Question

Cos(2theta)=11 sin(theta)-5

Answer 1

$\cos2\theta=11\sin\theta-5$
$1-2\sin^2\theta=11\sin\theta-5$
$2\sin^2\theta+11\sin\theta-6=0$
$(2\sin\theta-1)(\sin\theta+6)=0$
$\implies\begin{cases}2\sin\theta-1=0\\\sin\theta+6=0\end{cases}$

The second equation gives $\sin\theta=-6$, but $-1\le\sin\varphi\le1$ for any real $\varphi$, so we can ignore this equation.

This leaves us with

$2\sin\theta-1=0\implies \sin\theta=\dfrac12\implies\theta=\dfrac\pi6+2\pi k,\theta=\dfrac{5\pi}6+2\pi k$

where $k$is any integer.