Special Angles

We will first look into the trigonometric functions of the angles 30°, 45° and 60°.

Let us consider 30° and 60°.

These two angles form a 30°-60°-90° right triangle as shown.

The ratio of the sides of the triangle is
1 : √3 : 2

From the triangle we get the ratios as follows:

Next, we consider the 45˚ angle that forms a 45°-45°-90° right triangle as shown. The ratio of the sides of the triangle is

Combining the two tables we get:

Example:
Evaluate the following without using a calculator:
a) 2 sin 30˚ + 3 cos 60˚ – 3 tan 45˚
b) 3(cos 30˚)² + 2 (sin 30˚ )²

Solution:
a) 2 sin 30˚ + 3 cos 60˚ – 3 tan 45˚

b) 3(cos 30˚)² + 2 (sin 30˚)²