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Trigonometric Identities and Equations: Double, Power-Reducing, Half, Product-to-Sum, and Sum-to-Product.

Learn how to evaluate trigonometric expressions and solve equations using double-angle, power-reducing, half-angle, product-to-sum, and sum-to-product identities.

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Trigonometric Identities and Equations: Double, Power-Reducing, Half, Product-to-Sum, and Sum-to-Product.

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  1. Splash Screen

  2. Five-Minute Check (over Lesson 5-4) Then/Now Key Concept: Double-Angle Identities and Proof: Double-Angle Identity for Sine Example 1: Evaluate Expressions Involving Double Angles Example 2: Solve an Equation Using a Double-Angle Identity Key Concept: Power-Reducing Identities and Proof: Power-Reducing Identity for Sine Example 3: Use an Identity to Reduce a Power Example 4: Solve an Equation Using a Power-Reducing Identity Key Concept: Half-Angle Identities and Proof: Half-Angle Identity for Cosine Example 5: Evaluate an Expression Involving a Half Angle Example 6: Solve an Equation Using a Half-Angle Identity Key Concept: Product-to-Sum Identities and Proof: Product-to-Sum Identity for sin α cos β Example 7: Use an Identity to Write a Product as a Sum or Difference Key Concept: Sum-to-Product Identities and Proof: Sum-to-Product Identity for sin α + sin β Example 8: Use a Product-to-Sum or Sum-to-Product Identity Example 9: Solve an Equation Using a Sun-to-Product Identity Lesson Menu

  3. A. B. C. D. Find the exact value of sin 75°. 5–Minute Check 1

  4. Find the exact value of . A. B. C. D. 5–Minute Check 2

  5. Find the exact value of . A. B. C. D. 5–Minute Check 3

  6. Simplify . A.tan( + 19°) B.tan19 C.tan(19 )° D. 5–Minute Check 4

  7. Find the solution to = 1 in the interval [0, 2). A. B. C. D. 5–Minute Check 5

  8. You used sum and difference identities. (Lesson 5-4) • Use double-angle, power-reducing, and half-angle identities to evaluate trigonometric expressions and solve trigonometric equations. • Use product-to-sum identities to evaluate trigonometric expressions and solve trigonometric equations. Then/Now

  9. Key Concept 1

  10. If on the interval , find sin 2θ, cos 2θ, and tan 2θ. Since on the interval , one point on the terminal side of θ has y-coordinate 3 and a distance of 4 units from the origin, as shown. Evaluate Expressions Involving Double Angles Example 1

  11. The x-coordinate of this point is therefore or . Using this point, we find that cos θ = and tan θ = . Evaluate Expressions Involving Double Angles Use these values and the double-angle identities for sine and cosine to find sin 2θ and cos 2θ. Then find tan 2θ using either the tangent double-angle identity or the definition of tangent. sin 2θ = 2sin θ cos θ cos2θ = 2cos2θ – 1 Example 1

  12. Evaluate Expressions Involving Double Angles Method 1 Example 1

  13. Evaluate Expressions Involving Double Angles Method 2 Example 1

  14. Answer: Evaluate Expressions Involving Double Angles Example 1

  15. If on the interval , find, sin2, cos 2,and tan 2. A. B. C. D. Example 1

  16. cos θ = or cos θ = –1 Solve for cos θ. Solve an Equation Using a Double-Angle Identity Solve cos 2θ – cos θ = 2 on the interval [0, 2π). cos 2θ –cos θ = 2 Original equation 2 cos2θ –1 –cos θ – 2 = 0 Cosine Double-Angle Identity 2 cos2θ– cos θ – 3 = 0 Simplify. (2 cos θ – 3)( cos θ + 1) = 0 Factor. 2 cos θ –3 = 0 or cos θ + 1 = 0 Zero Product Property θ = πSolve for  Example 2

  17. Since cos θ = has no solution, the solution on the interval [0, 2π) is θ = π. Solve an Equation Using a Double-Angle Identity Answer:π Example 2

  18. A. B. C. D. Solve tan2 + tan = 0 on the interval [0, 2π). Example 2

  19. Key Concept 3

  20. csc4θ = (csc2θ)2 (csc2θ)2 = csc4θ Reciprocal Identity Pythagorean Identity Cosine Power-Reducing Identity Use an Identity to Reduce a Power Rewrite csc4θ in terms of cosines of multiple angles with no power greater than 1. Example 3

