3B2 . 1.2.3: cos A =1 2. Given the very acute angles A, B, and the side c to find the remaining parts. B3 sin A = A sin B=B1.2.3 ; cos B=11.2.3 1.2.3 (A+B)3 Hence, sin C=sin (A + B) = sin A cos B + cos A sin B=(A + B) 1.2.3 c sin A CA B(2A + B) Wherefore, a = It and sin (A + B) A+B 1.2.3 A(2B + A) 1+ 1.2.3 { { c sin B A}} 3. Given the two sides a, b, and the included angle C, which is very obtuse, to find the other parts of the triangle. a? a? c = a? — 2ab cos C + b2 =a? + 12 + 2ab 1.2 = (a + b)2 – ab a’. ab a? Whence c= n(a + b)2 - ab a’ = a + b nearly. a + b 2 a3 Again, sin A== sin C= 1.2.3 a a a } nearly; and с с a {a{ a a ab af с ab .a a)-1 1 a+b a +61 a + b 2 (a + b)2 2 ab 1+ atbl very nearly. (a + с a3 Sin A= ab a? a? 1.2.3 2(a+b)2 1.2.3 1 sin’A Also sin A = A sin’A 1.2.3' a 1 13 a a +65 '1.2.3 nearly. (a + b)21.2.3) In the same manner we have (a - b) a aa 1 nearly. atbl (a + b)2 1.2.3) These angles being understood to be in circular measure, as explained in the preceding note. For further information on subjects of this nature, the reader is referred to Bonnycastle's Trigonometry, Cagnoli Trigonométrie, several of the authors on Geodesy, especially Puissant, and to the second volume of this work. No. Square. C'ube. Sq. Root. C'ube Root No. | Square. Cube. Sq. Root. Cube Root 1 4 16 25 36 49 64 81 100 9 10 11 12 13 14 15 16 17 18 19 20 33 1 8 27 64 1:25 216 343 512 7:29 1000 1331 1728 2197 27-14 3375 4096 4913 5832 6859 8000 9:26] 10648 12167 13324 15025 17576 19683 21952 24389 27000 29791 32768 33937 39304 42875 466.56 50653 54872 59319 64000 68921 74088 79507 85184 91125 97336 103823 110592 117649 125000 132651 140608 148877 157464 166375 175616 185193 195112 205379 216000 226981 238328 250047 262144 274625 287496 300763 314432 328509 343000 357911 373248 1.0000000 1.000000 73 5329 1:4142136 1-259921 5176 1.7320508 1:44°250 75 5625 2:0000000 1.587 101 76 5776 22360680 1.709976 77 5929 2:4494897 1.817121 78 6084 2.6457513 1.912931 79 6241 2.8:84271 2.000000 80 6400 3.0000000 2-080084 6561 3:1692777 2.154135 82 6724 3:3160218 2.223980 83 6889 3.4641016 2:289429 84 7056 3-6055513 2:351335 85 7225 3.7416574 2:410142 86 7396 3.8729833 2:466212 87 7569 4.0000000 2.519842 88 7744 4:1231056 2:57 1282 89 7921 4:24:26-407 12 620741 90 8100 4:3588989 2-668102 91 8281 4.4721360 2.714418 92 8164 4.5825757 2.758924 93 8619 4.6904158 2.802039 94 8836 4.7958315 2-843867 95 9025 4.8989795 2.884499 96 9216 5-0000000 2.924018 97 9409 5-0990195 2.962496 98 9604 5.1961524 3.000000 99 9801 52915026 3.036589 100 10000 5.3851618 3.072317 101 10201 5.4772256 3:107232 102 10104 5 5677644 3:141381 103 10609 5.6568512 3:174802 104 10816 5.7445626 3.207534 105 | 11025 5-8309519 3:239612 106 11236 5.9160798 3.271066 || 107 11449 6:0000000 3 301927 108 11664 6:08:27625 3:33:1-22 109 11881 6.1644140 3.361975 110 12100 6-24-19980 3391211 111 12321 6.3245553 3.419952 112 12544 6.4031242 3:448217 113 12769 6:4807407 3:476027 114 12996 6.5574385 3.503398 115 | 13225 6.63324196 3:530348 116 13456 6.7082039 3:556893 117 13689 6.7823300 3:583048 118 | 13924 6.8556516 3.6088:26 119 14161 6.9282032 3.631241 120 14400 7.0000000 3659306 12114641 7.0710678 3681031 122 14884 7.1414284 3.708430 || 123 15129 7.2111026 3.732511 124 | 15376 7.2801099 3-756286 125 15625 7.3484692 3:779763 126 15876 7.4161985 3.802952 127 | 16129 7.4833148 3.825862 1:28 16384 7.5498314 3.848501 129 16641 7.6157731 3.870877 || 130 16900 7.6811457 3:892996 || 131 | 17161 7.7459667 3.914868 132 17424 7.8102497 3.936497 133 17689 7.8740079 3.957891 134 17956 