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

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

Fundamental Identities

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

Simplifying Trigonometric Expressions Claim: Proof:

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

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

Verifying Identities Verify right-to-left:

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:

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

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

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

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

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

Double-Angle Identities

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.

Half-Angle Identities

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

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

Product-Sum Identities

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

Sum-Product Identities

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