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=\frac12\implies\theta=\frac\pi6+2\pi k,\theta=\frac{5\pi}6+2\pi k`

where
`k`is any integer.