7.9372539 3:979057 135 18225 8.0000000 4.000000 136 18496 8:0622577 4:020726 137 18769 8.1240384 4.041240 138 19044 8.1853528 4.061548 139 19321 8.2462113 4.081655 140 19600 8:3066239 4:101566 141 19881 8.3666003 4:121285 142 20164 8-4261498 4.140818 143 20449 8.4852814 4.160168 144 20736 1:21 144 169 196 2:25 256 289 324 361 400 441 484 5:29 576 625 676 7:29 784 841 900 961 1024 1089 1156 12:25 1296 1369 1444 1521 1600 1681 1764 1849 1936 20:25 2116 2209 2304 2401 2500 2601 2704 2809 2916 3025 3136 3249 3364 3181 3600 3721 3844 3969 4096 4225 4356 4489 4624 4761 4900 5041 5184 389017 | 8:5410037 4:179339 405:"24 8.603253 4198336 421875 8.6602540 4:217163 438976 | 8.7177979 4:235824 456533 8.774.9644 4251321 474552 8.8317609 4.272659 493039 8.8881944 4.290810 51:2000 8.9442719 4.308869 531441 9.0000000 4:326749 551368 9:0553851 4.314481 571787 9:1104336 4:362071 592704 9:1651514 4:379519 610125 9.2195445 4.396830 636056 9.2736185 4:414005 658503 9-3273791 4:431048 681472 9:3808315 4:447960 704969 9.4339811 4:464745 729000 9:4868330 4.481405 753571 9-5393920 4:497941 778688 9:5916630 4:514357 804357 9.6436508 4:530655 830584 9.6953597 4.546836 857375 | 9.7467943 4:562903 884736 9.7979590 4.578857 912673 9-8488578 ! 4.594701 9411929-8994949 4:610436 970-299 | 9.9498744 4:626065 1000000 10.0000009 i 4.641589 1030301 10-0498756 4:657009 1061208 10-0995049 4:672329 1092727 10:1488916 4.687548 1124864 10.1980390 4.702669 1157625 10-24695084.717694 1191016 10:2956301 4.732623 1225013 10:3440804 4:747459 1259712 10:39:23048 4:762203 1295029 10.4403065 4.776856 1331000 10-4880885 4.791420 1367631 10-5356538 4.805895 1404928 10:5830052 4.820284 1442897 10-6301458 4:834588 1481544 10 6770783 4.848808 1520875 10.7238053 4.862944 1560896 10 7703296 4.876999 1601613 10.8166538 4.890973 1643032 10-8627805 4.904868 1685159 10.9087121 4.918685 1728000 10.9544512 4.932424 1771561 11.00000004.946087 1815848 11.0453610 4.959676 1860867 11.0905365 4.973190 1906624 11.1355287 4.986631 1953125 11•1803399 5.000000 2000376 11.2249722 5.013298 2048383 11:2694277 5.026526 2097152 11:3137085|5-039684 2146689 11:3578167 5.052774 2197000 11.4017543 5:065797 2248091 (11:4455231 5:078753 2299968 11:48912535.091643 2352637 11.5325626 5.104469 2406104 11:5758369 5:117230 2460375 11.6189500 5:129928 2515456 11.6619038 5:142563 2571353 11.7046999 5.155137 2628072 11.7473401 5°167649 2685619 11.7898261 5.180101 2744000 11.8321596 5:19-2494 2803221 11.8743422 5.204828 2863288 11.9163753 5'217103 2924207 11:9582607 5.229321 2985984 12:0000000 5.241483 37 39 40 41 42 43 44 45 46 48 49 50 51 52 53 54 55 56 57 58 59 60 61 63 64 65 66 67 68 69 70 71 72 VOL. I. M m 145 21025 3048625 12:0415946 5.253588 217 47089 10218313 14.7309199 | 6.009245 10648000 14:8323970 6•036811 289 8 349 No. Square. Cube. Sq. Root. Cube Root' No. Square. Cube. Sq. Root. Cube Root 289 83521 24137569 17-00000006611489 361 130321 47045881 |19:0000000 7•120367 94249 28934443 17:5214155 6.745997 379 | 143641 54439939 19.4679223 7.236797 324 104976 34012224 18:00000006 868285 396 156816 62099136 119.8997487 7.343420 3:26 106276 34645976 18:0554701 6:882389 398 158404 63044792 19.9499373 7.355762 345 119025 41063625 18:5741756 7:013579 || 417 173889 72511713 20:4205779 7.470999 350 122500 42875000 18.7082869 7.047299 422 178084 75151448 20:5426386 7.500741 2M2 No. Square. Cube. Sq. Root. Cube Root No. Square. Cube. Sq. Root. Cube Root 433 i 187489 81182737 | 208086520 7·565355 505 255025 128787625 | 22:4722051 7.963374 |
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