Q:

Solving $\cos x + 3^{1/2} \sin x = 1$ for $0\leq x \leq 360^{\circ}$

$\cos x + \sqrt3 \sin x = 1$
Not sure what the step would be to get an answer.

A:

Note that our expression is equal to $2\left(\frac{1}{2}\cos x+\frac{\sqrt{3}}{2}\sin x\right)$. Find an angle $\phi$ such that $\sin\phi=\frac{1}{2}$ and $\cos \phi=\frac{\sqrt{3}}{2}$. Then our expression is equal to $2\sin(\phi+x)$.
Remark: The start you mentioned in a comment will work, though it is a bit messier. Divide by $\cos x$. We get $1+\sqrt{3}\tan x=\sec x$. Square both sides. We get 
$$1+2\sqrt{3}\tan x+3\tan^2 x=\sec^2 x=\tan^2 x+1.$$ Solve for $\tan x$, and then for $x$. 
Since we squared, we may have introduced extraneous solutions, so we need to check that the candidate solutions we find are solutions of the original equation.

