How to solve?

$\sin 2 x - 2 \cos x + 0.25 = 0$

Jan 24, 2018

$\sin \left(2 x\right) = 2 \sin x \cos x$ then

$\sin 2 x - 2 \cos x + 0.25 = 2 \sin x \cos x - 2 \cos x + \frac{1}{4} = 0$

then

$\cos x = \frac{1}{8 \left(\sin x - 1\right)}$ but

${\sin}^{2} x + {\cos}^{2} x = 1$ so

${\sin}^{2} x + \frac{1}{64 {\left(\sin x - 1\right)}^{2}} = 1$ and now solving for $\sin x$ we have the real solutions as

$\sin x = \left\{\begin{matrix}- 0.998041 \\ 0.794264\end{matrix}\right.$

The next steps are left to the reader.