3. Rules for the Twelve Cases of Oblique Angled Spherical Triangles. Fig. 26, 27. Given, Case. | Sought. Rule. 1. Two sides AB, AC, and the included angle A. One of the Draw the perpendicular CD from C the unknown BD=AB - AD. different affection, as AD is less or greater than AB. The third cos AD: cos BD: : cos AC : cos BC, less or greater side BC. than 90°, as A and B D are of the same or of different affection. 2. 3. The side BC. Two angles A and ACB, and AC, the side between them. Draw the perpendicular CD, from C the end of the given side next to BC, the side sought. R : cos AC: : tun A : cot ACD, less or greater than 90°, as AC and A are of the same or of different affection. BCD=ACB, ACD. cos BCD: cos ACD :: tan AC : tan BC, less or greater than 90°, as A and BCD are of the same or of different affection. 4. The third sin ACD; sin BCD: : cos A : cos B, of the same or of different affection with A, as ACD is less or greater than ACB. Two sides AC and BC, and an angle A, opposite to one of them BC. 5. 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°. 6. The included R: cos AC: :tan A : cot ACD, less or greater than 90°, 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 90°; otherwise less than 900, 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. 7. The third R : cos A : : tan AC : tan AD, less or greater than 90°, side AB. 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 90°; 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. Given. Case. Sought. Rule. The side BC sin B : sin A : : sin AC : sin BC, doubtful; unless it opposite to the can be determined by this rule, that according as other given A+B is greater or less than 180°, AC+BC is also angle A. greater or less than 180°. The side AB R : cos A : : tan AC : tan AD, less or greater than 90°, adjacent to the as A and AC are of the same or of different affection, given angles tan B : tan A : : sin AD : sin BD, doubtful. AB= A, B. AD+BD. If AD+BD be greater than 180°, take their difference ; if B D is greater than AD, take their sum for AB. 10. The third angle ACB. R: cos AC: : tan A : cot ACD, less or greater than 90o, as AC and A are of the same or of different affec tion. cos A : cos B : : sin ACD : sin BCD, doubtful. ACB =ACD+BCD. If ACD+BCD exceed 180', take their difference; if BCD be greater than ACD, take their sum for ACB. 4. Rules for the first Ten Cases of Oblique Angled Spherical Triangles, expressed in general terins. 1. =tan I; One of the Find an arc x, so that cot side opposite angle sought tan given angle X sin x =tan angle sought. sin y The third side. Find an arc x, so that -=tan x; and cos first side Xcos y =cos side sought. COST Two angles and the side between them. 4. One of the Find an arc r, so that Scotc; and cos xXtan given side then will Elan side sought. 5. The angle opposite to sin side op. to ang. soughtX sin giv, ang:—sin ang. sought the other gi sin side opposite to given angle ven side. Two sides and an angle opposite to one of them. 6. The angle Find an arc x, so that =cot ; cot given angle given sides: and another y, so that tan same side Xcos x =cos Y ; then will x+y=angle sought. third Find an arc x, so that =lar ; and another y, so that sy; then will x+y=side sought. =sin side sought. other given sin angle opposite to given side angle. two Two angles and a side opposite to one of them. =sin yi 9. The side ad Find an arc , so that =lan I; given cot given side and another y, so that tan other angle then will x+y=side sought. cot and another y, so that cos other given angle =sin y; |