Copyright © 2014, 2010, 2007 Pearson Education, Inc. Chapter 5 Analytic Trigonometry 5.2 Sum and Difference Formulas Copyright © 2014, 2010, 2007 Pearson Education, Inc. 1

Objectives: Use the formula for the cosine of the difference of two angles. Use sum and difference formulas for cosines and sines. Use sum and difference formulas for tangents.

The Cosine of the Difference of Two Angles

Example: Using the Difference Formula for Cosines to Find an Exact Value We know that Obtain this exact value using and the difference formula for cosines.

Example: Using the Difference Formula for Cosines to Find an Exact Value Find the exact value of

Sum and Difference Formulas for Cosines and Sines

Example: Using the Sine of a Sum to Find an Exact Value Find the exact value of using the fact that

Example – Using the Addition and Subtraction Formulas Find the exact value of each expression. (a) cos 75 (b) cos

Example – Using the Addition and Subtraction Formulas Find the exact value of each expression. (a) cos 75 (b) cos Solution: (a) Notice that 75 = 45 + 30. Since we know the exact values of sine and cosine at 45 and 30, we use the Addition Formula for Cosine to get cos 75 = cos (45 + 30) = cos 45 cos 30 – sin 45 sin 30 =

(b) Since the Subtraction Formula for Cosine gives cos = cos Example – Solution cont’d (b) Since the Subtraction Formula for Cosine gives cos = cos = cos cos + sin sin

Example – Proving a Cofunction Identity Prove the cofunction identity cos = sin u. Solution: By the Subtraction Formula for Cosine, we have cos = cos cos u + sin sin u = 0  cos u + 1  sin u = sin u

Example: Verifying an Identity Verify the identity: Divide by cos α cos β.

Example: Verifying an Identity (continued) Verify the identity: We have divided by cos α cos β and simplified. The identity is verified.

Example: Finding Exact Values (continued) Suppose that for a quadrant II angle and for a quadrant I angle Find the exact value of From our previous work:

Example: Finding Exact Values (continued) Suppose that for a quadrant II angle and for a quadrant I angle Find the exact value of From our previous work:

Sum and Difference Formulas for Tangets

Example – Using the Addition and Subtraction Formulas  

Example: Finding Exact Values (continued)  

Example: Verifying an Identity Verify the identity: Simplify. The identity is verified.