Page:Spherical Trigonometry (1914).djvu/39

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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 $$ \begin{align} DE^2 &= AD^2 + AE^2 - 2AD \cdot AE \cos A,\\ DE^2 & = OD^2 + OE^2 - 2 OD \cdot OE \cos a; \end{align} $$ 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 $$ 0 = 2 OA^2 + 2AD \cdot AE \cos A - 2 OD \cdot OE \cos a; $$

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)