Page:Spherical Trigonometry (1914).djvu/42

24 $$ \begin{align} \cos a &= \cos b \cos c + \sin b \sin c \cos A, \\ \cos b &= \cos c \cos a + \sin c \sin a \cos B, \\ \cos c &= \cos a \cos b + \sin a \sin b \cos C. \dots (3) \end{align} $$

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

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

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.