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 anglea we have the following relationships:

Half angle formulas: Double angle formulas:

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

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

.

Examples

What is the exact value of cos(15º)?

What is your answer?

A quadrant three angle a has a cosine of -0.9. What is the exact value of tan(2a)?