine ? 3. Can the tangent of an angle be naught ? can it be smaller than 1 ? as large as 1 ? larger than 1 ? how large may the tangent be ? the cotangent ? the secant ? the cosecant ? 4. What relations as to sign have a segment and its pro- jections ? Draw figures in which : ' all three are positive ; all three negative ; the segment and major projection are positive and the minor projection negative ; the segment is negative and both projections positive. 5. If two lines be parallel and like directed, what is their angle.'' How long is the major projection of a segment of one of these lines as to the other ? the minor projection ? What are the ratios of this angle ? So, if two parallel lines have opposite directions ? So, if the terminal line be normal to the initial line ? So, if the initial line be normal to the terminal line ? 6. Construct the angles ^b, — |-k, |r, -fK, |k, — |e • ■ • and find their ratios. [K=a right angle. Which of these angles have the same sines ? the same cosines ? the same tangents? the same secants ? So, for the angles ^R, - Jr, |r, - fR, | R, - ^R, f R, - f R. 7. Construct sin-'J,-f, |, 1, 0; cos"' |, ± i, - 1 ; tan-'i, I, 0,-1, oo; cot-'i,-i,±l- 36 GENERAL PEOPERTIES OF PLANE ANGLES. [II, TH. ANGLES IN THE POOR QUARTERS. If there be two lines such that the second line is normal to the first, tlie plane of these lines is divided into four quar- ters. The first quarter lies between the positive ends of the two lines, the second quarter between the positive end of the normal and the negative end of the first line, the third quar- ter between their negative ends, the fourth quarter between the negative end of the normal and the positive end of the first line. An angle is an angle in the first quarter, in the second quar- ter, in the third quarter, or in the fourth quarter, according as its terminal line lies in the first, second, third, or fourth quarter, counting from the initial line. It is therefore an angle in the first quarter if its primary congruent angle be a positive acute angle ; in the second quarter, if a positive obtuse angle ; in the third quarter, if a negative obtuse angle ; in the fourth quarter, if a negative acute angle. E.g. of the figures above, the first angle and the eighth are angles in the first quarter, the second and seventh are angles in the second quarter, the third and sixth are angles in the third quarter, the fourth and fifth are angles in the fourth quarter. 2. §4.J TKIGONOMETEIC EATIOS. 37 POSITIVE AND KEGATIVB KATIOS. ^Theor. 2. The trigonometric ratios of an angle in the first qua'^r are all positive. Th^ine and cosecant of an angle in the second quarter are . ^osi^L- the cosine, secant, tangent, cotangent are negative. , The m^ent and cotangent of an angle in the third quarter' are positim ; the sine, cosecant, cosine, secant are negative. The cosine and secant of an angle in the fourth quarter are positive; the sm^, cosecant, tangent, cotangent are negative. /For if r be taken positive, and x, y be the major and minor projections of r ; XH 1 Va~ + V t/ 00 X' ai' ■ X y' \ \ f then'/ xoPj top' are equal angles, .•. the projection of r on ox equals that of r' on ot; and ■.• pot, p'ox' are equal angles, .'. the projection of r on ot equals that of r' on ox', and is the opposite of the projection of r' on ox ; i.e. r=r', x=y', y=~x'; .'. sin (r+ ff), =y' /r' — x/r, = CQ% a ; and so for the rest. 44 GENERAL PROPERTIES OF PLANE ANGLES. [II, THS. THE RATIOS OF THE SUPPLEMENT OF AN ANGLE. Theor. 8. If a he any plane angle, then : sin sup a = sin a, cos sup a— — cos a, tan sup a= —tan a, cot sup a= —cot a, sec sup a— —sec a, esc sup a=csc a. Let xop be any plane angle a, draw OT normal to ox, and OX' opposite to OX ; draw op', making the angle p'ox' eqn^l to a ; X' 00 ,Y x< X 1 y >^i v' ^\^ ^ P V x' X X' X X' x' o' x X X ^ Si 1 J y ^^ r\^ y , y^ ^' \ p then". ■ xop' + p'ox' = 2r, and xop = p'ox', [constr. .•. XOP, xop' are supplementary angles. On op, op', take equal segments r, r', and let their major and minor projections, i.e. their projections on ox, ot, be ^, y, a;', «/' ; then-.' xop, p'ox' are equal angles, .•. the projection of r on ox equals that of r' on ox', and is the opposite of the projection of r' on ox ; and •.• POY, top' are equal angles, .-. the projections of r, r' on OY are equal ; i.e. r = r', x= —x', — y=y' ; .'. sin sup a,=y'/r' = y/r, = sm a, cos sup a, = x' /r' — — x/r, = — cos a ; and so for the rest. q.e.d. 8,9,§6.} EATIOS OF RELATED ANGLES. 45 THE RATIOS OF 2R + a. Theor. 9. If a be anyplane angle, and e a right angle, tlieti : sin{2n + a)= —sin a, tan{2B, + a) — tan a, cos(2R + a)~-cosa, '>^-=- "^ cot(2n + a) — cota, few- = "Ce^ sec (3e + a)—— sec a, esc (2r + a) = — esc a. eau^ ^ - ^ Let xop be any plane angle a ; draw OY normal to ox, op' opposite to OP ; Y • ^ V ~^. r — V' 00 X ^ ■ i ^ Y y X' ^<>. >\ s. ^^^j ^-P- ■^J" >=^r >^: X' -Y y' then'.' pop' = 2r, .•. xop' = 2r + «. On op, op' take equal segments r, r', and let their major and minor projections, i.e. their projections on ox, oy, be then'." r, r' are opposite segments of op, .". their major projections are opposite, and bo are their ^ minor projections ; i.e. r=r', x——x', y=—y'; .-. sin (2r + a), =y' /r' = —y/r, = —sin a, cos (2R + a), =x'/r'= —x/r,= -cos a-; and so for the rest. Q.E.D. 46 GENERAL PROPERTIES OF PLAKE ANGLES. [II, TH. QUESTIONS. / 1. Given the ratios of ^n: by aid of theors. 5-9, find the ratios of |r, |r, |r- • •, and of — |-R, — fR, —^^i — |k • • •. 2. Given the ratios of -Jr: find those of fR, ^R, |-B • • ■, and of — -Jr, — |R, — fR, — fR • • •• 3. Given the ratios of R: find those of 0, 3r, 3r, 4r ■ • •, and of — R, — 3r, -3r, -4r • • •. 4. Find the ratios of n + a, as the complement of — or. 5. Find the ratios of 2u + a, as the supplement of —a. 6. Find the ratios of « — R, as the opposite of co-a. 7. Find the ratios of SR+ff, and of Sn — ex. 8. Find the ratios of 2r — a, as the complement of a — K. 9. Find the ratios of 4E + a', as supplement of —{2R + a). 10. Given cos a = -J : find sin~'^, sin^' — ^. 11. Given esc a = 2 : find sec~^3, sec'— 3. 13. What angles have the same sine as o' ? the same cosine? the same tangent ? the same secant ? the same cosecant ? In ratios of positive angles less than e, express the values of : 13. sin 135°, 335°, "535°, "7^5°, V^. "¥«, ¥e» "ff»- 14. cos 335°, 435°, -035°, -835°, ^R, -^^R, AjIr, -ffR. 15. tan 335°, 535°, -735°, "935°, ^e^ -^r, ^^-r, -^^r. 16. cot 435°, 635°, "835°, "1035°, ^Yn, "^/r, ^r, -^|r. 17. sec 535°, 735°, "935°, "1135°, ^'■-R, "Sy^R, A^E, -^|k, 18. esc 635°, 835°, "1035°, -1?35°, \tR, -s^r, ^r, -|fR. In ratios of positive angles not greater than JE, express the values of : ' 19. sin 50°, 150°, "^50°, "350°, ^\n, -4r. 30. cos 60°, 100°, -260°, "360°, ^E, -^^r. 31. tan 70°, 170°, -370°, -370°, ^'^R, '^R. 33. cot 80°, 180°, -380°, "380°, ^\r, -6e. 33. sec 90°, 190°, -290°, -390°, Un, -y^R. 34. cscl00°, 300*, -300°, "400°, \\r, '^r. 10, § 7.] PROJECTION OF A BROKEN LINE. 47 §7. PROJECTION OP A BROKEN LINE. The projection of a broken line upon a straight line is the sum of the projections upon it of the segments that consti- tute the broken line, and it is identical with the like projec- tiojL-of the single segment that reaches, from the initial to the terminal point of the broken line. * I.Y A ____ - ^^ , ^ \ E' ~~ir^ "N^ "^ /-' \ r" - ¥ A' / ,-<•'. t>^^ X T>" / >- I c F' !^^ ■p" J i-- I 3" A" ::j i iT 1 -^ p Theor. 10. The major projection of a segment of a line is equal to the product of the segment ly the cosine of the angle the line makes with the line of projection ; and the minor pro- jection is equal to the product of the segment iy the sine of this ^ angle. [df. sin^, cosine. Cor. The major projection of a broken line is the sum of the products of the segments each by the gosine of the angle its line makes tvith the line of projection, and the minor projec- tion is the sum of their products by the pines of these angles. E.g. in the figure above, let the br^en line abcdef be formed by the segments ab,,-b6, cd ■ • •, of the lines AL, BM, CK • • ■ , lei a, 13, y • ■ •(}> stand for the angles ox-al, ox-bm, ox-cisr • • • OX-AF. then maj-proj abcdef = a'b' + b'o' + c'd'- • • = AB-Cosa; + BC cos/S + CDcos;/- ■ • = a'f' = AF cos , min-proiABCDBF = A"B" + B"c" + c"D"- • • = AB sin a + BC sin /3 + CD sin 7 • ■ ■ = a"f" = af sin«5. and 48 GENERAL PROPERTIES OF PLANE ANGLES. [II, TH. § 8. RATIOS OF THE SUM, AND OP THE DIFFERENCE, OF TWO ANGLES. Theoh. 11. If a, 13 he any two plane angles, then : sin{a + fi) = s in acos fi + cos a sin A ( / >v■^^ siH(a — /3)=sinacosp — cosasin/3, cos {a + /3) — cos acos /3 — sin a sin ft, cos {a — fi) = cos acos ft + sin a sin ft. Foi% let xop, POQ be any two plane angles a, ft," so placed that their vertices coincide, and the terminal line, OP, of a, is the initial line of ft ; then X0Q=a + yS. On OQ take any segment oc, and draw cii normal to OP at B ; then'." the major projection of OC, as to ox, equals the like projection of the broken line OBC, i. e. maj-proj 0(3 = maj-proj ob + maj -pro j bo, [df . .•. maj-proj oc/oc = maj-proj OB/oc-f maj-proj bc/oc = maj-proj ob/ob • ob/oc + maj-proj bc/bc • bc/oc. 11, §8. J THE SUM AND DIFFERENCE OF TWO ANGLES. 49 But maj-proj oc/oc=cosxoQ = cos(a + /S), [df. maj-pro] ob/ob = cosxop = cos a ; and •.• OB, BC are the major and minor projections of oc, as to OP, . •. ob/oc = cos POQ = cos yS, Bc/oc = sin POQ = sin yS ; and •.•angle ox-BE = xop-|-PBR = ar+R, .•. maj-proj bc/bc = cos OX-BR = cos (a + r)= — sina,[th.7. .■. cos (ar + /?) = cosa;cosy5 — sinffsinjS. q.e.D. And^.^ a, (i may be any plane angles positive, or negative, and a — /?=«+( — /3) whatever thesignor magnitude of /S, .". cos (a ^ /?) = cos a cos ( — yS) — sin a sin ( — /S) [df. = c'os « cos yS + sin or sin yff. Q.e.d. [theor.5. So, •.• the minor projection of oc equals the like projection of the broken line OBC, .•. min-proj oc/oc = min-proj ob/oc + min-proj Bc/oc = min-proj ob/ob • ob/oc -f min-proj bc/bc • bc/oc, .•, sin (a -I- yS) = sin a. cos yS + sin (r + a) wa.fi = sina'COSyS-fcosasinyS, Q.E.D. and sin (or — /S) = sin«cos(— y8)-|-cosasin( — y6), = sin pr cos yS — cos or sin yS. q.e. D. Cor. 11 tan{a-{-^) = {tana + tan ^)/{l~tanatan^), I I tan{a—fi) — {iana — tanp)/{i + tfmatan/S). |- For tan {a + /3) = sin {a + /J)/cos (a + yS) [theor.4. = (sin ar cos yS + cos a sin yS)/(cos a cos yS — sin or sin yS). Divide both terms of this fraction by cos a cos yff ; then tan ( or + yS) = (tan a + tan /?)/( 1 - tan a tan y(3) ; and so for tan (a—/3). q.e.d. Cor. 2.\ sin {a + fi)+ sin {a—/3) — 2 sin a cos /S, sin {a + /3)— sin {a— fi) = 2 cos a sin /?, cos {a + fi)+cos (a— P) =2 cos a cos P, cos {a+P) — cos {a — /S)— —2 sin a sin yS. 50 GENERAL PROPERTIES OF PLANE ANGLES. [II, TH. CONVERSION FORMULA. TflEOR. 12. If a, file any two plane angles, then : sin a + sin /? = 2 sin^ (a + /3) cos 1^ {a — /3), sin a — sinfi= 2 cos J (a + /S) sin J (fl" — /?), cos a + cos /3 = 2cosi{a + /3)cos^{a—/3), cos a-cos /3= —2 sin ^ {a + fi) sin ^ (a — P). For let y, 6 be two plane angles such that r = K« + -^) and d = i{a-/3), then, ■y + S = a, and y — 6 = fi, and ".• sin(7 + 5)+sin (^ — (y) = 2sin/cos (sin a + sin /S)/(cos a — cos )8), ^ (sina — sin/J)/(cosa — cosyS), (sin a + sin /S)/(sin a — sin /S), (cos a + cos /3)/{cos a — cos /3). 18. Givena = 60°,jS = 45°: find tan 52° 30', tan 7° 30'. [th.l2. So, given a=45°,/?=30°: find tan37°30',tan7°30'. 19. Given sin 15° = .25882, sin45°= VJ: find cos 60°, cos30°. 30. Given cos 75° = .25883: find sin 30°. 21. Given cosl7°=.9563, sin23°=.3907: find tan7°,W 40°, sin 20°, cos 3° 30'. ' 52 GENERAL PEOPERTIES OF PLANE ANGLES. [II, THS. §9. RATIOS OP DOUBLE ANGLES AND OF HALF ANGLES. Theor. 13. If ale any plane angle, then : sin 2a:=2sir^tco^i = 2 tan a/(l + tanker), cos 2« = cos'a — sin'a — 1—2 sin' a -{1- tan'a)/{l + tan'a), tan2a = 2 tan a/{\ — tan'a). For sin2a = sin (a + «) = sin a cos a + cos or sin ^i{l- cos a)M^ cos ia— /\/i{l + cos a), ^ tan ia = sin a/{l + cos a) = (1 — cos a) /sin a = V [(1 — cos a)/(l + cos a)]. For •.• 3 sin' |a = 1 - cos a, [theor. 13, cor, .•. sin^a^ Vi(l — cosa). Q.e.d. So, ■.• 2 cos' ^a= 1 + cos a, [theor. 13, cor, .•; cos^a= Vi(H-cosa). q.e.d. So, ".■ tan-Ja=sin^«/cosiar, [theor. 4. .•. taiiia=3 sin-Ja cos^ff/2 cos'^a = sina'/(l + cosa) ; q.e.d. [theor. 13, cor. and tan ia — 2 iin' ia/3 sin ^a cos ^a =(1 — cos a) /sin or. q.e.d. So, tan^«= Vi(l— cosa')/\/Ml + cosa') = \/[(l— cos«)/(l+cosa)]. Q.E.D, QUESTIONS. y 1. From the known value of cos 30°, find the ratios of 15° from oos 15° find the ratios of 7°_aa'; from cos 7° 30' find tht ratios of 3° 45', and so on, each correct to four decimal places. If a + b + c = 2e, then : ^ 2. sin2A + sinj|2]^.+ sin2c= 4 sin A sinB sine. 3. sin A + sin B H-'Bin c "= 4 cos ^a. cos ^b cos ^c. -4.. cosA +cosB + cosc = 4sin^A sin^B sin^c + 1. Prove the identities : 7-5. esc 2a' + cot 2a = cot a; cosa = cos*^a — sin*^ar. 76., tana + coto'=2qsc2a; tan a — cot a= — 2cot2a. 7. tan (^B - J-a) + cSt (^K - ^a) = 2 sec a. _/■ 8. (cos a + sin «)/(cos a — sin a) = tjtn 2« + sec 2ar. ^ 9. tan''(iR+Ja')=(sec« + tana)/(8eca-tana). 54 GEITERAL PROPERTIES OF PLAKB ANGLES. [II, TH. §10. RATIOS OP THE SUM OF THREE OR MORE ANGLES, AND OP MULTIPLE ANGLES. Theor. 