# 3.12: Trigonometric Equations Using Half Angle Formulas

**At Grade**Created by: CK-12

**Practice**Trigonometric Equations Using Half Angle Formulas

As you've seen many times, the ability to find the values of trig functions for a variety of angles is a critical component to a course in Trigonometry. If you were given an angle as the argument of a trig function that was half of an angle you were familiar with, could you solve the trig function?

For example, if you were asked to find

would you be able to do it? Keep reading, and in this section you'll learn how to do this.

### Using Half Angle Formulas on Trigonometric Equations

It is easy to remember the values of trigonometric functions for certain common values of . However, sometimes there will be fractional values of known trig functions, such as wanting to know the sine of half of the angle that you are familiar with. In situations like that, a half angle identity can prove valuable to help compute the value of the trig function.

In addition, half angle identities can be used to simplify problems to solve for certain angles that satisfy an expression. To do this, first remember the half angle identities for sine and cosine:

if is located in either the first or second quadrant.

if is located in the third or fourth quadrant.

if is located in either the first or fourth quadrant.

if is located in either the second or fourth quadrant.

When attempting to solve equations using a half angle identity, look for a place to substitute using one of the above identities. This can help simplify the equation to be solved.

Let's look at some problems that use the half angle formula.

1. Solve the trigonometric equation over the interval .

Then or , which is .

.

2. Solve for

To solve , first we need to isolate cosine, then use the half angle formula.

when

3. Solve for

To solve , first isolate tangent, then use the half angle formula.

Using your graphing calculator, when

### Examples

#### Example 1

Earlier, you were asked to solve sin 22.5°.

Knowing the half angle formulas, you can compute easily:

#### Example 2

Find the exact value of

#### Example 3

Find the exact value of

#### Example 4

Find the exact value of

### Review

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

Use half angle identities to help solve each of the following equations on the interval .

### Review (Answers)

To see the Review answers, open this PDF file and look for section 3.12.

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Here you'll learn how to solve trig equations using the half angle formulas.

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