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

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

I can figure that another zero crossing occurs at , and observe that it is zero every . 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. . After much bashing of head against wall, I worked out that the solutions are all described by . 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.