15. If a, ^, y he any three plane angles, then : sin {a + /3 + y)= sin a cos ft cosy + sin fi cos y cos a + sin y cos a cos 'ft — sin a sin ft sin y = cos a cos ft cos y {tan a + tan ft + tan y — tan a tan ft tany). cos {a + ft + y) = cos a cos ft cos y — cos a sin ft sin y — cos ft sin y sin a — cosy sin a sin ft = cosacosft cosy {1 — tan ft tany — tany tana — tana tan ft). . . _ tan a + tan /? + tan y — tan a tan ft tan y \ P '' ~ \ — tan ft tany — tany tana — tana tan ft' Prove by expanding Bm{a + ft + y), cos {« + /? + /). [th. 11. OOR. 1. If a, ft, y, ■ ■ • be any plane angles, then : sin {a + ft + y--- )/cos a cos ft cos y- ■ ■ = stana — 2tana tan ft tany+ • • •, cos{^a + ft + y ■ •)/cos a cos ft cosy- ■ ■ = 1—2 tan a tan ft -\- 2 tan a tan ft tany tanS • • •, wherein 2 tan a stands for the sum of the tangents of all the angles, 2tan«tanyS for the sum of -their products taken two and two, and so on. Prove by induction. Cor. 2. sin 3a = 3 sin a cos'a — sin'a = 3 sin a — i sin'a, cos 3a = cos' a — 3 cos a siifa = — 3 cos a + 4 cos^a, sin 4a = 4 sin a cos" a — 4 sin" a cos a = 4 sin a cos a — 8 i cos 4a = cos*a — G sin'i = 1-8 cos'a + 8 cos^a, sinna^nsin a cos"'' a — 1« (w — 1) (w — 3) sin'a cos^~*a + • • • , cos na = cos°a—^n{n — l) sin'a cos''''a+ • • •, wherein the coefficients in the value of sin na are those of the second, fourth • • • terms of the expansion of (a + b)" ; and those in the value of cos na are the first, third • • • terms of the same expansion. Prove by induction. s a — 8 sii^acos a, nn'ix co^a%^i0u 15, § 10.] THREE OR MORE AlTGLESj MULTIPLE AKGLES. 55 QUESTIONS. 1. If a, ft, Y be any three plane angles, then : -sin (« + /?+ y)+sin (-« + /? + J/) +sin («-/? + ;/) + sin (« + /?— y) = 4 sin a sin /J sin;/, cos (or + /3 + >/) + cos ( — « + yS + y ) + cos (a — /S+ x) + cos (a + /S — 7) = 4 cos a cos yS cos y. If a + b + c = 2k, then : 3. tan |-A tan ^B + tan j-b tan ^c + tan ^0 tan jA = 1. 3. cot A + cot B + cot C = cot A COt B COt C + CSC A CSC B CSC 0. 4. tan A 4- tan B + tan c = tan A tans tan c. 5. tan 3 a = (sin « + sin 3 « + sin 5 a)/(cos a + cos3 a + cos 5 a). 6. If a, ft be any two plane angles, and n any integer, then : {sin a + sin (a; + /J) + sin (a + 3/S) + sin (a + 3y3) + ■ •' • + sin(5)— sin (a— ^/J). 7. Prom the results of ex. 6, p'rove that : \n any pos. integer. §ina + sin(a + 4E/w)+ • • • +sin [a' + 4B(w-l)/w]=0, cosa + cos(a + 4K/w)+ • • • +cos [a + 4K(w-l)/w] = 0. 8. In the results of ex. 7, take w = 3, and prove that : sin"a + sin 60° -a - sin 60° + a = 0, cos a- — cos 60° - a - cos 60° + a = 0. 9. In the results of ex. 7, take re = 5, and prove that : sin« + sin73° + a + sin36°-a-sin36° + a:-sin73°-af=0. cosar + cos73° + «-cos36°-a-cos36° + a' + cos72°-a'=0. 10. Show that when ?j = 3 the formula found in ex. 7 veri- fies the sines and cosines of all angles in the first quarter, if to a be given values from 0° to 30°. 11. In the results of ex. 7, take « = 9, 15, 35, 27, 45, in turn, and thence find other formulas of verification. 56 GENERAL PROPERTIES OF PLANE ANGLES. [II, §11. INVERSE FUNCTIONS. If a be a number, and a an angle such that a = sin a, this relation is expressed by the equation « = sin"*a, which is read a ii i\i& anti-sine oi a. So, the equation fi — cos'^b means that yS is an angle whose cosine is h, and y — i&w'^c, that y is an angle whose tangent is c. It is to be noted that, while the equations a = sina', S = cos^, c = tan y, give a, i, c single values for single values of a, ft, y, the equations « = sin"'a, ^=cos"'5, j/ = tan~*c g\YQa,ft,y many values for single values of a, h, c : for a, 2R — a, and all the congruents of these angles have the same sine ; /3, —ft, and all their congruents have the same cosine ; and y, 2r + y, and all their congruents have the same tangent. H.g. sin-i = 30°, 150°, 390°, 510° • • • -210°, -330° • • •. So, cos-Vi=45°, -45°, 315°, -315° •••. So, tan- V3 = 60°, 340°, 420°, 600° ■ ■ • - 1'^0°, - 300°, • • • . Many of the theorems of trigonometry may be expressed in terms of inverse functions ; and sometimes with advantage. Kg. if X, y, z stand for the sines of the angles a, ft, y, then si n (a ± /J) = sin a cos ft ± sin ft cos a, may be written 8in-'a;±sin-'y = sin-'[a;v'(l-y')+yV(l-a!')]- So, sin (a + /?+j/) = sina cos/J cosy + sinyS cosy cosa + sin y cos a cos y3 — sin a sin ft sin y, may be written sin-'a; + sin-'2/ + sin-l« = sin-* [x\J {1—y^-z^ + y^ z') +y^/{l-z''-x' + z'x')+z'^{l-x''-y' + x'y')-xyz]. So, sin 2a = 2 sin « cos a, may be written 2 8in-'a; = sin-'[2a;\/(l-a^)]. So, sin-^a'= Vi (1 —cos a) may be written isin-'a; = sin-Vi[l-V'(l-ar')]. These relations are always true : sin-'a:=csc-'l/x, co8-'a;=sec-*l/a;, tan-*a; = cot-'l/a;, sin-'a; + cos-'a;=B, 8ec-'a; + csc-'a; = R, tan-*2; + cot-'a;=R. §11.] INVERSE rUNCTIONS. 57 QUESTIOKS. Translate these formulae into inverse forms : 1. cos (a'±/3) = cosar cosySTsina: sinyS. 2. tan (a ± yS) = (tan a ± tan /?)/(! =f tan a tan /?). 3. cos 2a = CQs^a — sin'a = 2 cos'a — 1 = 1 — 3 sin'a. 4. tan3a=2tana/(l — tan'a). 5. cos ■Ja= Vi(l + cosa). 6. tan ^a = sin 0-/(1+ cos a) = (1 — cos a)/sinar = V [(1 — cos a)/(l + cos a)]. 7. cos {a + ^ + y)—cosa cos/3 cosy — sin« sin/S cosy — sin /5 sin y cos a — sin y sin a cos /?. 8. tan{a + /3 + y) _ tan (* + tan /? + tan y — tan a tan /3 tan y _ 1 — tan a tan/3 — tan/S tan y— tan y tan« 9. cos 3a = cos'or — 3 cos a sin^a. 10. tan 3a =: (3 tan a - tan»ar)/(l - 3 taii'a) . Show that 11. sin-*|+sin-'|^=R ; cos^'y^+cos"' J-|=K ; tan-'f + tan-'^ = K. 12. sin (3 sin ^ ' a;) = 3 « — 4 a:". [x any proper fraction. cos (3 cos "'.'!;)= — 3a; + 4a;'. tan (3 tan - ' a;) = (3 a; — a;') : ( 1 — 3 a") . [x any number. 13. tan-'i + tan-'J = iR. [Euler. 14. tan-*^ + tan-'^ + tan-'^=jR. [Dase. 15. 2tan-'^+tan-'i=iR. [Hutton. 16. 4tan-'|-tan-'T5^=iR. [Machin. 17. 4tan-'^-tan-'T5ij + tan-'^ = iR. [Rutherford. 18. 5tan-'| + 2tan-'^ = iR. [Euler. Solve the equations : 19. sin-'3a; + sin-'4a;=R. 20. tan-'2a; + tan-'3a; = iB. 58 GENERAL PEOPEETIES OF PLAKE ANGLES. [11, §12. GRAPHIC REPRESENTATION OP TRIGONOMETRIC RATIOS. Let xp be any arc with centre o and radius ox, and let py be tlie arc complementary to xp ; through p, X draw ap, xt normal to ox, and through Y, p draw BY, pt' normal to op, with t on op and t' on oy ; then AP, OA, XT, OT are the sine, cosine, tangent, and secant of the arc xp ; and BY, OB, pt', OT' are the sine, cosine, tangent, and secant of the complementary arc py, and the cosine, sine, cotan- gent, and cosecant of the arc xp. These lines are called line-functions of arcs as distinguished from the ratio-functions of angles ; and if they be divided by the radius, the ratios so found are the ratio-functions hereto- fore defined. With arcs of the same radius the ratios of their line-functions are equal to the ratios of the like ratio-fune- tions of their angles. §13. J GKAPHIC REPRESENTATION OF RATIOS. 59 CURVE OF SINES. Let OX be the radius of a circle, and divide the circumference into any convenient parts at Pi, Pj- • • ; draw AjPi, AjPj- ■ • normal to OX, and sines of the arcs xp„ xPj---; upon ox lay off xb„ xb,- • • equal to the arcs xPi, xPj- • •; at Bi, Bj- • • erect perpendiculars to ox and take Cj, Cj- • • such that BiCi = AiP„ BjCg = AjPj • • • ; through c„c, • • • draw a smooth curve ; it is the curve of sines, and the following relations are manifest : The sine is for the angle ; is nearly as long as the arc for a small angle ; increases more and more slowly ; is equal to the radius, and its ratio is + 1, its maximum, for a right angle ; decreases, at first slowly, but faster and faster as the angle approaches two right angles ; is for two right angles ; decreases from to the opposite of the radius, and its ratio is — 1, its minimum, as the angle grows from two right angles to three ; increases to as the angle grows from three right angles to four ; is again at the end of the first revolution; and so on.- The sine has all values between the radius and its opposite. If the revolution be continuous, the values of the sine are periodic, every successive revolution indicating a new cycle and a new wave in the curve. The sines are equal for pairs of angles symmetric about the normal at o. 60 GENEEAL PROPERTIES OF PLANE ANGLES. [II, OTHER TRIGONOMETRIC CURVES. The tangent is for the angle ; increases through the first quarter to +oo ; leaps to — oo ; increases through the second quarter to ; increases through the third quarter to + oo ; leaps to — oo ; increases through the fourth quarter to ; and so on. The tangent has all values from — oo to + cx> . Tangents are equal for pairs of angles that differ by a half revolution. The secant is equal to the radius, and its ratio is +1 for the angle ; increases through the first quarter to + oo ; leaps to — co ; increases through the second quarter to the opposite of the radius, and its ratio is — 1 ; decreases through the third quarter to — oo ; leaps to +w ; decreases through the fourth quarter to the value at the begin- ning ; and so on. The secant has no value smaller than the radius. Secants are equal for pairs of angles symmetric as to the initial line. The cosine, cotangent, cosecant have the same bounds as the sine, tangent, secant ; they go through like changes and are represented by like curves ; but they begin, for the angle 0, with different values, viz., the radius, oo, oo. QUESTIONS. 1. Show directly from the definitions what are the largest and what the smallest values that each function may have, and state for what angles tbe several functions take these values. So, what are the greatest and what the least values. 2. Draw the curve of tangents, curve of secants, curve of tosines, curve of cotangents, and curve of cosecants. 3. Trace the changes, when a increases from to iR, in : sin a + cos Of, tan cos JA = V [s {s — a) /be] , tan^A =^[{s-b){s-c)/s\{s-a)]. For '.' a, A are supplementary angles, .*. -J-or, ^A are complementary angles, and cos a = — cos a, cob ^a = sin |a, sinJ«=cos^A, cot Ja ='tan ^a. 66 PLANE TRIANGLES. [Ill, TH. LAW OF SINES. Theor. 3. If a,i, c he the sides of a plane triangle, and a, /3, y he the angles he, ca, ah, then : a /sin a — h/sin ^ = c/sin y. For, draw any normal to a a.nd project the closed broken line a + h-\-c on this normal ; \ 1 Y / ) i y \ \ b/ \-^, /"-■* N a C\ X then ■.• Za5= 7, lac- -13, and the projection of a on its normal is naught, .-. + 5 sin 7 + c sin ( - /J) = 0, .'. J-sin y = c-&m p, .: h/sm fi = c/smy. So, c/smy = a/sina, [project a + 5 + c on a normal to 5. and a/sma = h/smp. [project a + & + c on a normal toe. Cor. 1. {a + h)/c=cos^{a-fi)/cos^y. {a - b)/c - — sin i{a — /3)/sin \y. For •.• a/c = sin a/sin y, h/c = sin ^/sin y, [above. .-. (a + 5)/c=(sin a + sin/?)/sin j/ = 2sin^(a' + /3)cos^(a-/3)/2sin^XCOS^y = COS ^{a- y3)/cos ^y. [^(a + P) = snp^^. So, (a-J)/c = (sin a-sinyS)/sinx = 2cos^(« + /3)sini(«-y3J/2sin|-;/cos|->/ = -sinJ(a-/3)/sin^>'. Cor. 3, {a-h)/{a + h)= - tan ^(a - p)/ian I y. 2, §2.] GENEKAL PROPERTIES OF PLANE TRIANGLES. 67 Cor. 3. If a, I, c, a, /?, y be the parts of .an ideal triangle, and if K, b, c be the interior angles of the triangle, then : a /sin A. = b/sin^ — c/sina, (a + i) lo - cos f Ca - v,\/&hi,lj;i. {a — b)/c = sin i(A — b)/cos \c, (a - b) /{a + b\ — tan ^ (a — b) /cot \c. For •.• a, A are supplementary angles, and so are fi, B and y, c, .■.i{a-p)=-i{A-B), and iy, -Jc are complementary, .". sin a = sinA, sin/J = sinB, sin ;/ = sine, cos ^ ;)/ = sin ^C, sin|^ = cos|c, cot -J ;/ = tan |c, tan-^(ar — /?)= — tan J(a — b). questions. 1. What is the value, in terms of a, b, c, of : cos /J, cos ^13, sin ^/?, cot ^13 ? cosB, sin^B, cos^B, tan ^b ? cos y, cos \y, sin ^y, cot ^y ? cose, sin-Jc, cos^c, tan^c?. 2. What signs are to be given to the radicals in theor. 1, cors. 1, 2, 3j in case of an ideal triangle ? 3. Show that the values of cos ^a, sin ^a, • ■ • are impos- sible if one side be greater than the sum of the other two. In any plane triangle abc : 4. cos ^A cos |B/sin ^c = s/c. 5. cos^Asin^B/cos-Jc=i.(s — a)/c. 6. sin ^A cos ^b/cos ^c = (s — b)/c. ' 7. sin jA sin ^B/sin ^o = (s — c)/c. 8. acosB + 5cos A=c; acosB — Scos A=(a° — J')/c. 9. a cos E cos G + b cos c cos a +,c cos A cos b != a sin B sin c = J sin c sin a := c sin a sin b. 10. acosA + JcosB + c'cosa=2asinB sinc= ■ ■ • • C8 PLANE TRIANGLES. [Ill, PR. §3. SOLUTION OF PLANE TRIANGLES. PrOB. 1. To SOLVE AN OBLIQUE PLANE TRIANGLE. Apply such of the formulcB oftheors. 1, 2, and their corollaries, as serve to express the values of the unknown parts in terms of the known parts. Check : Form an equation involving the three computed parts ; hut use no part in the same way in the solution and the check. Cases of the general triangle appear in discussing the rela- tions of coplanar forces in mechanics, in particular when one of the forces is the resultant of the other two ; and in the solu- tion of such triangles, the general formulae given above may be used. For ordinary purposes the ideal triangle alone is suf- ficient, and in its solution it is convenient and in accord with usage to ignore the exterior angles a, ft, y, and to use the interior angles a, b, c. The rules may then take the form shown below. There are four cases. (a) Given a, I, c, the three sides : then cosA = (^l''^-c^-a')/2*c, co&^={c' + a^-V)/2ca, cos c = (a° + J" - c')/3 ab ; check : A + B + c = 2 R. These formulae are used if a, b, c be expressed in numbers so small that the squares, sums, and quotients are easily com- puted ; and the angles are then found from their natural co- sines. If a, h, c be expressed in large nu mbers use the formulae shown below, which are specially adapted to logarithmic work. tan|-A = V [(s - a) (s - S) (s - c)/s]/(s - a), tan j^B^V [(«-«) {«-*) («-c)A]/(s-5), tan ^c^V [(«-«) («-^) («-c)/s]/(s-c). For tan^A = V [(s - 5) (s - c)/s (s - a)] = ^l\_{s-b){s-c){s-a)/s{s-af'\ -\l\{s-a){s-b)(s-c)/s\/{s-a), and so for tan^^B, tan^c. 1, §3.J SOLUTION OF PLANE TRIANGLES. 