Unizor - Trigonometry - Conditional Identities - Problems 2
Channel: Zor Shekhtman on UNIZOR Education
Date: 2013-09-12

These are less trivial conditional identities and solving them requires some ingenuity.
We strongly suggest you to try to solve these problems by yourself before listening to the lecture with solutions.

Problem A
Given that solutions of an equation
x³+ax²+bx+c = 0
are tan α1, tan α2 and tan α3
and solutions of an equation
x³+cx²+bx+a = 0
are tan β1, tan β2 and tan β3.
Prove that
α1+α2+α3+β1+β2+β3 = K·π
where K is some integer number.
Hint:
Since tan α1, tan α2 and tan α3 are solutions of the equation x³+ax²+bx+c = 0, the following is true:
x³+ax²+bx+c =
(x−tan α1)(x−tan α2)(x−tan α3).

Problem B
Given angles α, β and γ that are angles of some triangle and
sin α : sin β : sin γ = 4:5:6.
Prove that
cos α : cos β : cos γ = 12:9:2.
Hint:
Use Law of Sines and Law of Cosines.