3. Rules for the Twelve Cases of Oblique Angled Spherical Triangles. 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 90°, as A and AC are of the same or of different affection. BD AB AD. sin BD sin AD: tan A: tan B, of the same or of different affection, as AD is less or greater than AB. cos AD: cos BD:: cos AC: cos BC, less or greater than 90°, as A and BD are of the same or of different affection. :: sin BC sin AC : sin A: sin B. The affection of B R: cos AC: tan A: cot ACD, less or greater than 90°, R: cos A tan AC: tan AD, less or greater than 90°, Rule. sin B: sin A :: sin AC: sin BC, doubtful; unless it can be determined by this rule, that according as A+B is greater or less than 180°, AC+BC is also greater or less than 180°. R: cos A tan AC: tan AD, less or greater than 90o, as A and AC are of the same or of different affection. tan B: tan A: sin AD: sin BD, doubtful. AB= AD+BD. If AD+BD be greater than 180°, take their difference; if BD is greater than AD, take their sum for AB. 12. A side AC. cos AC= sin AC. sin AB ̄ sin ( ¦ S—A C). sin (§ S—A B). rad3 sin S. sin (S-BC) Let S denote the sum of the three angles; then will ad2]. 4. Rules for the first Ten Cases of Oblique Angled Spherical Triangles, expressed in general terms. |