Half and Double Angle Formulas

Introduction: In this lesson, formulas involving half of and twice of an angle will be defined and applied to the fundamental trig functions.

The Lesson:

For any angle a we have the following relationships:

Half angle formulas:

1. 
2. 
3. 

Double angle formulas:

4. 
5. 
6. 

We will use these formulas to determine the exact values of trig functions of certain angles in terms of half or double values. Proofs are available in all trig and pre-calculus texts.

Two other formulas can be derived from

and

.

By squaring both sides of the equations we can obtain

and

 

If we let A = we have

.

Let's Practice:

1. What is the exact value of tan(15º)?

We can use a half angle formula noticing that .

We have tan(15º) = tan() =
.

2. A quadrant four angle A has a tangent of .
   What is the exact value of sin(2A)?

In the diagram of angle A shown below, the hypotenuse would be .

To find the sin(2A) we use the double angle formula

3. Find the and the for the angle in example (ii).

To find the we use the half angle formula

.

Since angle A is in quadrant four, we have . Dividing by 2 gives us which puts angle in quadrant two. Therefore the sine is positive and

To find the we use the half angle formula

.