Page:Spherical Trigonometry (1914).djvu/42

$${\begin{aligned}\cos a&=\cos b\cos c+\sin b\sin c\cos A,\\\cos b&=\cos c\cos a+\sin c\sin a\cos B,\\\cos c&=\cos a\cos b+\sin a\sin b\cos C.\dots (3)\end{aligned}}$$^[1]

These may be considered as the fundamental equations of Spherical Trigonometry: we shall deduce various formulae from them.

45. To express the sine of an angle of a spherical triangle in terms of trigonometrical functions of the sides.

We have $$\cos A={\frac {\cos a-\cos b\cos c}{\sin b\sin c}};$$

therefore

$${\begin{aligned}\sin ^{2}A&=1-\left({\frac {\cos a-\cos b\cos c}{\sin b\sin c}}\right)^{2}\\&={\frac {(1-\cos ^{2}b)(1-\cos ^{2}c)-(\cos a-\cos b\cos c)^{2}}{\sin ^{2}b\sin ^{2}c}}\\&={\frac {1-\cos ^{2}a-\cos ^{2}b-\cos ^{2}c+2\cos a\cos b\cos c}{\sin ^{2}b\sin ^{2}c}};\end{aligned}}$$

therefore

$$\sin A={\frac {\surd (1-\cos ^{2}a-\cos ^{2}b-\cos ^{2}c+2\cos a\cos b\cos c)}{\sin b\sin c}}\dots (4)$$

The radical on the right-hand side must be taken with the positive sign, because $\sin b$, $\sin c$, and $\sin A$ are all positive, owing to the restrictions of Arts. 22 and 23.

1. These formulae were discovered by ALBATEGNIUS, who made various applications of them. A demonstration of them is given by EULER (Mémoires de Berlin, 1753). All the other formulae of the spherical triangle may be deduced from them, as was shewn by LAGRANGE; Gauss, also, in an appendix to SCHUMACHER's translation of CARNOT's Géométrie de Position, derives all the other formulae from them (GAUSS, Ges. Werke, vol. IV, p. 401).