  21. Common denominator Simplify. Square the fraction. Cosine Power-Reducing Identity Use an Identity to Reduce a Power Example 3

  22. Common denominator Simplify. So, csc4θ = . Answer: Use an Identity to Reduce a Power Example 3

  23. A. B. C. D. Rewrite tan4x in terms of cosines of multiple angles with no power greater than 1. Example 3

  24. Sine Power-Reducing Identity Multiply each side by 2. Add like terms. Double-Angle Identity Simplify. Solve an Equation Using a Power-Reducing Identity Solve sin2θ+ cos 2θ– cos θ= 0. Solve Algebraically sin2θ+ cos 2θ–cos θ= 0 Original equation Example 4

  25. Factor.  =  = 0 Solve for θ on [0, 2π). The graph of y = sin2θ+cos 2θ–cos θ has a period of 2, so the solutions are Solve an Equation Using a Power-Reducing Identity 2cos θ= 0 cos θ – 1 = 0 Zero Product Property cos  = 0 cos  = 1 Solve for cos . Example 4

  26. The graph of y = sin2θ+ cos 2θ– cos θ has zeros at on the interval [0, 2π).  Answer: Solve an Equation Using a Power-Reducing Identity Support Graphically Example 4

  27. A. B. C. D. Solve cos 2x + 2cos2x = 0. Example 4

  28. Key Concept 5

  29. Sine Half-Angle Identity (Quadrant I angle) Evaluate an Expression Involving a Half Angle Find the exact value of sin 22.5°. Notice that 22.5° is half of 45°. Therefore, apply the half-angle identity for sine, noting that since 22.5° lies in Quadrant I, its sine is positive. Example 5

  30. Subtract and then divide. Quotient Property of Square Roots Answer: Evaluate an Expression Involving a Half Angle Example 5

  31. CHECK Use a calculator to support your assertion that sin 22.5° = . sin 22.5° = 0.3826834324 and = 0.3826834324  Evaluate an Expression Involving a Half Angle Example 5

  32. Find the exact value of . A. B. C. D. Example 5

  33. Solve on the interval [0, 2π). Original equation Sine and Cosine Half-Angle Identities Square each side. 1 – cos x = 1 + cos x Multiply each side by 2. Subtract 1 – cos x from each side. Solve an Equation Using a Half-Angle Identity Example 6

  34. Solve for x. The solutions on the interval [0, 2π) are . Answer: Solve for cos x. Solve an Equation Using a Half-Angle Identity Example 6

  35. Solve on the interval [0, 2π). A. B. C. D. Example 6

  36. Key Concept 7

  37. Product-to-Sum Identity Simplify. Distributive Property Answer: Use an Identity to Write a Product as a Sum or Difference Rewrite cos 6x cos 3x as a sum or difference. Example 7

  38. A. B. C. D. Rewrite sin 4x cos 2x as a sum or difference. Example 7

  39. Key Concept 8

  40. Sum-to-Product Identity Simplify. Use a Product-to-Sum or Sum-to-Product Identity Find the exact value of cos 255° + cos 195°. Example 8

  41. Simplify. The exact value of cos 255° + cos 195° is . Answer: Use a Product-to-Sum or Sum-to-Product Identity Example 8

  42. A. B. C. D. Find the exact value of sin 255° + sin 195°. Example 8

  43. Sine Sum-to-Product Identity Simplify. Solve an Equation Using a Sum-to-Product Identity Solve sin 8x – sin 2x = 0. Solve Algebraically sin 8x –sin 2x = 0 Original equation Set each factor equal to zero and find solutions on the interval [0, 2π). Example 9

  44. First factor set equal to 0 2cos 5x = 0 Divide each side by 2. Multiple angle solutions in [0, 2π). Divide each solution by 5. Second factor set equal to 0 sin 3x = 0 Multiple angle solutions in [0, 2π). Divide each solution by 3. Solve an Equation Using a Sum-to-Product Identity Example 9

  45. The period of y = cos 5x is and the period of y = sin 3x is , so the solutions are where n is an integer. Solve an Equation Using a Sum-to-Product Identity Example 9

  46. The graph of y = sin 8x –sin 2x has zeros at on the interval .  Solve an Equation Using a Sum-to-Product Identity Support Graphically Example 9

  47. Answer: Solve an Equation Using a Sum-to-Product Identity Example 9

  48. A. B. C. D. Solve sin 6x + sin 2x = 0. Example 9

  49. End of the Lesson

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