69 The special advantage of these formulae lies in this, that the radical part is the same for each of the three half angles. E.g. Let a, h, c be 3, 5, 7 ; then, using the upper formulae, the work may take this form : cos A = (25 + 49 - 9)/70 = 65/70 = . 9386, and A = 31° 47' cosB = (49 + 9-35)/42= 33/43= .7857, and b= 38° 13' cosc=(9 + 35-49)/30=-15/30=-.5000, and c = 120° Chech: a + b + c = 180°. 180° So, let a, h, the 357, 573, 735 ; then, using the lower formulee, the work may take this form : ,357 5 = 833.5 log, 3.9204- 573 s-a = 475.5 3.6773 + 735 s-5=359.5 2.4141 + 3) 1665 s-c= 97.5 1.9890 + 833.5 Chech: 1665. 3 )4.1599 3.0800 2.0800 3.0800 3.0800 - 2.6772 - 2.4141 - 1.9890 log-tan |-A = 9.4028 log-tan ^b = 9. 6659 log-tan ^0 = 0. 0910 |A = 14°ir iB = 24°51f ^0= 50° 58' A = 28° 33' B = 49°43' C = 101°56' chech : A + B + c = 180° nearly. QUESTIONS. 1. Show by the formulae that a triangle is possible only,when each side is less than the sum of the other two sides. What sign must be given to the radical in an ideal triangle ? 3. Solvethetriangle, givena, 127 m.; 5,64.9m.; c,153.16m. [55° 19.4', 34° 51.1', 99° 49.3'. 3, Solve the triangle; given a, 659.7 ; l, 318.3 ; c, 527.6. 4. Solve the triangle, given a, 625 ; i, 615 ; c, 11.' Before solving show which of the angles A, B, c are large, which small, and which smallest. Can an exact solution be made ? 70 PLANE TRIANGLES. [Ill, PR. (J) Given K, b, c, two angles and a side: then c = 180° — (a + b), « — sin A-c/siiio, & = sinB-c/sinc. check : sin ic — /^[(s — a) {s — b)/ab] . E.g. let A, B, c be 50°, 75°, 130 yards ; then the work may take this form : c = 180° -125° = 55° log 120 =2.0792 2.1658 3.1658 log-sin 55° = 9. 0134 log-sin 50° = 9. 8843 log-sin 75° = 9. 9849 cJicch : 2.1658 log a = 2.0501 logS = 2.1507 a = 112. 2 yards. J = 141.5 yards. c=120 a = 112.2 log, 2.0501- J=141.5 2.1507- 2)373.7 s = 180.85 s-a= 74.65 1.8730 + • s-b= 45.35 1.6566-1- s-c= 66.85 2)9.3238 ^0 = 27° 30' 373.7 9.6644 QUESTIONS. log-i sin ^0 = 9. 6644 1. In examples under this case, is there always a solution ? Is there ever more than one solution ? What limitations are there on the values of the two given angles A, B ? 2. Solve the triangle, given a, 34°; B, 95°; c, 1^9 ft. [51°, 9.995, 17.805. 3. Solve the triangle, given b, 58° 30'; c, 120° 13'; a, 5387 yds. Can an exact solution be made with the angles B, c so large, and A so small ? 4. Write out the formulae for the solution and the check when B, c, a are given. So, when c, a, b are given. So, when A, b, a are given. 5. Why may not more than three parts be given ? Rg. Why may not the data be A, 50°; b, 75°; a, 30 ; S, 30 ? l.§3.] SOLUTION OF PLANE TRIANGLES. 71 (c) Given a, i, c, tioo sides and their angle : then J(a + b) = 90°-^C!, tan^(A-B) = cot^c-(ffi-5)/(a + 5), , i(A + B) + J(A-B) = A, i(Ar+B)-|(A-B) = B, c = sinc-tt/sm A. check : (5 + c)/a = cos ^(b - c)/sin ^a. Kg. let a, b, c be 635, 361, 61° 17' ; then the work may take this form : FORMULA. NUMBERS. LOGARITHMS. cot^c 30° 38' 0.2275 + •a~b i 274 2.4378 + la + i 996 2.9983- = tanJ(A-B) 24° 54^' 9.6670 90° -^c = Ka + b) 59° 21 f A (sought) 84° 16' B (sought) ^34° 27' a 635 2.8028 + •sine 61° 17' '9.9430 + :sin,A 84° 16' 9.9978- = c (sought) \J. 559.75 2.7480 check : b + c :.: ■ 920.75 3.9642 + J: a 635 2.8028- 0.1614 = cos^(c-b) 13° 25' 9.9880 + : sin ^A 42° 8' QUESTIONS. 9.8266- 0.1614 1. In examples under this case, is there always a solution ? Is there ever more than one solution ? Are any limitations to be put upon the lengths of the sides or the magnitude of their angle? Between what limits do ^(a + b), ^(a — b) lie? VZ. Solvfe the triangle, given a, 25.3 ; b, 136 ; c, 98° 15'. [10° 10', 71° 35', 14;.86. 12 PLANE TRIANGLES. [Ill, PK. (d) Given h, c, b, tioo sides and an angle opposite one of them : then sinc = r-sinB/5, a = 180°-(b + c), « = sin a • J/sin b. check: (a + J)/c = cos^(a — B)/sin Jc. The angle c, found from the equation siu c = 5 • sin b/J, may in general have two supplementary values [sin sup « = sin a. and two triangles are then possible. But there are some limitations : 1. If B, b, c be so related that c- sin b > 5, then sin c > 1, which is impossible, and there is no triangle. 2. If c- sin 8 = 5, then sinc = l, c is a right angle, and there is one, a right triangle. 3. If either value of the angle c makes b>c when 5>c, or B < c when 5 < c, that value must be rejected. In particular : if b be acute, no triangle is possible if S <^, the perpendicular from A to the side a ; one right triangle if i=p; two triangles if pc. So, if b be right or obtuse, a triangle is possible only when b> c, and then but one. QUESTIONS. 1. Draw figures to show the several cases outlined above, and show how the geometric constructions interpret the facts as shown by the formulae, for the several cases. Solve these triangles, given : y2. h, 18; c, 30; b, 55°24'. [66° 9', 58° 37', 18.64, or 113° 51', 10° 45', 4.08. 3. a, 10 ; 5, 20 ; a, 30°. 4. b, 16 ; c, 30 ; B, 86° 40'. 5. c, 30 ; a, 20 ; c, 47° 9'. 6. a, 34 ; J, 20 ; A, 37° 36'. 7. a, 24; &, 20 ; A, 120°, [46° 12', 13° 48', 6.61. 8. a, 20; 5,30; A, 135°. 9. a, 16 ; 5,30; A, 150°. 10. Let o, p be two points 10 feet apart ; about o describe a circle with radius 4 feet ; through p draw a line making an angle of 30° with the line po : at what distance from p does this line cut the circle ? -1, §3.j SOLUTION OF PLANE TRIANGLES. 73 QUESTIONS FOR REVIEW. SoIto these triangles, given : n.. a, 40 ; S, 50 ; c, 60. 2. a, I; l},b; c, 6. 3. a, 411 ; b, 523; c, 633. 4. a, 60° ; b, 60° ; c, TO. 5. a, 24; B, 45° ; 0,24°. 6. A, 31° 26'; 5, 17.1; c, 47° 18'. 7. a, 14; *, 14; c, 60°. 8. «, 38.9; B, 9° 18'; c, 119.11. 9. A, 117° 23'; h, 6 ; c, 11.14,_ _10.^ «,_36 ; 5,40 ; a, 51° 16'. 11. If the three sides a, 5, c of a triangle be given, find'flie length of the perpendiculars from the vertices upon the oppo- site sides ; of the lines connecting the vertices, with the mid- points of the opposite sides ; of the segments of the bisectors of the angles, cut off by the opposite sides. 12. In ex. 10 of page 72, let the distance op be a, the radius of the circle i, and the angle poq, c : how many solutions are pos- sible when a>5 ? when A = B ? when o!<6 ? Show how the angle c is limited in each of these cases. 13. Discuss ex. 12 if c be negative. So, « or 5 be negative. 14. The sides of a triangle are 3, 4, v'38 : show, without solv- ing, that the largest angle is greater than 120°. 15. If a, 5, c be in arithmetic progression, 3 tan|^A'tan^c = l. 16. If c = 2b, then c^^i {a + h).. 17. Show by trigonometry that if an angle of a triangle be bisected, the segments of the opposite side are proportional to the other two sides. 18. If « cos A = 5 cos B, the triangle is either right-angled or isosceles. 19. If p be any point in an equilateral triangle abc, then cos (bpc - 60°) = (pb" -I- PC" - pa'')/2pb • PC. 30. Show how to solve a triangle from the three altitudes. 74 PLANE TRIANGLES. [Ill, -^ §4. SINES AND TANGENTS OP SMALL ANGLES. If an angle be very small, its sine and tangent are also very small ; but their logarithms are negative and very large, and they change rapidly and at rapidly varying rates. Such loga- rithms, therefore, are not convenient for use where interpola- tion is necessary, and in their stead the logarithms given below may be used ; they are based on the following considerations : An angle whose bounding arc is just as long as a radiys is a radian; it is equal to 57° 17' 44.8", i.e. to 206264.8", and the number of seconds in an angle is 206264.8 times the number of radians. The index for radian's is "■. For a small angle the number of radians in the bounding arc is a very small fraction, and it is a very little larger than the sine of the angle and a very little smaller than its tangent : it follows that, if a small angle be expressed in radians, the ratio sinAyA 'is a very little smaller, and the ratio tanA^A is a very little larger, than unity. These ratios approach unity closer and closer as the angle grows smaller. If the angle be expressed in seconds, the ratio sin a"/a is a very little smaller than the reciprocal of 206264.8, and the ratio tanA"/A is a very little larger than this reciprocal. These ratios change very slowly, and hence interpolation is always possible ; the table below gives their logarithms as far as 5°. Angle. log(8inA"/A). Angle. log(tanA"/A) Angle.^ log (tan a"/a). 0° -1° 4' 4.6856 0° -1°18' 4.6856 3° 87-3° 54' 4.6862. 1° 5' -2° 23' 4.6855 1° 19'-1° 59' 4.6857 3° 55'-4° 11' 4.6863 2° 24'-3° 11' 4.6854 2° -2° 29' 4.6858 4° 12'-4° 27' 4.6864 3° 12'-3° 50' 4.6853 2° 30'-3° 54' 4.6859 4° 28'-4° 41' 4.6865 3° 51'-4° 23' 4.6852 2° 55-3° 16', 4.6860 4°42'-4°55' 4.6866 4° 24'-4° 52' 4.6851 3° 17'-3° 36' 4.6861 4°55'-5°00' 4.6867 The cosine and cotangent of an angle near 90° are the sine and tangent of the complementary small angle. The logarithm of the cotangent of a small angle is found by subtracting the modified logarithm of the tangent of the angle from 10 ; that of the tangent of an angle near 90°, by subtracting the modi- fied logarithm of the tangent of the complementary small angle from 10. §4. J SINES ASD TANGENTS OF SMALL ANfiLES. 75 TO TAKE OUT THE SINE OR TANGENT OF A SMALL ANGLE. Take out the logarithm that c'oiTesponds to the number of degrees and minutes ; and add the logarithm of the whole num- ber of seconds in the angle. Let A be the number of seconds in an angle ; then •. • sin a" = (sin a"/a) • a, y .-. log-sin a" = log (sin a"/a) + log A ; and ■.• tan a" = (tan a"/a) ■ a, .-. log-tan a" = log (tan a"/a) -|- log A. Kg. log-sin 10' 30"=log (sin 63O"/630) -I- log G30 = 4.6856-1-2.7993 = 7.4849. So, log-tan 3° 13' 40" = log (tan Hfr307ll620)+log 11620 = 4.6860-1-4.0652 = 8.7512. The angle is found by a reverse process. Kff. to take out log-sin"' 8.4143 : Prom the table of sines and tangents, page xi, it appears that the angle sought lies just below 1° 30', and by the formula log A = log-sin a" -log (sin a"/a) ; and ■.■ 8.4143-4.6855 = 3.7288, .-. the angle is 5355"; i.e. 1° 29' 15". So, to take out log-sin~' 8.8062 : The angle sought lies near 3° 40', and •.•8.8062-4.6853 = 4.1210, .•. the angle is 13212"; i.e. 3° 40' 12". 1. Find log-sin 22', 43', 1° 11', 1° 27', 2° 24' 36". 2. Fincl log-tan 22', 43', 1° 11', 1° 27', 2° 24' 36". 8. Find log-sin -> 7.3146, 8.2719, 8.4185, ' 8.8927. 4. Find log-tan"' 7.3146, 8.2719, 8.4185, 8.8927. Solve these triangles, given : 5. a, 327 ; b, 328 ; c, 654. 6. a, 3279 ; b, 3280 ; C, 1° 76 PLANE TRIANGLES. [Ill, TH. §5. DIRECTED AREAS. If an elastic cord be stretched from a point o to a point a, and if while one end of this cord is fixed at o, the other end trace a line AB, straight, broken, or cnrved, the cord, now a radius vector of varying length, sweeps over the figure oab, and may be said to generate the area gab. It is convenient to call tlie area of the figure oab positive if the radius vector OA be positive and swing about o counter-clockwise, and neg- ative if it swing clockwise ; and this convention conforms to the conventions as to directed lines and angles already in use. r O AG AG A If after generating the area oab the cord swing back from OB to OA, and its end retrace the same line from b to A, then the area oab may be thought of as taken up and cancelled, and the sum of the areas oab, oba is naught. So, if C be any point on the line ab, then : area oab + area obc = area oac, and area oab + area obc + area oca = 0. Theor. 3. If ABC he an ideal triangle wJiose sides are a, b, c, and exterior angles a,fi, y, and if K be the area of this triangle, \ 1 Y / ^ 4. / \6 b/ \'A , /-^J N a o\ X then K = ^ab ■ sin y = iab- sine, = ^a' ■ sin j3 sin y/sin a = \a^ • sin b sin c/sin A, = V* (s -a) (s - *) (s - c)' 3, §5.] DIRECTED AKEAS. 77 For draw NA normal to bc, then'.* K = ^BC'NA, and NA = CAsin;/, .'. K = JBC ■ CA sin ;(/j i.e. K = Ja5 sin ;f = ^a5 sine. q.b.d. So, ■.■ 6 =asin/?/sina = asinB/sinA, .'. K = |-«''sin^ sin;'/sina = ^a'sinB sinc/sinA.Q.E.D. So, ".■ sin / = 3sin^/cos-^;/ = \/s (s — a) {s — b) {s — c)/ab, .•. K= Vs (s — «) (S — 5) (s — c). Q..E.D. ^ CoE. 1. If ABC be mi ideal triangle, any point, and k the area of abc, then : ABC = CAB + OBC + OCA, K = ^[OA • OB sin AOB + OB • oc Sin BOG + oc • OA sin coaJ. (a) within abc. For the three geometric triangles oab, obc, oca are together equal to ABC, as in the first figure, and their areas are all positive. {b) o without ABC. For •.• when two of the triangles oab, obc, oca are added and the third is taken away, the triangle abc remains as in the second figure, or when from one of them the other two are taken away, it remains as in the third figure, rs PLANE TRIANGLES. [Ill, THS. and ~while the areas of the two are positive or negative, the third is negative or positive, .-. the algebraic sum of the areas of these three triangles is that of ABO, in both cases ; .•. K = ^[OA-OB sin AOB + OB-OC sinBOC + oc-OA sincoAj. Cor. 2. If abc • • • l 5e any polygon, o any point in the plane of the polygon, and k the area, then: K = ^(OA-OB SIW AOB + OB-OC SW BOC + ■ ■ • + OL-0A sin 1,0 a). In the three theorems that follow, it is assumed that every motion of a point is the limit of some motion made up of small translations along successive lines, and every motion of a line is the limit of some motion made up of small rotations about successive points. Either motion is that of a point and a line through it, such that the point always slides along the line, while the line always swings about the point. JH.g. if a line roll round a circle, without sliding upon it, the line always swings about the point of contact, while the point of contact always slides along the tangent line. The area stvept over by a segment of a straight line is the alge- braic sum of the areas of all the infinitesimal quadrilaterals and triangles passed over, from instant to instant, by the segment. 