1. Chapter 4 Identities 4.1 Fundamental Identities and Their Use
4.2 Verifying Trigonometric Identities
4.3 Sum, Difference, and Cofunction Identities
4.4 Double-Angle and Half-Angle Identities
4.5 Product-Sum and Sum-Product Identities

2. Fundamental Identities and Their Use
Evaluating trigonometric identities
Converting to equivalent forms

3. Fundamental Identities

4. Evaluating Trigonometric Identities
Example
Find the other four trigonometric functions of x when
cos x = -4/5 and tan x = 3/4

5. Simplifying Trigonometric Expressions
Claim:
Proof:

6. 4.2 Verifying Trigonometric Identities
Verifying identities
Testing identities using a graphing calculator

7. Verifying Identities Verify csc(-x) = -csc x
Verify tan x sin x + cos x = sec x

8. Verifying Identities
Verify right-to-left:

9. Verifying Identities Using a Calculator
Graph both sides of the equation in the same viewing window. If they produce different graphs they are not identities. If they appear the same the identity must still be verified.
Example:

10. 4.3 Sum, Difference, and Cofunction Identities
Sum and difference identities for cosine
Cofunction identities
Sum and difference identities for sine and tangent
Summary and use

11. Sum and Difference Identities for Cosine
cos(x – y) = cos x cos y - sin x sin y
Claim: cos(p/2 – y) = siny
Proof:
cos(p/2 – y) = cos (p/2) cos y + sin(p/2) sin y
= 0 cos y + 1 sin y = sin y

12. Sum and Difference Formula for Sine and Tangent
sin (x- y) = sin x cos x + cos x sin y

13. Finding Exact Values
Find the exact value of cos 15º
Solution:

14. Double-Angle and Half-Angle Identities
Double-angle identities
Half-angle identities

15. Double-Angle Identities

16. Using Double-Angle Identities
Example:
Find the exact value of cos 2x if sin x = 4/5, p/2 < x < p
The reference angle is in the second quadrant.

17. Half-Angle Identities

18. Using a Half-Angle Identity
Example: Find cos 165º.

19. 4.5 Product-Sum and Sum-Product Identities
Product-sum identities
Sum-product identities
Application

20. Product-Sum Identities

21. Using Product-Sum Identities
Example: Evaluate sin 105º sin 15º.
Solution:

22. Sum-Product Identities

23. Using a Sum-Product Identity
Example: Write the difference sin 7q – sin 3q as a product.
Solution: