When hunting mathematical abstractions, your strategy is key. (Or maybe your tactics; we can never keep the two straight.)
If you want to catch a trigonometric expression, we use an angle as bait. We've done a lot of this already. "See that tangent up there? Caught that using an empty swimming pool, five rolls of duct tape, and 
."
Now we're going to turn things around: put on your safari hat, because we're going hunting for wild angles. We'll use the skills we've honed so far to track them back to their lairs, hiding within trigonometric equations.
The unit circle will continue to be our #1 tool of the trade for solving problems. By now, we can find the x- and y-coordinates for every variation of 
we care to name. We suggest drawing your own unit circle, with all the angles between 0 and 2π filled out, to use as a reference.
We also suggest drawing Batman fighting a dinosaur, because that is seriously cool.