I'm taking far too long to solve these trig equations. I can't seem to find anything that gives me a systematic way to finding all solutions to a trig equation.

For example, I have $\displaystyle f(x) = \sin (2\pi x)$

It's not particularly difficult differentiate and solve this for x, and end up with:

$\displaystyle x = \frac {\cos^{-1}(0)}{2\pi} = \frac {1}{4}$

I can figure that another zero crossing occurs at $\displaystyle -\frac {1}{4}$, and observe that it is zero every $\displaystyle \frac {1}{2}$. But I find myself lacking a more systematic way of approaching these kinds of problems and solving them more efficiently that I do, currently.

Another short example of one I did yesterday. $\displaystyle f(x) = x + 2 \sin x \implies f'(x) = cos (2\pi x) \implies x = \cos^{-1}(-\frac {1}{2})$. After much bashing of head against wall, I worked out that the solutions are all described by $\displaystyle (1 + 2n)\pi \pm \frac {\pi}{3}~ for ~n \in \mathbb{Z}$. But I can't help but think there's a better way to figure this stuff out. It took a lot of head scratching to figure that one out exactly.

So is there some way I can more easily and quickly find every point at which trig equations make a zero crossing, no matter the period and such?

Hope so, because they are becoming a bit of a pain.

Any help or pointers to this information extremely appreciated, and would save me much time and annoyance.