You want to find the exact value of . How could you find this value without using a calculator?

### Double Angle and Half Angle Formulas

In this concept, we will learn how to find the exact values of the trig functions for angles that are half or double of other angles. Here we will introduce the Double-Angle and Half-Angle Formulas.

#### Double-Angle and Half-Angle Formulas

The signs of and depend on which quadrant lies in. For and any formula can be used to solve for the exact value.

Let's find the exact value of .

is half of and in the first quadrant.

Now, let's find the exact value of if and .

To use the sine double-angle formula, we also need to find , which would be because is in the quadrant.

Finally, let's find the exact value of for from the previous problem.

Use to solve for .

### Examples

#### Example 1

Earlier, you were asked to find the value of without a calculator.

so we can use the formula for

If we simplify this expression, we get .

#### Example 2

Find the exact value of .

is in the quadrant.

#### Example 3

Given the function and , find .

First, find . , so

#### Example 4

Given the function and , find .

You can use either formula.

### Review

Find the exact value of the following angles.

The and . Find:

The and . Find:

### Answers for Review Problems

To see the Review answers, open this PDF file and look for section 14.15.