Page:Spherical Trigonometry (1914).djvu/41

(3) Suppose that one of the sides which contain the angle $A$ is a quadrant, for example $AB$; on $AC$, produced if necessary, take $AD$ equal to a quadrant and draw $BD$. If $BD$ is a quadrant, $B$ is a pole of

$AC$ (Art. 11); in this case $a={\tfrac {1}{2}}\pi$ and $A={\tfrac {1}{2}}\pi$ as well as $c={\tfrac {1}{2}}\pi$. Thus the formula to be verified reduces to the identity $0=0$. If $BD$ is not a quadrant, the triangle $BDC$ gives $$\cos a=\cos CD\cos BD+\sin CD\sin BD\cos C{\hat {D}}B,$$

and $\cos C{\hat {D}}B=0$, $\cos CD=\cos({\tfrac {1}{2}}\pi -b)=\sin b$, $\cos BD=\cos A$;

thus $\cos a=\sin b\cos A$;

and this is what the formula in Art. 42 becomes when $c={\tfrac {1}{2}}\pi$.

(4) Suppose that both the sides which contain the angle $A$ are quadrants. The formula then becomes $\cos a=\cos A$; and this is obviously true, for $A$ is now the pole of $BC$, and thus $A=a$.

Thus the formula in Art. 42 is proved to be universally true.

44. The formula in Art. 42 may be applied to express the cosine of any angle of a triangle in terms of sines and cosines of the sides; thus we have the three formulae,