# 14.5: Double and Half Angle Formulas

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

**Objective**

You’ll learn how to use the double and half angle formulas.

**Review Queue**

Use your calculator to find the value of the trig functions below. Round your answers to 4 decimal places.

1.

2.

Find the exact values of the trig expressions below.

3.

4.

## Finding Exact Trig Values using Double and Half Angle Formulas

**Objective**

Here you'll use the half and double angle formulas to find exact values of angles other than the critical angles.

**Guidance**

In the previous concept, we added two different angles together to find the exact values of trig functions. 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.

**Example A**

Find the exact value of .

**Solution:** is half of and in the first quadrant.

**Example B**

Find the exact value of if and .

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

**Example C**

Find the exact value of for from Example B.

**Solution:** Use to solve for .

**Guided Practice**

1. Find the exact value of .

2. and . Find:

a)

b)

**Answers**

1. is in the quadrant.

2. First, find . , so

a)

b) You can use either formula.

**Vocabulary**

- Double-Angle and Half-Angle Formulas

**Problem Set**

Find the exact value of the following angles.

The and . Find:

The and . Find:

## Simplifying Trig Expressions using Double and Half Angle Formulas

**Objective**

Here you'll use the half and double angle formulas to simplify more complicated expressions.

**Guidance**

We can also use the double-angle and half-angle formulas to simplify trigonometric expressions.

**Example A**

Simplify .

**Solution:** Use and then factor.

**Example B**

Find the formula for .

**Solution:** You will need to use the sum formula and the double-angle formula.

We will explore other possibilities for the because of the different formulas for in the Problem Set.

**Example C**

Verify the identity .

**Solution:** Simplify the left-hand side use the half-angle formula.

**Guided Practice**

1. Simplify .

2. Verify .

**Answers**

1.

2.

**Problem Set**

Simplify the following expressions.

Verify the following identities.

## Solving Trig Equations using Double and Half Angle Formulas

**Objective**

Here you'll solve trig equations using the half and double angle formulas.

**Guidance**

Lastly, we can use the half and double angle formulas to solve trigonometric equations.

**Example A**

Solve when .

**Solution:** Change and simplify.

Set each factor equal to zero and solve.

**Example B**

Solve when .

**Solution:** In this case, you do not have to use the half-angle formula. Solve for .

Now, let’s find and then solve for by dividing by 2.

Now, the second solution is not in our range, so the only solution is .

**Example C**

Solve for .

**Solution:** Pull a 2 out of the left-hand side and use the formula.

**Guided Practice**

Solve the following equations for .

1.

2.

**Answers**

1.

From this we can see that there are no solutions within our interval.

2.

Set each factor equal to zero and solve.

**Problem Set**

Solve the following equations for .

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