3, 4, §5.] DIKECTBD AREAS. 79 Theob. 4. If PQR • • • TP Je any closed figure traced hy the end of a radius vector, drawn from o, and varying in length if need be, the area of this figure is the area swept over by the radius vector, and is positive when the bounding line is traced in the positive direction of revolution, and negative when traced in the negative direction. (a) No reversals of motion of the vector, as in the first figure, or only one reversal, as in the second figure : For •.• there are no intermediate reversals, [l^JP- .". the figure enclosed by the boundary is swept over once, and but once, by the vector, when it swings in the direction in which the path is traced ; and ■.' all other figures swept over by the vector in one direc- tion are also swept over in the' other direction, and. cancelled, .*. the algebraic sum of the areas of all the figures swept over is the area of the figure enclosed by the boundary, and this area is jjositive when the path is traced in the posi- tive direction of rotation, and negative when it is traced in the negative direction. Q.b.d. (5) Intermediate reversals of motioti, as in the third figure : For •.■ intermediate reversals occur in consecutive pairs in op- posite directions, .•. if a point within the enclosure be swept over more than once, it is swept over an odd number of times so as to give an excess of just one passage in the forward direction ; 80 and PLANE TRIANGLES. [Ill, THS. ' every point without the enclosure is swept over, if at all, the same number of times in each direction, so that any outside area that may be generated is cancelled, .•. the algebraic sum of the areas of all the figures swept over is the area sought. q.e.d. Note 1. If the boundary cross itself, the figure is thus di- vided into two or more parts : the area of each part may be con- sidered separately, and the area of the whole is the algebraic sum of the areas of the several parts. JH.g. the area of the crossed quadrilateral abcd is the algebraic sum of the areas of the positive triangle aed and the negative angle ebc, and has the sign of the larger. Note 2. In adding two areas any common boundary trav- ersed in opposite directions may be erased. OoR. If a segment ab of a vector ob swing about o as centre into the position a'b', the area of the figure swept over by this segment is the area of the figure abb'a', bounded by the initial and terminal positions of the segment and the paths of its ends. ' For •.* AB = OB-OA, .■. the area k of the figure swept over by the segment AB is the area of the figure swept over by vector ob less the area of the figure swept over by vector oa. 4j5, §5.] DIRECTED AKEAS. 81 .•. K = OBB'-OAA' = OBB' + 0A'a = ABB'a'A. Q.E.D. [th. 4. Theob. 5. If two joints A., B move (forward or backward in any way) along any paths aa', bb' to a', b', then the area swept over iy the straight line AB (varying in length if need be) is the area of the figure abb' a'. For let the motion of the generator ab be made np of infini- tesimal rotations about successive instantaneous centres Ci. Oo, Ga • • • I then-.- AB sweeps over figures abb, a,, AiBiBjAs- • •, [th. 4, cor. and •.' all the intermediate positions AiB,, AaB,- • • of ab are common boundaries of these figures traced in oppo- site directions, .-. the sum of all the areas swept over is the area of the figure bounded by the path abb'a'a. [th. 4, nt. 3. Cor. 1. The area swept over ly any straight line ab is the sum of the excess of the area of the figure subtended (from any origin) by the path of the terminal point b over that subtended by the path of the initial point A and the excess of the area of the triangle subtended by the initial line ab over that subtended by the terminal line a'b'; i. e. abb'a'a = (obb' - oaa') + (oab - oa'b'). 6 82 PLANE TRIANGLES. [Ill, TIIS. Cor. 2. If the generator ab return to its initial position, the area swept over is the excess of the area of the figure hounded by the path of the terminal point B over that of the figure hounded by the path of the^initial point a. Cor. 3. If the generator ab return to its initial position, and the initial point a trace out the same path, to and fro, then the area swept over is the area of the figure bounded by the path of the terminal point B. Theor. 6. If a wheel be affixed to its axis at the mid-point, and if this wheel roll and slifle in any way upon a plane ivhile its axis remains parallel to the plane, the area swept over by the axis is the product of its length into the distance rolled by the wheel. For, let AB be the axis and m the mid-point ; let the axis turn about an instantaneous centre 0, through an infinitesimal angle 6, and at the same time let the axis slide along its line an infinitesimal distance, to a'b'; then-.' oa = oa', ob = ob', om = om', sm6=0, .: area abb'a' = obb' - oa a' = J(ob'' - o a") • 6^ =^(ob-oa) (ob + oa)-^=ab-om:-^ = AB-the distance rolled by the wheel at m, 5j6, §5.J DIKECTED AREAS. 83 .•. the area swept over by any number of such successive rotations is the product of ab by the distance rolled by the wlieel at m. q.e.d. CoE. 1. If the wheel he affixed to its axis at any other point, c, and the axis turn through an angle, a, between its first and last positions, the area swept over is AB • the distance rolled ly the wheel a^ c + ab • cm • a. For ■.• in the infinitesimal rotation above, area ab-om-^=ab- (oc + cm)-^ = AB ■ the distance rolled by the wheel at c + AB • om • 6, .: the sum of all such rotations is AB-the distance rolled by the wheel at c + AB • cm • a. [a=0+d'+ ■ ■ ■. Cor. 2. If the axis return to its first position without malcing a complete revolution, the area swept over is ab • the distance rolled by the wheel affixed at any point c. [a = 0. QUESTIONS. 1. If A, b, C be fixed points on a line that turns in a plane thyough an angle a, then BC areaAB — AB areaBC = ^AB-BC-CA-a. 2. If the line in ex. 1 return to its first position : (fl) without making a complete revolution, . area B — (ab area c + BC area a) : AC ; {b) after making a complete revolution, area b + ;r • ab ■ BC = (ab area c + BC area a) : ac. 3. If the chord AC, in ex. 1, slide round an oval, the area between the oval and the path of B is ^t-ab-bc. 4. Find the area of the curve traced by a point on the con- necting rod of a piston and crank in one revolution ; also the distance a small wheel attached at the same point would roll if a plane surface pressed against it. 84 PLANE TRIANGLES. [Ill, PR. amslee's PLANIMETER. Let the axis ab, above noted, be pivoted at a to an arm OA of fixed length tbat turns about a fixed centre 0, so that A traces a fixed circle while B traces any pf|,th whatever ; let the wheel be affixed to ab at any point c, but let it be impossible for ab to sweep past OA so that AB, OA can take but one position for one position of B, and, if A encircle o, ab also encircles in the same direction : 1. If A return to its first position without encircling o, then'." A traces out the same path, to and fro, .•. the area encircled by B is the area swept over by AB, [theor. 5, cor. 3. i.e. the area is the product of the number of turns of the wheel into the constant area %7ir • AB, [theor. 6, cor. 2. wherein r is the radius of the wheel. 2. If A encircle o counter-clockwise, then the area encircled by b is the area swept over by AB + the area of the circle OA, [theor. 5, cor. 2. i.e. the area encircled by B is ^nr-AB-the number of turns of the wheel (positive or negative) + ab • cm • a + tt ■ oa', wherein a is 2/t'. [theor. 6, cor. 1. The constanis of the planimeter 27rr-AB, 7r(2AB • cm + oa') can be found once for all. The first is the area due to one turn of the wheel ; the second is that due to the swinging of the arms oa, ab about o. 2, §6.j INSCKIBBD, ESCRIBED, CIKCUMSCEIBED CIRCLES. 85 §6. INSCRIBED, ESCRIBED, AND CIRCUMSCRIBED CIRCLES. PrOB. 2. To FIND THE RADII OF THE CIRCLES INSCRIBED IN, ESCRIBED, AND CIRCUMSCRIBED ABOUT, ANT TRIANGLE. For the radius of the inscribed circle, divide the area ly half the perimeter. For the radius of an escribed circle, divide the area by half the perimeter less the side beyond which the circle lies. For the radius of the circumscribed circle, divide half of either side by the sine of the opposite angle. For, let ABC be any triangle, and let ?• = radius of inscribed circle, r', r", r"'s radii of escribed circles whose centres are o', o", o'", and rs radius of circamscribed circle ; then-.' K = -^r (a + 5 + c) = rs, [geom. .-.r-K/s. Q.E.D. So, ■.■■s. = ^r' {-a + b + c) = r' {s-a), [geom. ,-. r'=K/{s-a) ; and so for r", r'". q.e.d. Checks : 1/r = 1/r' + 1/r" + 1/r'", K' = r-r'- r" -r'". 8G PLANE TRIANGLES. [Ill, PR. About ABC draw a circle and draw ca', a diameter ; join a'b ; then'.- a = a', and angle abc is a right angle, [geom. and ca' = a/sin a' = «/sin A, .•. E = |a/sinA • • ■. Q.E.I). Note, a/sin a = 5/sin b = c/sin c = 3r. QUESTIONS. Find the radii of the inscribed, escribed, and circumscribed circles, and check the work, given : 1. a, 13.7; I, 33.8; c, 51.5. 3. A, 64° 19' ; B, 100° 3' ; c, 51.35. 3. a, 136 ; I, 95.3; c, 11° 37'. 4. In a right triangle, 8r + ?' = s. 5. If E:=3r, the triangle is equilateral. 6. In the ambiguous case the two values of E are equal. 7. The distances from the centre of the inscribed circle to the centres of the three escribed circles are equal to 4k sin j-a ■ • • , and to a sec ^A • • • . 8. The square of the distance between the centres of the inscribed and circumscribed circles is e' — 2Er. Prove the equations : 9. r =(s — a)tan JA. 10. r =s tan ^-a tan -^b tan -jC. 11. ?'=a/(cot^B + cot Jc), r' = a/(tan|-B + tan^c). 2, § 6. J INSCEIBED, ESCRIBED, CIRCUMSCRIBED CIRCLES. 87 Prove the equations : 12. R=a5c/4K. 13. R = s/(sin A + sinB + sinc). 14. r' + r" + r"'-r = 4:ii; rr'/r'W" = tan" |-A. 15. K = 4;Rr COSj^A COS^B cos|o. 16. E + r=:E(cOSA + COSB + COSC). 17. 4e sin A sin B sin c = a cos a + 5 cos b + c cos c. 18. In the figure on page 85 co'" is perpendicular to o'o", Ao' to o"o"', Bo" to o"'o'. The point o, the co-poiut of these three perpendiculars, is'the orthocentre of the triangle o'o"o"'. The triangle abc, whose sides join the feet of the perpendic- ulars two and two, is the pedal triangle of o'o"o"'. 19. The circle circumscribed about abc passes through the mid-points of the triangle o'o"o"', and through the mid-points of the segments oo', oo", oo'". This circle is the nine-point circle of the triangle o'o"o"'. 30. The nine-point circle of a triangle circumscribes its pedal triangle, passes through the mid-point of each side, and bisects the lines joining the vertices to the orthocentre. 21. If a, I, c be the sides of a triangle, and p be the radius of the circle inscribed in a triangle whose sides are 5 -He, c-\-a, a-t-5, then p^ = %nr. 32. If a, I, c be the sides of a triangle, and in, n, p be the altitudes, then mnp={a + l + cyr'/al)C. 23. If u, V, w be the distances between the excentres of a triangle, then uviu sin A sin b sin c = 8r V V". 24. Find the radii of the circles that touch two sides of a triangle and the inscribed circle. So, of those that touch the circumscribed circle. 25. Find the relation which exists between the angles of a triangle whose orthocentre lies on the inscribed circle. 88 DEllIVATIVES, SEKIES, AND TABLES. [IV, IV. DERIVATIVES, SEEIES, AND TABLES. §1. CIRCULAR MEASURE OF -ANGLES. In the applications of trigonometry to numerical problems, e.g. the solution of triangles, the most convenient unit of an- gular measure is the right angle, or the degree, the ninetieth part of a right angle ; but in certain other problems, e.g. the " computation of trigonometric ratios and their logarithms, that angle which lies at the centre of a circle, and whose bounding arc is just as long as the radius of the circle, is a better unit. This unit angle is called a radian, and its magnitude is inde- pendent of the length of the radius. [geom. Eadians may be indicated by the signl7 just as degrees, min- utes, and seconds are indicated by the signs °, ', " ; and since the ratio of the half circumference of a circle to its radius is n, [3.141592- • •] and angles at the centre are proportional to their arcs, two right angles are equal to n radians. The primary equation expressing the relation between de- grees and radians is 7f^—l%Q°: from this it follows that ^^i'-tgO", i7zt''>=45°, ^7!f^t30°, •••, t'-tl807;r = 57°17'M.8", l°=7rV180=. 0174533'-, 1' = .0002909'-, • • •, and the measure of other angles is expressed by the ratio of the bounding arc to the radius. §!•] CIKCULAK MEASUEE OF ANGLES. S9 QUESTIONS. 1. Prove that' the number of radians in an angle is ex- pressed by the ratio of the arc subtending it to the radius of the circle, i.e. by the number of radii in the arc. 3. Express in degree-measure the angles : \7t, i[7t, \n, |;r, 3.1416^ .7854'-, V, LS^ -%"■, {rv + lf^ 3. Express in radius-measure the angles : 14°, 15°, 34°, 130°, 137° 15', "4800°, 13', 34". 4. If the radius be an inch, find the length' of the arcs : 14°, 15°, 130°, 57° 17' 44.8", 1°, -^^r, i;r, a*-, tc + V. So, if the radius be five inches. 5. How many radii in an arc of : 30°, 180°, 3'" ? 6. If the radius be 10 inches, find the number of radians subtended by an arc of : 13 inches, n inches, 10°, 5' 13", three quadrants. 