It’s hard to explain without pictures, but I’ll try.

Well, first off. Law of cosines:

a^2 = b^2 + c^2 – 2bc*cos(A)

The lowercase letters are the lengths of the sides, and A is the angle that’s opposite of side a. (Yes, this works for all triangles, not just right triangles)

So you plug in the side lengths, get the cos(A), and stick that into a calculator with inverse cosine (some books call it arc-cosine or something) to get the angle A.

I think it’d be easiest to draw the triangle so that 7 is horizontal and on the bottom, then draw a line perpendicular to it and going through the other angle. This will divide your triangle into two right triangles. Angle A can either be the angle between 5 and 7 or between 3 and 7.

Now you have the angle and the hypotenuse. The height is the opposite, so you use inverse sine (arc-sine) to get the height.