TITLE: Solve $\sin(x)+2\sin(x)\cos(x)=\pi/4$
QUESTION [6 upvotes]: Is it possible to solve (not approximate) the following trigonometric equation by hand? 
$$\sin(x)+2\sin(x)\cos(x)=\pi/4.$$

REPLY [3 votes]: If you are willing to use $\arcsin$ (or $\sin^{-1}$), then yes.
This can be rewritten as a quartic in $\sin x$ which is theoretically solveable by hand (thought it might be very tedious).
Now you take the appropriate roots of those...