# Chapter 3: Trigonometric Identities and Equations

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

## Chapter Outline

- 3.1. Fundamental Identities
- 3.2. Proving Identities
- 3.3. Solving Trigonometric Equations
- 3.4. Sum and Difference Identities
- 3.5. Double Angle Identities
- 3.6. Half-Angle Identities
- 3.7. Products, Sums, Linear Combinations, and Applications

### Chapter Summary

## Chapter Summary

Here are the identities studied in this chapter:

Quotient & Reciprocal Identities

Pythagorean Identities

Even & Odd Identities

Co-Function Identities

Sum and Difference Identities

Double Angle Identities

Half Angle Identities

Product to Sum & Sum to Product Identities

Linear Combination Formula

, where and

## Review Questions

- Find the sine, cosine, and tangent of an angle with terminal side on .
- If and , find .
- Simplify: .
- Verify the identity:

For problems 5-8, find all the solutions in the interval .

- Solve the trigonometric equation over the interval .
- Solve the trigonometric equation over the interval .
- Solve the trigonometric equation for all real values of .

Find the exact value of:

- Write as a product:
- Simplify:
- Simplify:
- Derive a formula for .
- If you solve for , you would get . This new formula is used to reduce powers of cosine by substituting in the right part of the equation for . Try writing in terms of the first power of cosine.
- If you solve for , you would get . Similar to the new formula above, this one is used to reduce powers of sine. Try writing in terms of the first power of cosine.
- Rewrite in terms of the first power of cosine:

## Texas Instruments Resources

*In the CK-12 Texas Instruments Trigonometry FlexBook, there are graphing calculator activities designed to supplement the objectives for some of the lessons in this chapter. See http://www.ck12.org/flexr/chapter/9701.*