How to find missing side lengths with an angle and one length?

In order to solve a triangle, you need three of the six bits of information that define the triangle (3 angles, 3 sides), including at least one side. If you have one angle and one side, you only have two, not three. The triangle cannot be solved.

However, if your triangle is a right triangle, and the angle referred to in the problem statement is not the right angle, then you have the required information. In general, you make use of the definitions of the trigonometric functions. These can be remembered by the mnemonic SOH CAH TOA.