7. The angle 3.43'' is subtended by an arc of 5.71 inches: find the radius ; the arcs opposite ^Tf^, V, 5° ; the angles in radians and in degrees opposite a one-inch arc, a two-radius arc, five quadrants. 8. If the circumference of a circle be 30 inches, find the arcs opposite n'', 30°, S"". 9. How many radians and how many degrees are subtended by : 3J radius arcs, n radius arcs, 3J quadrants ? 10. How many radians in 17° 13' 15" ? in 10° ? 11. An angle of three radians at the centre of a sphere sub- tends a two-foot arc of a great circle : find the radius. 13. 'The apparent diameter of the sun, as seen from the earth, is half a degree ; a planet crosses the sun's disk in a straight line at a distance from its centre equal to three-fifths of the sun's diameter : show that the angle subtended at the earth by the part of the planet's path projected on the sun is 7r'-/4.f;0. 90 DERIVATIVES, SERIES, AND TABLES. [IV,TH.1,2, §2. DERIVATIVES OP TRIGONOMETRIC RATIOS. Theor. 1. If 6 he the circular measure of a positive acute angle, then sin 0<6 GEOMETRIC PROOF. Let 0-xp be any circle, and q a point on this circle near p ; bisect arc PQ at R, and join ox, op, oq, or ; draw AP, BQ normal to ox ; join p, Q, and tlirougli p draw a parallel to ox meeting bq in d ; let ^=/xop, 6''=/F0Q, {d + ^0')= ZxQR, rsradius of circle ; O B A X then-.- sin 6=AP/r, sin {6 + 6') = BQ/r, 0' = arc vq/r, and /DQP=:/X0B, [geom, .■. [sin (^+6'')-sm^]/^'=:DQ/arc.pQ = (DQ/chord pq) • (chord pq/arc pq) = cos {d+iO') ■ (chord pq/arc pq), .•. lim(incsin^/inc ^),=Dg sin 6*, = cos^; q.e.d. [th. 1. and so for Dg cos 0, Dq tan 6 ■ ■ ■. DERIVATIVES OF ANTIFUNCTIONS. Theob. 3. Dx siw.~'a;=l/V(l-«'), Dx cos~^x— -l/\/(l -a;"), Dj /!a«-'a; = 1/(1 4- a;'), D^ co1r'^x= -l/(l+a;*), D^ secr^x = l/x\/{x^ — l), Dx cscr^x= —l/x»J{a? — l'), For let 6 — sin"'a;; then'.' sin 6 = x, .-. Dx sin ^, =eos6-D^d, =1, .•.Dx6' = l/cos^ = l/V(l-a;'); Q.B.D. 2,3,§3.] DERIVATIVES OF TRIGONOMETIUC RATIOS, 93 Note. 'When x stands for sin 6 or cos 0, x may have any value positive or negative not larger ^than unity; when x stands for tan or cot 6, x may have any value whatever ; and when X stands for sec 6 or esc 6, x may have any value not smaller than unity : for if, in the formulae above, x exceeds the bounds named, the function is imaginary. QUESTIOiTS. 1. If ^ be any plane angle and &' be the increment of 6, then : inc' sin ^= -(2 sin^-^')" sin (^+ d'), inc^ cos 6*= - (2 sin ^d')' cos ((9 + d'), inc* sin d- (2 sin -^8')* sin {d + 2d'), inc*cos^= (2sinJ^')* cos ((9 + 25'), wherein inc'sin 5=the increment of the increment of sin 6, i.e. [sin (5 + 2(9') -sin {d+ 6')] - [sin ((9+ 61') - sin (9], or sin (5 + 25') -2 sin (5 + 5')+ sin (9; and inc* sin 5 = inc inc inc in c sin 5, i.e. sin(5 + 45')-4sin(5 + 35') + 6sin (5 + 2(9') -4sin(5 + 5') + sin5. , 2. If Sa, db, 6c, 6a, 6b be any simultaneous small changes in the values of a, h, c, a, b, that are consistent with the known relations of the parts of a right triangle [a + b = 90°, a' + 5'=e', fl! = csinA, 5 = ccosa], then 6b=—6a, 6c=a/c-6a + h/c-Sb = &mA.-6a-\-(iosx-6b, 8a=s,hi K-6c + c cos A • t^A, 6b ^ cos a • dc — c sin a ■ 6a., dnd [eliminate 6c from the last two equations] 6A=(ios a/c • 6a — sin a/c • 6b = {b6a — adb)/{a' + b'). 3. If, in a right triangle, only the values of a, b be given, and if these have the possible errors *«;', *&'; i.e. if a may pos- sibly diflEer from its assumed value by either +«' or ~a', and b by either +5' or "J'; show from ex.2 that the resulting values of c, A will have the possible errors ± {aa' + bb')/c — ± (a' sin A + 5' cos a), {a', b' positive, and ± (ab' + ba!)/& — ± {b' sin A + a' cos a)/c. 94 DERIVATIVES, SERIES, AND TABLES. [IV,LM.7-11, So, if only h, c be given, witli the possible errors *J', *c', find' the possible error? of the other sides and angles. So, if only 5, .4. be given, or only c, A, with the possible errors ^y, *a', or *c', *a'. 4. From the known relations of the parts of an oblique tri- angle [a + B + c = 180°, ffi sin B = 5 sin A, • • • j prove that (a) (JA + (yB + (yc = 0, (b) I cosA-(yA — acosB-tyB — sinB-(Ja + sin A'c55 = 0, c cos B • (Jb — 5 cos c • (5c — sin c • (55 + sin B • tfc = 0, a cos c-6c — c cos A- (5a — sin A- (5c + sine- ^a = 0. From these equations, by elimination and reduction, derive (c) 5 • (5c + c cos A ■ (5b — sin A • (5c + sin (3 • (5(Z = 0, c • B(5 + 5 cos A • (5o — sin a ■ (55 + sin B • (5a = 0, with four symmetric equations ; and {d) . 5 sin c • (5a — (5ff + cos C • (55 + cos B • (5c = 0, with two symmetric equations. 5. If in an oblique triangle only a, B, c be given, and if their possible errors be *a/10000, *10", * 15", find the possible errors of A [ex. 4, a]; of 5 [ex. 4, c]; of c [ex. 4, c]. Find the values of these possible errors when ABC is very nearly equilateral, 5000 feet on each side. 6. Given the values of c, a, i, with the possible errors ^c', ' *fl', *5', find the possible errors of b, a, c [ex. 4, c, cf]. 7. Given A, a, h, with the possible errors *a', *a', *5', find the possible errors of B [ex. 4, 5]; of c, c. 8. Given a, b, b, with possible errors *a', *b', *5', fi^d the possible errors of c, «, c : first, when, as in all the above case.s, the computation is assumed to be exact ; second, whenc, a, c have the further possible errors "=c", *a", *c" from decimal fig- ures omitted in the computation. 9. Given a, b, c, with possible errors *«', *5', *c': find the possible error of A, with a possible error in computation of ''a". TH.4,§3.] EXPANSION OF TRIGONOMETRIC RATIOS. 95 §3. EXPANSION OP TRIGONOMETRIC RATIOS. In the expansion of trigonometric ratios the following prop- erties of series are made use of : they are all proved in works on algebra, and are quoted here for convenient reference. Lem. 7. If, after a given term, the terms of a series form, a decreasing geometric progression, the series is convergent. Lem. 8. If one series he convergent, and if the term.s of another series he not larger than the corresponding terms of the first series, the second series is convergent. Lem. 9. If, after a given term, the ratio of each term of a series to the term hefore it he smaller than some fixed number that is itself smaller than unity, the series is convergent. CoK. The series Ao + KtX + AjO;^ + AjK' + • ■ ■ is convergent for all values of x that make the limit of the ratio of the (n + 1)* term to the w"' term s?naller than unity when n becomes very great. Lem. 10. If in the series A„ + kiX + a^x' + AiX'+ ■ ■ -jthe limit of the ratio of the {n 4- 1)'* term to the w* term, for any value of X, he smaller than unity, then, in the derivative series Ai + 2Aja; -|-3Aaa;''+ • ■ •, the limit of the (w + 1)"' term to the w'* term, for this value of x, is smaller than unity, and this series is con- vergent. Lem. 11. If tioo series arranged to rinsing powers of any same variable he equal for all values of the variable that make them both convergent, the coefficients of like powers of the vari- able are equal. Theor. 4. If 6 he the circular measure of any plane angle, then : sin e= B- 1)'/^ ! + ^V5 ! - ff/l !+•••, cos e^i~ey2 !+ey4 i-^ye !+•••. For assume sin d=Xo-\-&.iB + i>-iS' + Aid'-\ , wherein the a's are unknown but constant, and has such values as make the series convergent. 9G DERIVATIVES, SERIES, AND TABLES. [IV, TH. and find the first two ^-derivatives of both members of the equation ; then cos^ = Ai + 3As^ + 3a,6''-|-- • •, and -sin(9 = 3A5-l-2-3A,6'+- • •, i.e. sin^= -2Aj-2-3a8(9; and both of these derivative series are convergent, [lem 9, cor. Take as one of the values of 6 ; then-.-sinO = and A,, + AiO + AjO' + AjO' + • • • = A,,, .-. A„=0. So, ■.•cosO=:l and Ai + SAsO + SAsO'-I- • • • =Ai, .•.Ai=l. So, •.■Ao-l-A,i9-|-A,e"-l-A,(9»+- •• = -2Aj-3-3a8^-3-4a4^,- • ■ • for all values of 6 that make both series conver- gent, .-. Ao=-2Aj, Aj = 3-4A4, A4 = 5-6Aa---, and A,= — 2-3Aa, A3=-4-5As, Aj=: — 6-7a,- ■ • ; [lera.ll. .'. Ao, Aj, A,, Aj, Aj, • • • = 0, and A, = l, A3=-l/3!, A5 = 1/5!, a,= -1/7---; .-. sin(9=^-i9'/3! + eV5!-(9V7!+---, and oos(9 = l-i9"/2! + 6'*/4!-eV6!-|----. Note. These series are convergent for all finite values of 6, For the ratios of successive terms, in that for the sine, are ^7(2-3), ^7(4-5), ^7(6-7) •■•; ' and, in that for the cosine, ^7(1.2), ^7(3- 4), ey {p. &),...; i.e. series of fractions such that the limit of the (ra + 1)"' term to the w* term is smaller than unity whatever be the value of 6. But they converge rapidly only when is quite small. Cor. 1. tan(9=i9+^/3 + 2e7(3-5) + 176'y(3''-5-7) + 62^/(3*-5-7) + ---, cot (9 =1/^-^/3-^/(3". 5) -26)7(3'- 5 -7) -^/(3»-5=-7) , 4, §3.] EXPANSION OF TKIGONOMETEIC KATIOS. 97 Bece = l + eV3 + 5^/(3»-3) + 61^/(3«.3'-5) + 277(9y(3'.3»-7) + ---, cscfii = \/e + ^/(3 • 3) + 7«V(2» • 3' • 5) + 31^V(2*-3»-5-7)+137(9'/(3'.3'-5-7) + ---. For the tangent, divide the series for the sine by that for the cosine ; for the cotangent, divide the series for the cosine by that for the sine ; for the secant, divide unity by the series for the cosine ; for the cosecant, divide unity by the series for the sine. Note. The series for tan B and sec Q are convergent only when 6 -^ \n, for tan 6 and sec 6 are finite and continuous functions of 6 for all values of B smaller than \n ; but when 6 = ^7t their values are infinite. So, the series for 6 cot 6 and 6 CSC B are convergent only when B -^tt. CoE. 2. loff-sin e = logB- ^/(3 • 3) - ^/(2» • 3' • 5) -6y(3*-5-7) , log-cos 6=-{0'/%+ By {2' ■ 3) + ^y(3= • 5) + 17eV(2»-3»-5-7)+---]. For •-• Da log-sin ^= cos ^/sin B-CQid= 1/B—6/Z -(97(3" -5) , [lem. .■.log-sin^=logl9-(9y(2-3)-^V(2«-3''.5) -eV{3*-5-7) , [lem. i.e. log-sin ^ = the series whose ^-derivative is the above series for cot^, and which, as ^=0, approaches to log 6 as log-sin d must do. So, ■.• Dfllog-cosl9= -sme/cosd= -i&vi.6=-{6+B'/Z+ ■ ■ ■), .: log-cos e=- [^/2 -h ^/(3' • 3) + ^V(3? • 5) + 17^V(2»-3''-5-7)-F---]. Note. The series for log-sin 6 is convergent for all values of 6 smaller than n ; that for log-cos B for all values smaller than ^7t. 7 98 DERIVATIVES, SERIES, AND TABLES. [IV, TH. 5, GREGORY'S THEOREM. Theor. 5. Ifxhe any number smaller than unity, then tan'^x=x-lz'+\x^-iiX^ + \x'--^x^^+ ■■■. For, assume ta,n-^x=A„ + A^x + A,x^ + A^x' + A^* + A,fl;'-\ , and take the a-derivative of both members ; then Dj; tan " 'a; = A, + 2A,a; + SAjK" + dA^a;' + SAsX* + • • • ; and ■.• D^tan-^x = l/{l+x') = l-x'' + x*-x'A , [theor. 3. .: A^ + 2A^X + 3A,X' + 4:AiCt^+ ■ ■ ■ =l-x'i-x*-x'+-- •, for all values of x that make both series convergent, .-. A„ A„ A„---=0, [lem.ll. and A. = l, A3=-l/3, A, = l/5, A,= -l/l---. So, take as a value of x, then tan-'0 = A„ + A,0 + A,0' + A30'+---,- and A„ = 0; .•. tan-^x=x-i'r'+^x' — ifX''+lx'- ■ • •. q.e.d. Note. This sferies is convergent when a; B.g. let ox, ot, oz be three directed lines such that ox is per- pendicular to ox, oz, and normal to the plane Yoz, that OT is perpendicular to oz, ox, and normal to zox, and that oz is perpendicular to ox, oy, and normal to xot. 106 SPACE TRIGONOMETRY. [V. QUESTIONS. 1. If a rod project above a liorizontal plane in a direction parallel to the earth's axis, in what direction will its shadow on the plane swing in the northern hemisphere ? in the south- ern hemisphere ? So upon a vertical plane ? . In what order will the numbers be placed on a horizontal sun-dial ? on a vertical sun-dial ? 3. To an observer standing behind the transparent dial of a tower clock, what is the direction of rotation of the clock hands ? is it the same for all four faces ? is the actual direc- tion of rotation the same in two opposite faces ? 3. What is a right-hand screw ? 4. In turning on the nuts that keep the wheels of a carriage upon the axles, is the motion clockwise or counter-clockwise ? is it the same motion on both sides of the carriage ? 5. As a carriage is driven forward, how do the wheels turn, to one standing on the right side of tlie roadway ? to one standing on the left side ? 6. If when a carriage is driven forward the rotation of the wheels be positive, what is the rotation when the carriage is backing ? 7. If a carriage drive past, on which side of the roadway must one stand that the normal to the plane of rotation of the wheels may reach towards him ? away from him ? 8. How must a line of shafting be directed so that it shall be normal to the pulleys that are fixed upon and revolve with it ? 9. If two wheels with parallel axes be so geared that they revolve in opposite directions, what relation have the normals to their planes of rotation ? 10. In the figure of theor. 1, how may a point be placed so as to be in front of all the planes XOY, YOZ, zox ? in front of XOY, yoz, and back of zox ? What other positions may a point have ? §2.] DIEDRAL ANGLES. 107 §3. DIEDRAL ANGLES. If two directed planes meet in a directed line, their co-line, and one of them, the initial plane, swing about this co-line till it coincides with the other, the terminal plane, both in position and direction, the diedral angle so generated is the angle of the two planes. This angle is directed and measured by the plane angle that is generated by a normal to the co-line of the two planes, lying in the initial plane and carried by this plane as it swings about the co-line till it becomes normal to the co-line in the terminal plane. The co-line may be directed at pleasure, but however it is directed the plane of the swinging normal must be taken normal to this line. E.g. let the directed planes a, i meet in the directed line x'x, and let a'a, b'e be normal, in a, I, to x'x at o ; then the diedral angle ab is directed and measured by the plane angle AOB, in the plane normal to x'x at 0. So, if the directed co-line be xx' ; then AA', bb' are normal in a, I, to xx', at o, the diedral angle ah is directed and measured by a'ob', in the plane nor- mal to xx' at 0. 