# 3.9: Applications of Sum and Difference Formulas

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

**Practice**Applications of Sum and Difference Formulas

You are quite likely familiar with the values of trig functions for a variety of angles. Angles such as , , and are common. However, if you were asked to find the value of a trig function for a more rarely used angle, could you do so? Or what if you were asked to find the value of a trig function for a sum of angles? For example, if you were asked to find could you?

Read on, and in this section, you'll get practice with simplifying trig functions of angles using the sum and difference formulas.

### Watch This

James Sousa Example: Simplify a Trig Expression Using the Sum and Difference Identities

### Guidance

Quite frequently one of the main obstacles to solving a problem in trigonometry is the inability to transform the problem into a form that makes it easier to solve. Sum and difference formulas can be very valuable in helping with this.

Here we'll get some extra practice putting the sum and difference formulas to good use. If you haven't gone through them yet, you might want to review the Concepts on the Sum and Difference Formulas for sine, cosine, and tangent.

#### Example A

Verify the identity

#### Example B

Solve in the interval .

**
Solution:
**
First, get
by itself, by dividing both sides by
.

Now, expand the left side using the sine difference formula.

The when is .

#### Example C

Find all the solutions for in the interval .

**
Solution:
**
Get the
by itself and then take the square root.

Now, use the cosine sum formula to expand and solve.

The is in Quadrants III and IV, so and .

### Vocabulary

**
Difference Formula:
**
A
**
difference formula
**
is a formula to help simplify a trigonometric function of the difference of two angles, such as
.

**
Sum Formula:
**
A
**
sum formula
**
is a formula to help simplify a trigonometric function of the sum of two angles, such as
.

### Guided Practice

1. Find all solutions to , when is between .

2. Solve for all values of between for .

3. Find all solutions to , when is between .

**
Solutions:
**

1. To find all the solutions, between , we need to expand using the sum formula and isolate the .

This is true when , or

2. First, solve for .

Now, use the tangent sum formula to expand for when .

This is true when or .

If the tangent sum formula to expand for when , we get no solution as shown.

Therefore, the tangent sum formula cannot be used in this case. However, since we know that when or , we can solve for as follows.

Therefore, all of the solutions are

3. To solve, expand each side:

Set the two sides equal to each other:

As a decimal, this is , so and .

### Concept Problem Solution

To find , use the sine sum formula:

### Practice

Prove each identity.

Use sum and difference formulas to help you graph each function.

Solve each equation on the interval .

Difference Formula

Trigonometric function difference formulas exist for each of the primary trigonometric functions. For example, the cosine difference formula is .Sum Formula

A sum formula is a formula to help simplify a trigonometric function of the sum of two angles, such as .### Image Attributions

## Description

## Learning Objectives

Here you'll learn to solve problems using the sum and difference formulas for trig functions, including the sine sum/difference formulas, cosine sum/difference formulas, and tangent sum/difference formulas.