Question

Given the product p of sines of the angles of a triangle and the product q of their cosines, find the cubic equation whose coefficients are functions of p and q and whose roots are the tangents of the angles of the triangle.

Answer 1

$\sin A\sin B\sin C=p$, $\cos A\cos B\cos C=q$
$\therefore \tan A\tan B\tan C=p/q=S_3$
$\therefore S_1=S_3=\tan A+\tan B+\tan C$
If $S_2=\tan A\tan B+\tan B\tan C+\tan C\tan A$
$=\frac{\sin A\sin B\cos C+\sin B\sin C\cos A+\sin C\sin A\cos B}{\cos A\cos B\cos C}$
$N^r$ is $\sum \left(twosinesone\cos \right)$.
$=\frac{1+\cos A\cos B\cos C}{\cos A\cos B\cos C}=\frac{1+q}{q}$
Hence the required equation whose roots are $\tan A,\tan B,\tan C$ is
$x^3-x^2{S}_1+x{S}_2-S_3=0$
or $x^3-\frac{p}{q}{x}^2+\left(\frac{1+q}{q}\right)x-\frac{p}{q}=0$
or $q{x}^3-p{x}^2+(1+q)x-p=0$.