Question

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

Answer 1

Given sin A sin B sin C = p ; cos A cos B cos C = q

Hence tan A tan B tan C = tan A + tan B + tan C = p/q

Hence equation of cubic is

now Σ tan A tan B = sin A Sin B cos C + sin B sin C Cos A+ Sin C Sin A  Cos B / Cos A Cos B Cos C

We know that A + B + C =π

cos(A+B+C) = –1; cos(A+B) cos C – sin(A+B) sin C = –1

( cos A cos B – sin A sin B) cos C – sin C (sin A cos B + cos A sin B) = –1

1+ cos A cos B cos C= sin A sin B cos C + sin B sin C cos A + sin C sin A cos B

dividing by cos A cos B cos C

Hence (i) becomes qx³ – px² + (1 + q)x – p = 0