Page:Spherical Trigonometry (1914).djvu/39

Case. I.—Three sides and an angle.

42. To express the cosine of an angle of a triangle in terms of sines and cosines of the sides.

Let $ABC$ be a spherical triangle, $O$ the centre of the sphere. Let the tangent at $A$ to the arc $AC$ meet $OC$ produced at $E$, and let the tangent at $A$ to the arc $AB$ meet $OB$ produced at $D$; join $ED$. Thus the angle $EAD$ is the angle $A$ of the spherical triangle, and the angle $EOD$ measures the side $a$.

From the triangles $ADE$ and $ODE$ we have $${\begin{aligned}DE^{2}&=AD^{2}+AE^{2}-2AD\cdot AE\cos A,\\DE^{2}&=OD^{2}+OE^{2}-2OD\cdot OE\cos a;\end{aligned}}$$ also the angles $OAD$ and $OAE$ are right angles, so that $OD^{2}=OA^{2}+AD^{2}$ and $OE^{2}=OA^{2}+AE^{2}$. Hence by subtraction we have $$0=2OA^{2}+2AD\cdot AE\cos A-2OD\cdot OE\cos a;$$

therefore $\cos a={\frac {OA}{OE}}\cdot {\frac {OA}{OD}}+{\frac {AE}{OE}}\cdot {\frac {AD}{OD}}\cos A$;

that is $\cos a=\cos b\cos c+\sin b\sin c\cos A$; ...... (1)

therefore $\cos A={\frac {\cos a-\cos b\cos c}{\sin b\sin c}}$. ....... (2)