Linear equations in sine and cosine

Linear equations in $\sin$ and $\cos$ take the form $a\sin x + b\cos x + c = 0$, with $a,b,c\in\mathbb{R}$ and $(a,b)\ne(0,0)$. There are two classical methods, both valid, for solving them: the parametric-formulae method, which uses the substitution $t=\tan(x/2)$, and the auxiliary-angle method, which rewrites $a\sin x + b\cos x$ as a single sine function.

• The parametric-formulae method
• The auxiliary-angle method