TITLE: Trigonometry: Solve $(1-\cos\alpha)^2 + \sin^2\alpha = d^2$ for $\alpha$
QUESTION [6 upvotes]: My next step in implementing my algorithm in Java is following.
It is quite difficult to explain, but I know what I need. I have this equation:

Given: d
Asked: $\alpha$
$$(1-\cos\alpha)^2 + \sin^2\alpha = d^2 $$

Which I "simplified" to this, using this formula's:
$$2+2\cos\alpha+\frac{\cos2\alpha}{2}-\frac{\sin\alpha}{2} = d^2$$
But now, I'm stuck. This is probably pretty easy, but I'm 15 years old at the moment. I didn't see that much trigonometry in school yet.
Can you help me?

REPLY [5 votes]: A slightly more general tip that I sometimes tell my students:
If an equation contains more than one trig function, it might be nice if there were a way to rewrite it using only one trig function.
Sometimes that's easy and sometimes that's hard. But if you see precisely $\sin^2 \alpha$ or $\cos^2 \alpha$, those are very easy to rewrite using another trig function.
(The identity $\sin^2 \alpha + \cos^2 \alpha = 1$ is far and away the most important trig identity of them all.)