Copyright © Cengage Learning. All rights reserved. 5.4 Sum and Difference Formulas

2 What You Should Learn Use sum and difference formulas to evaluate trigonometric functions, verify trigonometric identities, and solve trigonometric equations.

3 Using Sum and Difference Formulas

4 In this section and the following section, you will study the uses of several trigonometric identities and formulas. You need to memorize these formulas ASAP.

5 Example 1 – Evaluating a Trigonometric Function Find the exact value of (a) cos 75 ° and (b) sin. Solution: a. Using the fact that 75 ° = 30 ° + 45 ° with the formula for cos(u + v) yields cos 75 ° = cos(30 ° + 45 °) = cos 30  cos 45  – sin 30  sin 45  = =

6 Example 1 – Solution Try checking this result on your calculator. You will find that cos 75   b. Using the fact that with the formula for sin (u – v) yields cont’d

7 Example 1 – Solution cont’d

8 Example 3 – An Application of a Sum Formula Write cos(arctan 1 + arccos x) as an algebraic expression. Solution: This expression fits the formula for cos(u + v). Angles u = arctan 1 and v = arccos x are shown in Figures 5.21 and 5.22, respectively. Figure 5.22Figure 5.21

9 Example 3 – An Application of a Sum Formula cos(u + v) = cos(arctan 1) cos(arccos x) – sin(arctan 1) sin(arccos x)