Find the $\theta$ of $\frac{a}{\cos\theta}+\frac{b}{\sin\theta}=c$

Question

I am wondering how to find the $\theta$ of $\frac{a}{\cos\theta}+\frac{b}{\sin\theta}=c$, given $a, b, c\in\mathbb{R}_{>0}$, and $a+b\leq c$. Thanks.

Answer 1

Write your equation in the form $$a\sin(x)+b\cos(x)=c\sin(x)\cos(x)$$ and Substitute $$\sin(x)=\frac{2t}{1+t^2}$$ and $$\cos(x)=\frac{1-t^2}{1+t^2}$$