108 SPACE TRIGONOMETRY. [V, TH. It is to be noted tliat the angle a'ob' as seen from x' is the opposite of AOB as seen from x, and that the angle b'oa' is the opposite of boa ; i.e. a reversal of the co-liue of the two planes reverses their angle. Theor. 2. The angle of two directed planes is equal to the angle of their normals, as seen from the positive end of the di- rected co-line of the two planes. For, in the figure above, draw op, oq normal to the planes a, b; then".' op is normal to a'a in the plane AOB, and OQ to b'b, .'. the angle poq is equal in magnitude to the angle Aob ; [geom. and •.• these angles have the same direction in the same plane, and the plane angle aob directs and measures the diedral angle db, .-. the angle of the two planes is equal to the angle of their normals. q.e.d. Cor. 1. If the angle ab be a positive right angle, so is the angle poa ; op lies in the plane I and coincides with ob, and oq lies in the plane a and coincides with oa'. 2j§!2J DIEDKAL ANGLES. 109 Note. The student of the geometry and trigonometry of space must train himself to see his figures as figures in space, though shown only, by diagrams on a flat surface. For the most part these diagrams are made up of straight lines and curves, and when he looks at the points and lines of his dia- grams, he must see the points, lines, and surfaces in space which they represent. It will help him to do this if he will close one eye and, without moving his head, look steadily at his diagram with the other eye : presently it will stand out. It will help him, also, if he will hold some object, his book for example, or a card, or a wire cage, between the light and the wall : he will learn that the shadows are the pictures, pro- jections, of his space figures on a plane. Among other things, he will see that right angles are rarely projected into right angles, that circles are commonly projected into ellipses and sometimes into straight lines, and that lines of the same length are often unequal ; and he will learn to look back from the picture to the figure in space. E.g. in the diagram on page 104, the horizontal circle seems to be but half as broad as it is long, and the right angles XOY, Yoz are drawn as angles of 60°, while the right angle zox is drawn as an angle of 120° and appears to be the sum of the other two. So, in the figure on page 108, there are three non co-planar straight lines a'a, b'b, x'x, that meet in a point o and deter- mine three planes that meet in the same point. Three circles lie in these planes and have o as their common centre ; and these circles determine a sphere whose centre is 0. To make this figure stand out more clearly arcs that lie on the front of the sphere are shown by full lines, while those that are behind either of the other planes are shown by broken lines ; and so for the diameters. The front edge of the horizontal circle is tipped down, while the normal op is tipped" forward and does not show its full length. 110 SPACE TRIGONOMETRY. §3. PROJECTIONS. [V, TH. The projection of a point on a line was defined on page 31. The projection of a point on a plane is the foot of the perpen- dicular from the point to the plane. The projection of a directed line on a plane is the co-line of the given plane and a plane perpendicular to it through the projected line. The plane of projection is that plane on which the projec- tion is made, the perpendicular plane is the projecting plane, and the .co-line of the two planes is the line of projection. The angle of a line and a plane is the angle of the line of projection on the plane, when directed, and the given line. The projection of a segment of a directed line on a plane, or on another directed line, is the segment of the line of projec- tion that reaches from the projection of the initial point of the given segment to that of the terminal point. It is a positive segment if it roach forward, in the direction of the line of projection, a negative segment if it reach backward. The projection of a broken line upon a directed line is the sum of the like projections of the segments that constitute the broken line, and it is equal to the projection of the segment that reaches from the first initial to the last terminal point. The angle of two directed lines that do not meet is that of any two lines parallel to the given lines that meet and reach forward in the same dircistions as the lines. 3, §3.] PROJECTIOXS. Ill Theok. ^. If a segment of a directed line be projected on another directed line, the projection is equal to the product of the segment ty the cosine of the angle of the two lines. {a) The two lines co-planar. [II, theor. 10. (5) The two lines not co-planar. For, let I, m be two directed lines not co-planar, and let AB be a segment ;0f I, and a'b' be its projection on m ; through a' draw l' a line parallel to I and like directed, and through A, B draw planes perpendicular to the line m ; then'." these planes are parallel, and a'b", ab are segihents of parallel lines cut ofE by parallel planes, .-. a'b" = ab; [geom. and ■. • angle I'm = angle Im, [df . ang. two lines. and a'b' = a'b" cos I'm, [II, theor. 10. .-. A'B' = ABCOSto. Q.B.D. CoK. The projection of a broken line upon a directed line is the sum of the products of the segments that constitute the broken line by the cosines of their angles with the line of pro- jection. p. Kg. in the figure above, let ox, ot, oz be three directed lines, each normal to the plane of the other two ; let p be any point in space, and project p on the plane OXY at B, and B on ox at A ; then the projection of the broken line oabp on oi' is op, and OP = A cos xop + AB cos TOP + BP cos ZOP. 112 SPACE TRIGONOMETRY. [V, TH. §4. TRIEDRAL ANGLES AND SPHERICAL TRIANGLES. If three planes meet at a point, they form a triedral angle. The three face angles and the three diedrals of a triedral are its six parts. If three directed lines be given that meet at a point, they may be taken in such order and their three co-planes may be so directed that all the parts of the triedral shall be positive and less than two right angles ; and so, if three directed planes be given, their co-lines may be so taken and directed that all the parts shall be positive and less than two right angles. E.g. if BOO, COA, AOB be three planes whose directed co-lines are OA, ob, oc, and if these three planes be so directed that the three face angles bog, coa, aob, and the three diedrals COA-AOB, aob-boc, boc-coa are all positive and less than two right angles ; then the triedral o-abc maybe called the ideal triedral of the points A, b, c, as to the centre o. The three directed planes of a triedral Boc, COA, aob may be named by the three Eoman letters a, I, c, and so may the three plane angles BOC, COA, aob ; and the three diedrals coa-aob, aob-boc, boc-coa by the three Greek letters a, /i, y, and so may their three co-lines OA, ob, 00. 4,§4.] TEIEDEAL ANGLES AND SPHERICAL TRIANGLES. 113 POLAR TEIEDRALS. If through any point normals be drawn to the three faces of a triedral, these normals lie, two and two, in planes perpen- dicular to the three edges of the triedral [geom.], and if these new planes be so directed that they are normal to the edges of the first triedral, a new triedral is formed so related to the other that the edges of either of them are normal to the faces of the other. Two such triedrekh form a. Tpairot polar triedrals. The simplest case of such a' pair of triedrals is where the six planes all pass through the same point. Theoe. 4. In any pair of polar triedrals, the face angles of one of them are equal to- the diedrals of the other. For the angle of a pair of directed planes is equal to that of their normals. , [theor. 3. 8 114 SPACE TRIGONOMETRY. [V, SPHERICAL TRIANGLES. If any point be taken as the centre of rotation of a directed plane, and a sphere be described about this point as centre, tlie co-line of the plane and sphere is a circle of rotation of the plane, and so it is a directed great circle of the sphere that has the same direction as the plane. If any diameter of this circle be directed, the tangent at its positive end reaching forward in the direction of the circle is normal to the diameter, and that at its negative end is contra- normal. That diameter of the sphere which is normal to the plane is the axis of the great circle, and its ends are ih.Q posi- tive and negative poles of this circle. E.g. the earth's north and south poles are the positive and negative poles of the plane of the equator. If two directed planes pass through the centre of a sphere, they cut it in two directed great circles ; and if their co-diam- eter be directed, tangents at its positive end that reach for- ward in the direction of the circles are normal to this diameter, and their angle, in a plane facing the positive end of the diam- eter, measures the diedral angle of the planes. So the tan- gents at the negative end of this diameter are contra-normal, and their angle is equal to the other in a plane facing the same way. The angle of the axes of the two circles is equal to that of the two planes. [theor. 3. E.g. the angle between the plane of the equator and that of the ecliptic, both west- to-east planes, is 33° 3?'. § 4] TKIEDRAL ANGLES AND SPHERICAL TRIANGLES. 115 If about the vertex of a triedral angle as centre, a sphere be described, the co-lines of this sphere with the three directed planes are three directed great circles, and together they form a spherical triangle whose sides subtend the face angles and whose angles, when viewed from the positive ends of the edges, measure the diedrals of the triedral. The sides meet on the co-diameters of the great circles, i.e. on the edges of the trie- dral, and these points are the vertices of the triangle. If two polar triedrals have a common vertex, and a sphere be described about this vertex as centre, the six directed circles cut from the six directed planes by this sphere form &pair of polar spherical triangles, such that the vertices of the one are the positive poles of the sides of the other, and the sides of the one, measuring the face angles of the triedral, are equal to the angles of the other. In this figure, the line oa is normal to the plane b'oc', ob to C'OA', OC to a'OB' ; oa' to BOC, ob' to COA, OC' to AOB. 116 SPACE TEiaONOMETEY. [V, THE SIXTY-FOUR TEIBDKALS OF THREE CO-POINTAE LINES. If a'a, b'b, c'c be three diametei's of a sphere that do not lie in the same plane, each of these lines may have either of two directions. It follows that either a or a' may be taken as the positive end of the diameter a'a, and so for B, b' and for c, c', and that there may be eight distinct sets of three points on the surface of the sphere : A, B, C, a', B, C, a, B', C, a, B, C', a', b', c, a', b, c', a, b', c', a', b', c', i.e. that these three diameters form eight distinct spherical triangles, and eight distinct triedrals, in the geometric sense. So, each of the three planes of these three diameters, taken two and two, may have either of two directions, and the tri- angle of one set of points may have eight distinct forms. Sixty-four triedrals and sixty-four spherical triangles are thus formed with the same three diameters of a sphere, whose sides are all positive and less than four right angles, and whose angles may be positive or negative. These triangles are called the primary triangles, and other triangles congruent with these may be formed by adding mul- tiples of four right angles to either angle, or one or more great circles to either side. § L] TIUEDEAL ANGLES AND SPHERICAL TRIANGLES. 117 118 SPACE TRIGONOMETET. [V, LM. §5. GENERAL PROPERTIES OP TRIEDBAL ANGLES. Lem. 1. If at any point of an edge of a triedral a normal he drawn to the opposite face, and if through this normal a plane be drawn normal to another edge, the co-lines of this plane with the jilanes adjacent to the edge are perpendicular to the edge. [geom. If these lines be so directed that they are normal to the edge, each in its own plane, the angle of these two normals is equal to the diedral of the two planes. ' [df. ang. of two planes. The normatfirst drawn is normal to that one of the two nor- mals which lies in the opposite face. Av E.g. let o-ABC be a triedral angle, through d any point on the edge OA draw ED normal to the opposite face Boc, and through ed, draw planes normal to the edges ob, oc, cut- ting OB in F, and oc in G ; then the lines df, fe are perpendicular to OB, and eg, gd to oc : and if df, fe be directed normal to OB, and eg, gd to oc ; then the plane angle df-fe is equal to the diedral aob-boc, and EG-GD to BOC-COA. [df. So the line ED, drawn normal to the face BOc, is normal to the line FE in the plane efd and to the line EG in GBD. 1, § 5.] GENERAL PROPERTIES OE TRIEDRAL ANGLES. 