I found what I believe to be an effortless way of finding the solutions for θ of a multi-angled trigonometry equation based in the format of: $$ a\sin{\left(b\theta\right)}=c$$ $$a\cos{\left(b\theta\right)}=c$$
through the formulas
$$ \theta =\frac{1}{b}\Bigl(\frac{3\pi}{2}n\pm\cos^{-1}{\frac{c}{a}\Bigr)} $$ $$ \theta =\frac{1}{b}\Bigl(2\pi\pm\cos^{-1}{\frac{c}{a}\Bigr)} $$ for the sin & cos equations respectively, where n is a set of integers ranging from plus or minus infinity.
My question is, is there anything else like this method & better? I can't seem to find anything like it. If not, why doesn't it exist? It seems that we're taught to solve these sort of equations in school through a longer method (by using the CAST circle). I've often found myself using these equations in physics exams relating to simple harmonic motion out of pure convenience.
This is $$\sin(b\theta)+\cos(b\theta)=\frac{2c}{a}$$
$$\frac{\sqrt{2}}{2}\sin(b\theta)+\frac{\sqrt{2}}{2}\cos(b\theta)=\frac{\sqrt{2}c}{a}$$ $$\sin(b\theta+\pi/4)=\frac{\sqrt{2}c}{a}$$