I am attempting to answer a set of questions where I need to find all possible values of $x$, in the range $0 < x < 2\pi$, as a fraction of $\pi$, in a question such as:
$$\cos x = 1$$
For this one, and a few others, I could figure out how to reach the answer - normally either by 'special' triangles or using the unit circle. I reached this question, though, and could only come up with one part of the answer:
$$\sin x = \frac{1}{2}$$
I can see how to do this from a right angled triangle with sides $1$, $\sqrt{3}$, and $2$ - it's relatively straight forward to see that the sides with length $1$ and $2$ are the $O$ and $H$ part of $\sin \theta = \frac{O}{H}$. My problem here is how I'd figure out the other value of $x$ in the range $0 < x < 2\pi$ - I can't see how I can use a right angled triangle, and on the unit circle, I know that $sin \theta$ corresponds to the $y$ coordinate for points on the circle - but I can't see how I can use this to determine the angle.
Is there a way that makes sense to figure this out, other than just using a calculator?
