How do you find the hypotenuse of a right triangle provided with one side and one angle length.

It depends upon where the side is compared to where the angle is:

If the side is opposite the angle (across the triangle from the angle) then use sin:  sin(A) = a / hyp                    where you know the values for the angle A and the opposite side a.

If the side is adjacent to the angle (next to it), then cos:  cos(A) = b /                                                             w;here you know the values for the angle A and the adjacent side b.