30°- 60°- 90° and 45°- 45°- 90° Triangles

Before we jump to several practice problems dealing with calculating trig functions, we must remind ourselves of the information we have learned in the past concerning 30°- 60°- 90° and 45°- 45°- 90° triangles.

Recall that there are very special relationships between the lengths of the legs of these two triangles. In particular, in the case of a 45°- 45°- 90° triangle, we know that the legs are both the same length (which makes this an isosceles triangle) and that the hypotenuse is exactly times the length of either of the legs.

So, for example, if a 45°- 45°- 90° triangle has legs of length 7, then the hypotenuse has length .

The case for the 30°- 60°- 90° is somewhat different, especially since it is not an isosceles triangle. If the side opposite the 30° angle has length a, then the length of the hypotenuse is 2a and the leg opposite the 60° angle has length a.

Thus, for example, if the length of the shortest leg in a 30°- 60°- 90° triangle is 5, then the length of the hypotenuse is 10 and the length of the leg opposite the 60° angle is .