After all of your experience with trig functions, you are feeling pretty good. You know the values of trig functions for a lot of common angles, such as etc. And for other angles, you regularly use your calculator. Suppose someone gave you an equation like this:

Could you solve it without the calculator? You might notice that this is half of . This might give you a hint!

### Half Angle Formulas

Here we'll attempt to derive and use formulas for trig functions of angles that are half of some particular value.

To do this, we'll start with the double angle formula for cosine: . Set , so the equation above becomes .

Solving this for , we get:

if is located in either the first or second quadrant.

if is located in the third or fourth quadrant.

This formula shows how to find the sine of half of some particular angle.

One of the other formulas that was derived for the cosine of a double angle is:

. Set , so the equation becomes . Solving this for , we get:

if is located in either the first or fourth quadrant.

if is located in either the second or fourth quadrant.

This formula shows how to find the cosine of half of some particular angle.

Let's see some examples of these two formulas (sine and cosine of half angles) in action.

1. Determine the exact value of .

Using the half angle identity, , and is located in the first quadrant. Therefore, .

Plugging this into a calculator, . Using the sine function on your calculator will validate that this answer is correct.

2. Use the half angle identity to find exact value of

Since , use the half angle formula for sine, where . In this example, the angle is a second quadrant angle, and the sin of a second quadrant angle is positive.

3. Use the half angle formula for the cosine function to prove that the following expression is an identity:

Use the formula and substitute it on the left-hand side of the expression.

### Examples

#### Example 1

Earlier, you were asked you to find . If you use the half angle formula, then

Substituting this into the half angle formula:

#### Example 2

Prove the identity:

Step 1: Change right side into sine and cosine.

Step 2: At the last step above, we have simplified the right side as much as possible, now we simplify the left side, using the half angle formula.

#### Example 3

Verify the identity:

Step 1: change cotangent to cosine over sine, then cross-multiply.

#### Example 4

Prove that

### Review

Use half angle identities to find the exact values of each expression.

- Use the two half angle identities presented in this section to prove that .
- Use the result of the previous problem to show that .
- Use the result of the previous problem to show that .

Use half angle identities to help you find all solutions to the following equations in the interval .

### Review (Answers)

To see the Review answers, open this PDF file and look for section 3.11.