119 1:20 SPACE TEIGONOMETET. [V, PR. To make clear the relations of the parts of the figures on page 118, construct a space model as follows : Use card-board or stiff paper, and with any centre o and any convenient radius draw a circle ; draw the radii oa, ob, oc, oa', making the angles aob, boc, coa' equal to the given face angles c, a, h ; • on oa, oa' take OD, od' equal, and draw df, gd' normal to ob, oc at F, G, and meeting each other in e ; cut out the figure, and fold along ob, oc ; bring oa, oa' together, and join e, d with a thread : ED is normal to the plane boo and to the lines fb, eg. — -^.o The right triangles fed, egd are shown in the figure as hinged at fe, eg, and folded down into the plane of the draw- ing. The point d is shown at it, h'. These triangles turn up when the two faces aob, coa' are turned up, and with them they form a solid figure. Of the six figures on page 118, the upper middle figure is a space figure, the lower middle figure shows the base of this figure in its own plane, and the right triangles, at the right and left, are the right triangles of the sjiace figure, each shown of its true size and in its own plane. The eight figures on page 119 show the eight spherical tri- angles of page 117, with the lines ed, df, fe, eg, gd drawn as in the figure on page 118. The reader will note the directions of these lines, and the consequent directions of the diedrals a, 13, y. The lemma applies to all the figures alike. 1, §6.] GRAPHIC SOLUTION OF TRIEDRAL ANGLES. 131 §6. GRAPHIC SOLUTION OF TRIEDRAL ANGLES. By a graphic solution is meant a geometrical constructiou. of the required figure, such that the parts sought are deter- mined and shown without the use of algebraic formulae and without computation. Such a solution, often useful of itself and quickly made, serves also as an effective check on the results of numerical computation. PiioB. 1. Given three parts of a triedral angle, to CONSTRUCT THE OTHER THREE PARTS : (a) Given the tliree face angles, a, 5, c : Through any point o of the plane of the paper draw rays OA, OB, DC, oa', making the angles aob, boc, coa' equal to the given ^ace angles c, a, i ; with as centre and any radius, cut OA, oa' in d, d'; draw DF, gd' normal to OB, oc, and meeting each other in e ; through E draw normals to df, gd', and cut these normals, on their positive ends, by the circles fd, gd', i.e. by the cir- cles whose centres are f, g, and whose radii are fd, gd', in H, h'; join FH, GH'; then the plane angle hf-fb is equal to the diedral /3, and the plane angle eg-gh' is equal to the diedral y. For, revolve the right triangles feh, h'eg about fe, eg till EH, eh' are both normal to the plane a and coincide ; and revolve the right triangles dfo, ogd' about ob, oc till OA, oa' coincide in front of the plane a ; then-.- the' right triangles feh, fed have coincident planes, the same base fb, and equal hypotenuse hf, df, .•.the perpendiculars eh, ed are equal. So, eh', ed' are equal, the points h,d, h',d' coincide, and the figure of lem.l is reproduced ; .-. the plane angles hf-fb, eg-gh' are equal to the die- drals /?, y. 122 SPACE TRIGONOMETRY. [V, PR. To construct the diedral a, arrange the face angles in the order a, I, c or h, c, a, and then on as above. {b) Given the three diedrals, a, ft, y : Construct the polar triedral, taking angles equal to a, fi, y for the face angles ; then the three diedrals that are found are equal to the three face angles a, b, c that are songht. (c) Given a diedral and the two adjacent face angles, ft, c, a: Through any point o in the plane of the paper, draw rays OA, OB, oc, making the angles aob^ boc equal to c, a ; with o as centre and any radius, cut OA in D, and draw DF normal to OB at F ; "-■-^o through F draw a line such that the angle of df with this line is equal to the diedral /3, and cut this line on its nega- tive end by the circle fd, in H ; through H draw the normal to df at e ; draw EG normal to oc, cutting the circle OD at d', and through d' draw oa'; then the angle coa is equal to the face angle b. The diedrals y, a may be constructed as in case (a). {d) Given a face angle and the two adjacent diedrals, b,y, a : Construct the polar triedral, taking angles equal to b, y, a for a diedral and the two adjacent face angles ; •then the face angle and two diedrals that are found are equal to the diedral and two face angles /?, c, a that are sought. 1,§6.] GRAPHIC SOLUTIOK OF TRIEDEAL AlfGLES. 123 (e) Given two face angles and an opposite diedral, h, c, § : Through any point o in the plane of the paper, draw the rays 00, OA, OB, making the angles COA, aob equal to the face angles i, c ; with as centre and any radius, cut OA in D ; through D draw GD normal to oc, and dp normal to ob ; through F draw a line such that the angle df makes with this line is the angle /3, and with circle fd cut this Una on its negative end in H ; 'A' through H draw eh normal to df, and with H as centre and radius gd cut de in k ; and through o draw oc ', oc" tangent to the circle ek at g', g"; then either Boc' or BOc" is the face angle a ; and the other parts may be constructed as above. There is no triangle if gdgd>eh; one, an isosceles triangle, if GD = HF; one, if GD>hf. (/) Given two diedrals and an opposite face angle, ft, y, I : Construct the polar triangle, taking angles equal to ft, y, I for two sides and a diedral opposite one of them ; then the face angle and the two diedrals that are found are equal to the diedral and two face angles a, c, a, that are aouffht. 134 SPACE TEIGONOMETRY. 17. FOUR-PART FORJIUL^. [V, TH. The reader will note the complete generality of the proof of the LAW OF COSINES and of the law of sines, no limitation of the sign or magnitude Of any part being imposed, and the con- sequent generality of the formulae that depend upon these laws. THE LAW OF COSINES. Theoe. 5. In a triedral angle : (a) Tlie cosine of a face angle is equal to the product of the cosines of the other two face angles less the product of their sines by the cosine of the opposite diedral : i.e. cos a — cos h cos c — sin b sin c cos a, cos b = cosc cos a — sin c sin a cos /3, cos c = cosa cos b — sin a sinb cos y. For, let o-ABC be a triedral angle, through D any point on the edge OA draw ed normal to the opposite face Boc, and through ed, draw planes normal to the edges OB, oc, cut- ting OB in F, and oc in G ; then the lines df, fe are perpendicular to OB, and eg, gd to oc, 5, §7.j FOUR-PART FORMULA. 125 and •.• the projections on oc of od and of the broken line ofed are equal, [df. and proj ed = 0, [ed perp. to oc. .'. proj OD =proj OF + proj fe, .-. proj OD/oD = proj OF/oD + proj fe/od, .•. proj OD/oD=proj of/of- of/od + proj fe/fe • fe/fd • fd/od, .'. cos COA =: cos cob ■ COS BO A + COS OG-FB • COS FE-DF • sin BOA, i.e. cos5 = cos(-fl)-cos( — c) + cos(-« + R)-cos/3-sin( — c); .■. cos 5 = cose cos a — sine sin« cos/3. [II, theors. 5, 6. Q. E. D. So, •.•the projections on ob of od and the broken line oged are equal, '.•. cos c = cos a cos i — sin a sin i cos y. q. e. d. So, if D be taken a point on oc, and ed be normal to the face c ; then cos « = cos 5 cose — sin J sine cos a. q.e.d. The reader may well examine this proof with care : he will see that it is conclusire ; but he may ask what suggested the several steps in the tenth and eleventh lines. Only this: it was necessary to eliminate the lines which appear in the equation proj OD = proj OF + proj fe, and to bring in the ratios. Dividing by od, the first ratio proj od/od appears at once as one of the ratios sought. But proj of/od is not such a ratio, and the line of that joins the projection of of to OD is used as an intermediary line ; then the ratio proj of/od is written as the product of the two ratios proj of/of, of/od, which can be interpreted . So, the ratio proj fe/od cannot be interpreted, and the two lines fe, fd that join the projection of fe to od are used as intermediary lines ; then the ratio proj fe/od is written as the product of the three ratios proj fe/fe, Fe/fd, fd/od, which can be interpreted. 136 SPACE TRIGONOMETRY. [V, TH. {b) The cosine of a diedral angle is equal to the product of the cosines of the other two diedrals less the product of their sines by the cosine of the opposite face angle : i. e. cos a — cos ft cos y — sin /S sin y cos a, cos ft = cos y cos a — sin y sin a cos h, cos y = cos a cos ft — sin a sin ft cos c. For, let a', V , c', a', ft', y' be the parts of a triedral polar to the given triedral, then-.-a' = a', b'-ft, c' = y, a' = a, ft' = b, y' = c, [theor. 4. and cos b' = cos c' cos a' — sin c' siu a' cos ft', [above. .". cos/?=cos;' cos a — sin y sin a cosb. So, cos ;^ = cos a cos yS — sin a sin /? cos c, cos a = cos ft cos;' — sin yS sin y cos a. Q.e.d. Cor. 1. cos ^^ = V [sin {s — b) sin (s — c)/sin b sin c], [s = i{a + b + c). cos^a= \/\_sin {a — ft) sin {a — y)/ sin ft sinyl, [ff=:^{a + ft + y). For ■.■2cos'\a = \+cosa [II, theor. 13, cor. = 1 + (cos b cos c — cos a)/sin b sin c [(«). = (cos5 cosc + sinS sine — cos «)/sin& sine = [cos {b - c) — cos a']/ sin h sin c [II, theor. 11. = — 2 sin ^{b — c-{-a) sin ^ (b — c — a) /sin b sin c [II, theor. 12. = 2 sin i (a — 5 + c) sin I (« + S — c)/sin b sin c = 3 sin (s — b) sin (s — c)/sin b sin c, .•. cos^a= V[sin {s — b) sin (s — c)/sin& sine] q.e.d. So, ■.■2cos''|a=l + cosa = 1 + (cos ft cos y — cos a)/sin ft sin y, .•.co&^a=i^[sins sin {s — a)/sin {s — b) sin (s — e)], ian^a= »^\sin/ = — cos c, sin a = sin A, sin /3 — sin B, . sin ;/ = sin c, a- = supE, ff— a=A — E, ff — /y = B — E, ff — y = c — 'E, 1-^8 SPACE TRIGONOMETRY. [V, TH. THE LA"VV OF SINES. Theok. 6. In a triedral angle, the sines of the face angles are proportional to the sines of the opposite diedrals. i.e. sin a/ sin a = sin l/sin ft = sin c/sin y. For let o-ABC be a triedral angle, through d any point on the edge OA draw ed normal to the opposite face boo, and through ed, draw planes normal to the edges OB, oo, cut- ting OB in E, and oc in G ; then the lines df, fe are perpendicular to ob, and eg, gd to oc ; and ".• the projections of the broken lines ofd, ogd on ed are equal, [df. proj. and proj OF = 0, projOG=:0, [of, OG pcrp. to ed. .•. proj FD = proj GD, .-. proj FD/oD = proj gd/od, .•. proj FD/FD-FD/oD = proj gd/gd-gd/od, i.e. sinFE-DF-sinOB-OD = sinEG-GD-sinoc-OD ; .". sin ( — /S) sin ( — c) =: sin ;^ sin 5, .'. sin/J sinc = sin ;k sinS and sin 5/sin /3 = sin c/sin /. G, §7.] FOUR-PART FOEMULiE. 139 So, if D 1)6 taken a point on oc, and ed be normal to the face c, then sin a/sin a = sin 5/sin /3 ; .•. sin a/sin a = sin 5/sin yS= sin c/sin y. q.e.d. Cor. If a, l, c, a, p,y le all positive and less than tioo right angles, and a, b, c be the interior diedrals, then : sin a/ sin A = sin h/sin b = sin c/sin c. For ■ . • the angles a, A are supplementary, and so are /?, B and ;>/, C, .'. sinA = sina, sin B = sin/?, sinc = sin ;k- [H theor. 8. Note 1. If the theorem be regarded as relating to a spher- ical triangle, it may be written : The sines of the sides of a spher- ical triangle are proportional to the sines of the opposite angles; and the law of cosines may be expressed in like form. QUESTIONS. If ABC be any spherical triangle, then : 1. sin ^a sin ■^y8/cos^^= sins/sine, 2. sin-|a sin ^i /oos^c = sin + j''V = (cos c cos « — cos 5)/sin c sin a, .'. cos 6 =cos c cos rt — sin c sin a cosyS. So, •.■ cos c = W + mm' + nn' = (cos a + cos y3 — cos y)/sm a sin /?, .•. cos y = cos a cos fi — sin a sin /? cos c ; and so for cos a, cos /?. §9. FIVE-PART FORMULA. Theor. 10. In a triedral angle whose parts are a, i, c, a, ^, y, sin b cos y + cos c sina + sin c cos a cos /S = 0, sin c cos f3 + cos b sin a + sin b cos a cos y = ; sin c cos a + cos a sin b + sin a cos b cos y = 0, sin a cosy + cos c sin b + si7i c cosb cos a = ; sin a cos /3 + cos b sin c + sin b cos c cos a — 0, sin b cos a + cos a sin c + sin a cos c cos /? = ; For, project the broken line gofe ok eg ; then".' jjroj GO = 0, [GOperp. toGE, .■.proj OF + proj FE = GE, .-. proj of/od + proj fe/od = ge/od, .•. proj of/of • of/od + proj fe/fe • fe/fd ■ fd/od ■ =ge/gd-gd/od, .•. cos EG-OF • COS BOA + COS EG-FE • COS FE-FD • sin BOA = COS E6-GD • sin COA, i. e. cos ( - R - «) • COS ( - c) + cos ( - a) • cos ( - /3) ■ sin ( - c) = cos y-&mb ; •.■ — sin a cos c — cos a cos /J sin c = cos y sin I, .'. sin b cos y + cos c sin a + sin c cos a cos /? = 0. Q. e. d. 9, 10, §9.] FIVE-PAET FORMULA. So, project the broken line jogb on fb j 135 E G then-.' proj fo = 0, [fo perp. to fe. .•. proj OG + proj GE = FE, .•. proj OG/oD + proj ge/od = fe/od, .-. proj og/og • og/od + proj ge/ge • ge/gd • gd/od = fe/fd • fd/od, i. e. cos (k — a) • cos coa + cos fe-ge • cos ge-gd • sin ooa = cos fe-fd • sin boa ; .•. sin a-cosS + cos (3E + a)-cos ( — sup y) -sinb = cos{ — /3)-sm{ — c), .'. sin a cos 5 + cos a cosy sinS= — cosyS sine, .'. sine cos/3 + cosy8 sina + sin5 cosa cos/^O. q.e.d. So, with normals drawn to the planes b, c, in turn, the other four formulas may be proved directly ; or they may be inferred by symmetry, from the two formnlse just proved, i.e. the third and fifth from the first, and the fourth and sixth from the second. 136 SPACE TRIGONOMETRY. [V, THS. Cor. sin ficosy + cos c sin a + sin y cos a cos ^ = 0, sin y cos 13 + cos h sin a + sin p cosa cos y = Q ; sin y cos a-\- cos a sin fi + sin a cos I cos y = 0, sin a cos y + cos c sin /S + sin y cosi cos a=0 ; sin a cos j3 + cos i sin y + sin /3 cos c cos a = 0, sin /3 cos a + cos a sin y + sin a cos c cos ft = 0. For •.• sin a/sin a = sin 5/siny3 = sine/sin /, [law of sines. .". sin a, sin i, sin c may be replaced by sin a, sin /?, sin, / in the formulae of the theorem, and those of the corol- lary result directly. Note 1. The formulse of the theorem and those of the corol- lary may be paired in such manner that, if one of them be taken as applying to a triangle, the other is seen to be true for the polar triangle. E.g. the fourth formula of the corollary may be paired with the first formula of the theorem. Such a pair of formulae may be called a pair oi polar formuIcB. Note 2. The law of cosines may be proved by aid of the for- mulae given above, and without the polar triedral. E.g. Multiply the first formula of the corollary by cos/3 and subtract the product from the last formula ; then cos c is eliminated ; and ■.