1. 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

2. 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.

3. The Cosine of the Difference of Two Angles

4. 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.

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

6. Sum and Difference Formulas for Cosines and Sines

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

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

9. 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 =

10. (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

11. 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

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

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

14. 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:

15. 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:

16. Sum and Difference Formulas for Tangets

17. Example – Using the Addition and Subtraction Formulas

18. Example: Finding Exact Values (continued)

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