Copyright © Cengage Learning. All rights reserved. 5.1 USING FUNDAMENTAL IDENTITIES Copyright © Cengage Learning. All rights reserved.

What You Should Learn Recognize and write the fundamental trigonometric identities. Use the fundamental trigonometric identities to evaluate trigonometric functions, simplify trigonometric expressions, and rewrite trigonometric expressions.

Introduction We will learn how to use the fundamental identities to do the following. 1. Evaluate trigonometric functions. 2. Simplify trigonometric expressions. 3. Develop additional trigonometric identities. 4. Solve trigonometric equations.

Introduction

Introduction cont’d

Using the Fundamental Identities One common application of trigonometric identities is to use given values of trigonometric functions to evaluate other trigonometric functions.

Example 1a – Using Identities to Evaluate a Function Use the values and tan u  0 to find the values of all six trigonometric functions.

Example 1b – Using Identities to Evaluate a Function Use the values of sin x = ½ and cos x > 0 to find the values of all six trigonometric functions.

Example 2a – Simplifying a Trigonometric Expression Simplify sin x cos2 x – sin x.

Example 2b – Simplifying a Trigonometric Expression Simplify cos2x csc x – csc x.

Examples 3 and 4 Factor each expression. 1 – cos2 x 2 csc2 x – 7csc x + 6 sec2 x + 3 tan x + 1

Examples 5 and 6 Simplify csc t – cos t cot t. Perform the addition and simplify:

Example 7 – Rewriting a Trigonometric Expression Rewrite so that it is not in fractional form.

Example 8 Use the substitution x = 5 sin , 0 <  < /2 to write as a trigonometric function of  .

Example 9 Rewrite ln |sec  | + ln |cot  | as a single logarithm and simplify the result.