Determining the general solution for the trigonometric equation $ 5\cos(x)-12\sin (x) = 13 $ Given that
$$5\cos(x)-12\sin (x) = 13 $$
I'm trying to evaluate the general solution for that expression. It reminds me of $5-12-13$ triangle. Since we don't know the degree of $x$, I couldn't proceed further. Specifically, let's take its derivate, which yields 
$$\dfrac{d}{dx} 5\cos(x)-12\sin (x) = 13  = 0$$

Hint
By Cauchy Schwarz you have
$$169=13^2=(5\cos(x)-12\sin (x))^2 \leq (5^2+12^2)(\cos^2(x)+\sin^2(x))=169$$
Therefore, you must have equality in CS, and hence 
$$\frac{\cos(x)}{5}=\frac{\sin(x)}{-12}$$
Now combine this equality with $$\sin^2(x)+\cos^2(x)=1$$
