I bet this question has been asked a million times, but I can't find a straight answer. I need to find the length of the hypotenuse in a triangle where I have one side and all the angles.
Example:
Now in the above triangle I have the length of a = 20 and all the angles. How do I - from here - get the length of the hypotenuse (c)?
just use Law of sines (https://en.wikipedia.org/wiki/Law_of_sines): it states that
$$\frac{a}{\sin{\alpha}} = \frac{b}{\sin{\beta}} = \frac{c}{\sin{\gamma}}$$
where $\alpha, \beta, \gamma$ are the angles opposited to sides $a, b, c$ respectively. Since $\gamma$ is a right angle, $\sin{\gamma} = 1$, and therefore in your example $c = \frac{a}{\sin{60°}}$.