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Chapter 4 Identities

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.

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Chapter 4 Identities

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  1. Chapter 4Identities 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 • Fundamental identities • 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: • 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:

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