Special Angles - 30o, 45o, 60o

The values of the six trigonometric ratios cannot be calculated exactly for most angles. Nor can the exact value of an angle generally be found given the value of one of the ratios. There are, however, three special angles that lend themselves nicely to ratio calculation. They are 30o, 45o and 60o. Notice that 30o and 60o angles are complementary and that a 45o angle is its own complement.

Special Angles - 45o

The trick to these calculations is picking a right triangle containing the desired angle and making the side lengths numbers that turn out to be easy to manipulate. Here's how that works for 45o. Since we can pick a triangle of any size we can make the legs of the right angle both of length 1 and see what that leads to. The computation is contained in the next table.

Ratios for 45o

Special Angles - 30o and 60o

These two angles can be done simultaneously since they are complementary. We start with an equilateral triangle with side length 2. That makes each angle 60o and it can be split into two 30-60-90o triangles. The computations follow.

Ratios for 30o and 60o

You should learn to reproduce these diagrams on paper yourself for future reference. In fact, after drawing them a few times by hand you'll probably have all the values memorized. But it's still good to be able to create them on demand just in case a value is forgotten or an application arises that involves a 45-45-90o or a 30-60-90o triangle.