• sin/3 cos a — sin /S cosy3cos;/-l-eosa sin;K(l — cos''/3) = 0, .-. cos a=cos /3 COS;?/ — sin/3 sin^ cos a. q.e.d. So, conversely, the formulae of theor. 10 may be "found from the law of cosines by retracing these steps. questions. Which formula of the corollary may be paired with the second formula of the theorem ? with the third ? with the fourth ? with the fifth ? with the sixth ? Eewrite the formula of the corollary so as to show their cor- relation with those of the theorem, letter for letter and term for term. • 10, 11, §9.j five-pakt formula. 137 napiee's analogies. Theok. 11. In a triedral angle ivliose parts are a, i, c, a, fi, y, tan ^{/3 + y)/tan ^a = — cos ^{b — c) /cos ^{b + c) , tan ^{13 — y)/ tan ^a = —sin^{b — c)/sin^{b-\-c), tan ^{b + c)/tan ^a— — cos ^{/3 — y)/cos ^(/3 + y), tan ^{b — c)/tan Ja = — sin ^{/S — y)/sin l{l3 + y). For, add the fourth and fifth equations of theor. 10, then sin a (cos /3 + cosy) + (l + cos a) sin {b + c) — Q; and ■." sin a/sin a = sin 5/sin j3 = sin c/sin y — (sin b + sin c)/(sin y3+ sin y), .'. sin fl = (sin 5 + sine) sina'/(sinyff + sin^), .". (1 + cos a) sin {b + c) = — (sin b + sin c) sin a (cos y? + cos ;')/(sin /3 + sin y), .". (sin ^ + sin ;(/)/(cos /? + cos y)- (1 + cos a)/sin'iar = — (sin b '+ sin c)/sin (b + c) ; .-. 3 sin !(/? + ;/) cos ^(/?-X) /2 cos ii^ + y) cos l{l3 — y)- cot ^a = — 2sin|-(5 + c) cos^{b — c) /2 sin ^(b + c) cos|-(5 + c), [II, theor. 12. .". tan J(y8 + 7)/tan^a'= — cos^(J — c)/cos^(5 + c). q.e.d. So, ".■ sina/sin a= (sin J — sine) /(sin yS — sin ;k), .•. (1 + cos a) sin (b + c) = — (sin 5 — sin c) sin a (cos yS + cos 7)/(sin /J — sin y), and tan ^(/? - 7)/tan Ja = -sin^(J-c)/sin^(S + c). So, add the first and second equations of theor. 10, cor., then sin a (cos 5 + cos e) + (1 + cos a) sin (^ + ^) = ; and ■.■ sin a = sin a (sin /? + sin 7)/(sin b + sin c), .*. (1 + cosffi) sin {/3 + y) = — (sin /? + sin y) sin a (cos b + cos c)/(sin 5 + sin c), and tan|-(S + c)/tan^a=: -cos|^(/J-;/)/cos^(/3 + X)- 138 SPACE TRIGONOMETRY. [V, THS. So, •.• sin a = sma (sin/J-sin y)/(smb — smc), .: (1 +cosffl) sin {/3 + y) = — (sin ^ — sin y)/sm a (cos ^ + cos y)/{sm h — sin c), and tan|^(J — c)/tan Ja= — sin^(/3-;/)/sin J(/3 + j/). CoE. i/'a, b, c, a, /?, 7 be all positive and less than two right angles, and A, b, c be the interior diedrals, then : tan }{b + c)/cot ^A = cosi{b-e)/cos ^{b + c), tan ^(b — c)/cot |-a = sin^(b — c)/sin J(5 + c), tan ^{b + c)/tan ^a = cos J(b — c)/cos ^(b + c), tan ^{b — c)/tan ^a — si7i^{B — c)/sin^{B + a). Note. Another proof of Napier's analogies is given below : it does not use the formula of theor. 10, and it has the single defect that it employs radicals, and so is not free from ambig- uous signs. tun ^ {/3+y) /tun l-a [II, theor. 11, cor. 1. = (tan J/S + tan ^y)/ta.T:i ^a (1 — tan ^/? tan ^y) / sing sin (s — b) / sins sin (s — c) _' sin (s — c) sin (s — a) ' sin (s — a) sin (s — J) / sins sin (s — «) F sins "1 ' sin(s — S) sin(s — c) [_ sin(s — a)J [theor.5, cor.3. Strike out the common factor Vsins, and multiply both numerator and denominator by \/sin (s — a) sin (s — b) £in (s — c); then tan|-(/3-|-;/)/tan-^a = [sin (s — 5) + sin (s — c)]/[sin (s — a)— sins] = sin la cos j{b- c)/ - sin ^a cos i{b + c) [II, th. 12. = —cos ^{b — c)/ cos -^{b + c) ; q.b.d. and so for the rest. 11, 12, §10.J SIX-PAET FORMTLJE. 139 §10. SIX-PART FOBMUL^. delambre's formula. Theor. 12. In a triedral angle whose parts are a, h, c, a, /?, y, sin^a/sin ^a = =f sin\{'b — c)/sin \{f3 — y), sin^a/cos \a=± sin ^{h + v)/cos i{/3 — y), cos \a/sin \a— ± cos i{b — c)/sin i{/3 + y), cos ^a/cos ia = =F cos i{b + c)/cos ^{13 + y), with like formulse if a, a be replaced by b, § or by c, y: For •.• sin^a/sin''«=(l + cosa) (1 — cosa)/(H-cosa) (1-cosa), aad siri'a/siii'« = sin b sin c/sin yS sin y, [law of sines. .". (1 — cos a)/(l — cos a) = (1 + cos a) sin J sin c/(l + cos a) sin /3 sin y = [(1 — cos «) — (! + cos a) sin b sin c] /[(I — cos a) — (1 + cos a) sin^ siny] [prop. = [1 — (cos ft + sin b sin c cos a) —sin b sin c] /[I -(cos a + sin /3 sin y cosrt)-sin/J siny] = [1 — cos b cos c — sin b sin c] /[I — cos /3 cos y - sin y5 sin y] [law of cos. = [l-cos(5-c)]/[l-cos(^-X)], [add. theor. .-. 2 sin' ia/2 sin" ia = 2 sin" ^(6 - c)/2 sin' ^(/? - y), .•. sin^a/sin|-a;= ^sml{b — c)/sm^{fi-y). So, (1 — cosa)/(l + cosa) = (1 — cosa) sin 5 sin c/(l+cosa) sin>8 s'my, and sin|^a/cos^a= ±sin^(6 + c) cosi{^-y). So, (l + cos«)/(l-cosa) = (1 + cos «) sin h sin c/(l - cos a) sin /? sin >', and cos|a/sinJa= ±cos|-(5-c)/sin|(/? + j/). So, (H-cosa)/(l + cosa) = (1 - cos a) sin 5 sin c/(l - cos a) sin /S sin y, and cos ^a/cos ^a= =f cos f (5 + c)/cos ^(^ + y). 140 SPACE TRIGONOMETRY. [V, TH. Note 1. For any triangle the second members of tliese equa- tions must all have their upper signs or all their lower signs, since Napier's analogies may be got from Delambre's formulEe by division, and must accord with them : first Nap. anal, from third Del. form, by fourth Del. form., second Nap. anal, from first Del. form, by second Del. form., tliird Nap. anal, from second Del. form, by fourth Del. form., fourth Nap. anal, from first Del. form, by third Del. form. Cor. If a, 5, c, a, fi, y ie all positive and less tlian tivo right angles, and a, b, c be tlie interior diedrals, then : sin \a/cos ^a = + sin^{h — c)/sin ^(b — c), sin \a/sin iA = + sin \{h + c)/cos ^(b — c), cos ^a/cos Ja ~ +cos^{b — c)/sin |(b + c), cos ia/siniA= +cos 7^(b + c)/cos ^{b + g). For •-■ A, B, c are the supplements of «, /J, y, .-. sin Ya'^cos-J-A- • -, sin-^(/3— ;k)= — sin|-(B — c)- • •, and "." the first members of these equations are positive, .•. the second members are position. Note 3. For any triangle, the equations sin^a/sin^«= =Fsin J(6 ~c)/sm^{/3 — y), sin ^ J/sin |-/S = =Fsin^(c —a)/sm^{y — a), sin |-e/sin ^y=--^ sin -^{a — &)/sin J{a — /?), must be taken all with the upper sign or all with the lower sign ; and so of the other three groups of like equations. For ■.• sin ^a : sin |« = — sin ^{h — c) : sin ^(/? — y) [up. signs. = sin^« - sin J(5 - c) : sm^a + sin^(y3 - y) = sin^a + sin|-{5 — c) : sin ^a — sin^( /3 — y), .: cos^(s-c) siu^(s-&)/sin J((T — ;/) cos|-(o'-/3) = sin ^(s — c) cos^{s — b)/ COS ^{(T — y) sin^((T — /3), 1] .■. tan^(s-5) cot^(s-c)=cot^(o' — /?) ta,n^{a-y). So, •.■ sin^a : cos^a=: +sinj(5 + c) : 003^{fi — y), [up. signs. 2] .•. tan^s cot j(s — a) = cot^((7 — yS) eot^{ff — y). 13j § lO.J SIX-PART FORMULA. 141 So, •.• COS Ja : sin|a!= +cos^(5-c) : sin ^{/3 + y), [up. signs. 3J .-. cot ^(5 -5) cot J(s - c) = tan ^a cot^{ff-a). So, •.• cos Ja : cos^a= -cosJ(5 + c) : cos i(/3 + y), [up. signs. 4] .-. tan^s tan J(s-a) = cot|-a: cot^{ff — a). So, when the lower signs are taken : 6] cot^(s-5) tan^(s-c)=:cot^(o--/J) ta,Ti^{(}~y), 6] cot^s tan J(s-a) = cot|-(<5' — /3) oot^{ff — y), 7] tan^(s-6) tan|-(.'?-c) = tan|- both right, or both obtuse. It has been shown in geometry that in an ideal triedral : 1. The sum of the three face angles lies between naught and four right angles. 2. The sum of the three interior diedrals lies between two right angles and six right angles. 3. Each face angle is less tlian the sum of the other two face angles, and so of the exterior diedrals. 4. Each interior diedral is greater than the difference be- tween two right angles and the sum of the o'ther two interior diedrals. 5. Of two unequal face angles the greater lies opposite the greater interior diedral, and so opposite the less exterior die- dral, and conversely. 6. If two face angles be equal, so are the opposite diedrals (both exterior and interior), and conversely. 7. A plane through the vertical edge of an isosceles triangle perpendicular to the opposite face, bisects the interior vertical diedral and the opposite face angle. Certain other facts relating to ideal triedrals are manifest, and still others appear by examining formulae already proved. 8. The sine of every part is positive. 9. Every half part is positive and acute, and all its ratios are positive. 10. The half sum of two parts is positive and less than two right angles. 11. The half difference of two parts is acute and its cosine is positive. 14, 15, § 13.] THE IDEAL TEIEDKAL. 147 THE IDEAL TEIAITGLE. Theor. 14. In an ideal triangle, that one of two unequal sides which is nearer right lies opposite the angle which is nearer right, and conversely. For ■.■ that angle which lies nearest to a right angle has the greatest sine, and ■." sin a/sin A = sin 5/sin b = sin c/sin c, .". if sin a > sin i, then also sin a > sin b ; and so of the others. Theoe. 15. In an ideal triangle, a side and its opposite in- terior angle are of the same species, if another side be as near right as the given side, or if another angle be as near right as the given angle. For let the side c be as near right as the given side a ; then'.'coses^ cosa, and cos51, which is impossible. If B, b be equal but not right, the triangle is biquadrantal, for then sin a, sin a, sine are all 1, and a, a, c are all right. If B, b be both right, the triangle is biquadrantal, for then e also is right, and a, a are indeterminate. 2, § 14.] SOLUTION OF IDEAL RIGHT TRIANGLES. 153 If B be nearer right than b and of the same species, there are two triangles. Por ■.• A, a, c are all found from their sines, and sin A, sin a, sin c are all positive, .•. to each of these sines correspond two possible angles, supplementary to each other, both positive and less than two right angles. But A, a must be of the same species ; and if c be" acute. A, a are of the same species with b, b. So, if c be obtuse, a, a are of species opposite to that of b, b, .'. two triangles, and but two, are possible. Geometrically. Let bdb', b'eb be half circles forming lines whose angle is the given angle B of the triangle ; draw an arc CA normal to bob' and equal to the given arc b ; with its initial point c sliding over bob' jDush the arc-normal ca to the right and left till the terminal point A rests on the circle b'eb ; then if b be acute and bB, no triangle. So, if B be obtuse and b>B, two triangles are formed ; if & = B, one triangle, biquad rautal ; itb sin I, then sin c> 1, which is impossible, and there is no triangle. If sin c sin B = sin h, then sin c = 1, c is right, and there is one (a right) triangle if I, B be of the same species, but no triangle if they be of opposite species. If sin c sin B < sin b, then sin C < 1, and C may be either of two supplementary angles ; but these angles must be taken subject to the la w, the greater angle lies opposite the greater side. In particular : If c be nearer right than h, there are two triangles if I, B be of the same species, but none if they be of opposite species. If c be just as near right as b, there is one (an isosceles) tri- angle if h, B be of the same species, but no triangle if they be of opposite species ; and c, c are also of the same species. If c be less near right than b, there is one triangle and c, are of the same species. Geometrically. Lay off an arc ab equal to the given side c ; at B turn by the angle yS, the supplement of B, and lay off the indefinite arc bc ; with A as pole and an arc-radius equal to b describe a small circle : if this small circle neither cuts nor touches the circle BC, there is no triangle ; if it touches the arc BC at a point c, such that BC is positive and less than two right angles, there is a right triangle ; if it cuts the arc bc at one point c, such that bc is a limited arc, there is one triangle ; if it cuts the arc bc in two points Cj, c^, such that BCi, BCj are both limited arcs, there are two triangles. (d) Given b, c, b, two angles and a side opposite one of them : then sinc = sin c-sin J/sin B, tan ^a = tan j{b + c) cos^ (b + c)/cos |- (b — c), cot |-A = tan ^ (b + c) cos ^{b + c)/cos ^(b — c). Check : one of Delambre's formulse. If B, c, b be all right, the triangle is biquadrantal, and a, A are indeterminate and equal. 3, § 15.] SOLUTION OF IDEAL OBLIQUE TRIANGLES. 159 If sin c sin 5 > sin B then sin c > 1, which is impossible, and there is no triangle. If sin sin 6 = sin B, then sinc = l, c is right and there is one (a'quadrantal) triangle if B, b be of the same species, but no triangle if they be of opposite species. If sin c sin 5 < sin B, then sinc Tt-fi, Tt-y- Tt—a, b, Tt—c, Tt—a, fi, Tt-y. Tt—a, Tt — b, c, TC-a, f-A y. §18.J THE GENERAL SPHEKICAL TRIANGLE. 171 DETEEMINATION OF THE SPECIES OF THE PAETS. These sixty-four triangles have been divided into two classes called ^rojjer triangles and improper triangles. To the first class belong the ideal triangle ... 1 and those got from the ideal triangle by reversing : one side or one angle, 6 one side and the opposite angle, 3 two sides and one opposite angle, 6 two angles and one opposite side, 6 two sides and the two opposite angles, ... 3 the three sides and two angles, 3 the three angles and two sides, 3 all the sides and angles, _J^ in all, thirty-two proper triangles. 33 The other thirty-two triangles are improper triangles ; and it will appear that the upper signs of Delambre's formulae must be used in solving proper triangles, and the lower signs in solv- ing improper triangles. Certain limitations must also be observed, as below : Let a, h, c, a, /?, y be the parts of the ideal triangle, and a', V , c', a! , y3', /' the parts of any primary triangle ; let s' s i(«' + &' -f- d) and ff' s i(a' + /?' + r') ; then : 1. If the data make a solution possible, the products sins' sin («'-«') sin (s'-5') sin (s' -c), sine- sin ((?'-«') sin(cr'-^') sin (