2. Given the very acute angles A, B, and the side c to find the remaining parts. Since A and B are very small, we have very nearly Hence, sin C = sin (A + B) = sin A cos B + cos A sin B = (A+B) 1.2.3 (A+B)3 ̧ 3. Given the two sides a, b, and the included angle C, which is very obtuse, to find the other parts of the triangle. Put C 180° . -a: then, since C is very obtuse, a is very small, and a2 c2 = a2 — 2ab cos C + b2 = a2 + b2 + 2ab { 1 — 1.3 } = (a + b)2 — ab a2. Whence c√ (a + b)2. · ab a2 = a + b α Again, sin A== sin C = = sin a == { C 1.2 ab a2 a+b 2 απ {1+ a3 1.2.3 ab nearly. } nearly; and -1 these values, we have }{a α a2 — ab + b2 a2 A3 (a + b) 1.2.3 = A 1.2.3) sin3A 1.2.3 a2-ab+b2 a2 2 ab a2 2(a+b)2 a2 1 1.2.3 nearly; or Asin A + 2(a+b)3 1.2.3} + {a (a - b) b a2 nearly. (a - b) a a2 nearly. In the same manner we have sin A 1.2.3' (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. 5-19615243-000000 99 9801 9702999-9498744 4-626065 1000000 10-0000009 4641589 1030301 10-0498756 4-657009 1061208 10-0995049 4-672329 1092727 10-14889164-687548 1124864 10-1980390 4-702669 1157625 10-2469508 | 4-717694 1191016 10-2956301 4-732623 1225043 10:3440804 4-747459 1259712 10-3923048 4-762203 1295029 10-4403065 4-776856 1331000 10-4880885 4-791420 1367631 10:5356538 4-805895 1404928 10-58300524-820284 1442897 10-6301458 4·834588 1481544 10-6770783 4848808 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 11-0000000 4-946087 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 11-4891253 | 5-091643 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 5·192494 2803221 11·8743422 | 5·204828 2863288 11-9163753 5 217103 2924207 11-9582607 5-229321 2985984 12 0000000 5-241483 VOL. I. M m No. Square. 145 21025 148 21904 149 22201 155 24025 156 24336 157 24649 158 24964 159 25281 165 27225 166 27556 167 27889 168 28224 169 28561 170 28900 171 29241 172 29584 173 29929 Sq. Root.. Cube Root No. Square. 237 56169 238 56644 Cube. 204 41616 206 42436 10218313 14-7309199 6·009245 No. Square. Cube. Sq. Root. Cube Root No. Square. Cube. 4943086319-1572441 49836032 19-1833261 5024340919-2093727 7-159599 7·166096 7-172581| Sq. Root. Cube Root 305 306 3338624817-9443584 | 6-854124 50653000 19-2353841 7 179054| 7-185516| 7-191966- 7·198405 5189511719-3132079 52313624 19-3390796 7·204832| 371 137641 379143641 380 144400 392 153664 393 154449 54010152 19-4422221 7-230427 381 145161 55306341 19-5192213 7-249504 339 114921 38958219 18:4119526 6.972683 411 168921 3930400018·4390889 | 6·979532 412 169744 2M2 No. Square. Cube. Sq. Root. Cube Root No. Square. Cube. Sq. Root. Cube Root 557 310249 172808693 23-6008474 8-227825 |