Chapter 4 Identities 4.1 Fundamental Identities and Their Use 4.2 Verifying Trigonometric Identities 4.3 Sum, Difference, and Cofunction Identities 4.4

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  • Chapter 4Identities4.1 Fundamental Identities and Their Use4.2 Verifying Trigonometric Identities4.3 Sum, Difference, and Cofunction Identities4.4 Double-Angle and Half-Angle Identities4.5 Product-Sum and Sum-Product Identities

  • Fundamental Identities and Their UseFundamental identitiesEvaluating trigonometric identitiesConverting to equivalent forms

  • Fundamental Identities

  • Evaluating Trigonometric IdentitiesExampleFind the other four trigonometric functions of x whencos x = -4/5 and tan x = 3/4

  • Simplifying Trigonometric ExpressionsClaim:

    Proof:

    Claim:

    Proof:

  • 4.2 Verifying Trigonometric IdentitiesVerifying identitiesTesting identities using a graphing calculator

  • Verifying IdentitiesVerify csc(-x) = -csc xVerify tan x sin x + cos x = sec x

  • Verifying IdentitiesVerify right-to-left:

  • Verifying Identities Using a CalculatorGraph 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 IdentitiesSum and difference identities for cosineCofunction identitiesSum and difference identities for sine and tangentSummary and use

  • Sum and Difference Identities for Cosinecos(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 Tangentsin (x- y) = sin x cos x + cos x sin y

  • Finding Exact ValuesFind the exact value of cos 15Solution:

  • Double-Angle and Half-Angle IdentitiesDouble-angle identitiesHalf-angle identities

  • Double-Angle Identities

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

  • Half-Angle Identities

  • Using a Half-Angle IdentityExample: Find cos 165.

  • 4.5 Product-Sum and Sum-Product IdentitiesProduct-sum identitiesSum-product identitiesApplication

  • Product-Sum Identities

  • Using Product-Sum IdentitiesExample: Evaluate sin 105 sin 15.Solution:

  • Sum-Product Identities

  • Using a Sum-Product IdentityExample: Write the difference sin 7q sin 3q as a product.Solution: