For, the rectangle contained by the sines of any two sides is to the square of the radius, as the rectangle contained by the sines of half the perimeter and its excess above the base, to the square of the cosine of half the angle opposite to the base, by Prop. 30; and as the rectangle contained by the sines of the two excesses of half the perimeter above each of the sides to the square of the sine of half the angle opposite to the base, by Prop. 29: therefore, the rectangle contained by the sines of half the perimeter and its excess above the base, is to the rectangle contained by the sines of the two excesses of half the perimeter above each of the sides, (as the square of the cosine to the square of the sine of half the angle opposite to the base, that is,) as the square of the radius to the square of the tangent of half the angle opposite to the base. Q. E. D. * Solution of the several Cases of Right Angled and Oblique Angled Spherical Triangles. GENERAL PROPOSITION. In a right angled spherical triangle, of the three sides and three angles, any two being given, besides the right angle, the other three may be found. 1. Rules for the sixteen Cases of Right Angled Spherical Trigonometry. See Fig. 16. sin C: cos B *:: R:cos AC* (22.) \tan B:cot C :: R:cos BC (19.) 15. The two quantities, in the same analogy, marked with asteriscs, are both of the same affection ; that is, both at the same time less, or both at the same time greater, than 90°. Cases 7, 8, 9, are doubtful, for two triangles may have the given things, but have the things sought in one of them the supplements of the things sought in the other. In the remaining cases, if the two given quantities which occur in the previous terms of the same analogy are both of the same affection, the last term will be less than 90°; but if they are of different affection, the last term will be greater than 90°. These limitations are founded on the 13th, 14th and 15th Propositions. 2. Rules for the sixteen Cases of Right Angled Spherical Trigonometry, expressed in general terms. Given. Case. Sought. Rule. rad The hypotenuse 2 side adjacent to 1 rad Xcos given angle =tangent. and one angle. given angle. Si cot hypotenuse =cotangent. rad X sin given side The other side. =tangent. cot given angle A side and its adjacent angle. / 5. The hypotenuse. rad X cos given angle =cotangent. tan given side cos given side X sin given angle_onsine. The other angle. cos gi rad tan given side x cot given angle_sine. The other side. rad A side and its radxsin giveu side The bypotenuse. opposite angle. 8. sin given angle =sine. 9. The other angle. rad Xcos given angle =sine. cos given side 10. The other side. rad Xcos hypotenuse =cosine. cos given side The bypotenuse u. lAngle opposite rad X sin given side and a side. =sine. rad =cosine. The two sides. rad =cotangent. tan other side 222 14. A side. 15. The two angles. | 16. rad xcos opposite angle =cosine. sin other angle rad The hypotenuse. 3. Rules for the Twelve Cases of Oblique Angled Spherical Triangles. Fig. 26, 27. Given. Case. Sought. Rule. Draw the perpendicular CD from C the unknown angle, not required, on AB, R: cos A :: tan AC : tan AD, less or greater than 900, as A and AC are of the same or of different affection. BD=AB, AD. sin BD : sin AD: :lan A : tan B, of the same or of different affection, as AD is less or greater than A B. cos AD: cos BD : : cos AC : cos BC, less or greater than 90°, as A and B D are of the same or of different affection. The third The angle B, sin BC: sin AC: : sin A : sin B. The affection of B opposite to the is doubtful; unless it can be determined by this rule, other given that according as AC+BC is greater or less than 180°, side AC. | A+B is also greater or less than 180°. The included R:cos AC :: tan A : cot ACD, less or greater than 900, angle ACB. as AC and A are of the same or of different affection. tan BC: tan AC :: cos ACD: cos BCD greater than 90°, if one or all of the 3 terms ACD, AC, BC, are greater than 900; otherwise less than 90°. ACD + BCD=ACB, doubtful. If ACD+BCD ex ceed 180°, take their difference ; if BCD is greater than ACD, take their sum, for the angle ACB. The third R: cos A : : tan AC : tan AD, less or greater than 90°, side AB. L as A and AC are of the same or of different affection. cos AC : cos BC::cos AD:cos BD, greater than 90°, if one or all of the 3 terms AD, AC, BC, are greater than 900; otherwise less than 90°. AD+BD=AB, doubtful. If AD+BD exceed 180°, take their difference ; if BD is greater than AD, take their sum for AB. 7. The 4. Rules for the first Ten Cases of Oblique Angled Spherical Triangles, expressed in general terins. Given. Rule. Case. Sought. One of the Find an arc x, so that cot side opposite angle sought =tan angle sought. sin y Two sides and the included angle. |