1 / 41

Splash Screen

Splash Screen. Five-Minute Check (over Lesson 5-3) Then/Now New Vocabulary Key Concept: Sum and Difference Identities Example 1: Evaluate a Trigonometric Expression Example 2: Real-World Example: Use a Sum or Difference Identity Example 3: Rewrite as a Single Trigonometric Expression

Download Presentation

Splash Screen

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  1. Splash Screen

  2. Five-Minute Check (over Lesson 5-3) Then/Now New Vocabulary Key Concept: Sum and Difference Identities Example 1: Evaluate a Trigonometric Expression Example 2: Real-World Example: Use a Sum or Difference Identity Example 3: Rewrite as a Single Trigonometric Expression Example 4: Write as an Algebraic Expression Example 5: Verify Cofunction Identities Example 6: Verify Reduction Identities Example 7: Solve a Trigonometric Equation Lesson Menu

  3. Solve for all values of x. A. B. C. D. 5–Minute Check 1

  4. A. B. C. D. Find all solutions of 2cos2x + 3cos x + 1 = 0 in the interval [0, 2π). 5–Minute Check 2

  5. A. B. C. D. Find all solutions of 4 cos2 x= 5 – 4 sin x in the interval [0, 2π). 5–Minute Check 3

  6. A. B. C. D. Find all solutions of sin x + cos x = 1 in the interval [0, 2π). 5–Minute Check 4

  7. A. B. C. D. Solve 4 sin θ – 1 = 2 sin θ for all values of θ. 5–Minute Check 5

  8. You found values of trigonometric functions using the unit circle. (Lesson 4-3) • Use sum and difference identities to evaluate trigonometric functions. • Use sum and difference identities to solve trigonometric equations. Then/Now

  9. reduction identity Vocabulary

  10. Key Concept 1

  11. 30° + 45° = 75° Cosine Sum Identity Evaluate a Trigonometric Expression A. Find the exact value of cos 75°. Example 1

  12. Multiply. Combine the fractions. Answer: Evaluate a Trigonometric Expression Example 1

  13. B. Find the exact value of tan . Write as the sum or difference of angle measures with tangents that you know. Evaluate a Trigonometric Expression Example 1

  14. Tangent Sum Identity Simplify. Rationalize the denominator. Evaluate a Trigonometric Expression Example 1

  15. Multiply. Simplify. Simplify. Answer: Evaluate a Trigonometric Expression Example 1

  16. A. B. C. D. Find the exact value of tan 15°. Example 1

  17. Use a Sum or Difference Identity A. ELECTRICITY An alternating current i in amperes in a certain circuit can be found after t seconds using i = 4 sin 255t, where 255 is a degree measure. Rewrite the formula in terms of the sum of two angle measures. Rewrite the formula in terms of the sum of two angle measures. i = 4 sin 255t Original equation = 4 sin (210t + 45t) 255t = 210t + 45t The formula is i = 4 sin (210t + 45t). Answer:i = 4 sin (210t + 45t) Example 2

  18. Use a Sum or Difference Identity B. ELECTRICITY An alternating current i in amperes in a certain circuit can be found after t seconds using i = 4 sin 255t. Use a sum identity to find the exact current after 1 second. Use a sum identity to find the exact current after 1 second. i = 4 sin (210t + 45t) Rewritten equation = 4 sin (210 + 45) t = 1 = 4[sin(210)cos(45) + cos(210)sin(45)] Sine Sum Identity Example 2

  19. Substitute. Multiply. Simplify. The exact current after 1 second is amperes. Answer:amperes Use a Sum or Difference Identity Example 2

  20. A. ELECTRICITY An alternating current i in amperes in a certain circuit can be found after t seconds using i = 4 sin 210t, where 210 is a degree measure. Rewrite the formula in terms of the sum of two angle measures. A. i = 4 sin (240t – 30t) B. i = 4 sin (180 + 30) C. i = 4 sin [7(30t)] D. i = 4 sin (150t + 60t) Example 2

  21. B. ELECTRICITY An alternating current i in amperes in a certain circuit can be found after t seconds using i = 4 sin 210t, where 210 is a degree measure. Use a sum identity to find the exact current after 1 second. A. –1 ampere B. –2 amperes C. 1 ampere D. 2 amperes Example 2

  22. A. Find the exact value of Tangent Difference Identity Simplify. Substitute. Answer: Rewrite as a Single Trigonometric Expression Example 3

  23. B. Simplify Sine Sum Identity Rewrite as fractions with a common denominator. Simplify. Answer: Rewrite as a Single Trigonometric Expression Example 3

  24. Find the exact value of . A. 0 B. C. D. 1 Example 3

  25. Write as an algebraic expression of x that does not involve trigonometric functions. Applying the Cosine Sum Identity, we find that Write as an Algebraic Expression Example 4

  26. If we let α = and β = arccos x, then sin α = and cos β = x. Sketch one right triangle with an acute angle α and another with an acute angle β. Label the sides such that sin α = and cos β = x. Then use the Pythagorean Theorem to express the length of each third side. Write as an Algebraic Expression Example 4

  27. Using these triangles, we find that = cos α or , cos (arccos x) = cos β or x, = sin α or , and sin (arccos x) = sin β or . Write as an Algebraic Expression Example 4

  28. Write as an Algebraic Expression Now apply substitution and simplify. Example 4

  29. Answer: Write as an Algebraic Expression Example 4

  30. A. B. C. D. Write sin(arccos 2x + arcsin x) as an algebraic expression of x does not involve trigonometric functions. Example 4

  31. Verify Cofunction Identities Verify cos (–θ) = cos θ. cos (–θ) = cos (0 –θ) Rewrite as a difference. = cos 0 cos θ + sin 0 sin θ Cosine Difference Identity = 1 cos θ + 0sin θ cos 0 = 1 and sin 0 = 0 = cos θ Multiply. Answer:cos (–θ) = cos (0 –θ) = cos 0 cos θ+ sin 0 sin θ= 1 cos θ+ 0 sin θ = cos θ Example 5

  32. A. B. C. D. Verify tan (–) = –tan . Example 5

  33. A. Verify . Cosine Difference Identity Simplify.  Verify Reduction Identities Example 6

  34. Answer: Verify Reduction Identities Example 6

  35. Tangent Difference Identity tan 360° = 0 Simplify.  Answer: Verify Reduction Identities B. Verify tan (x – 360°) = tan x. Example 6

  36. Verify the cofunction identity . A. B. C. D. Example 6

  37. Find the solutions ofon the interval [ 0, 2). Original equation Sine Sum Identity and Sine Difference Identity Solve a Trigonometric Equation Example 7

  38. Simplify. Divide each side by 2. Substitute. Solve for cos x. Solve a Trigonometric Equation Example 7

  39. On the interval [0, 2π), cos x = 0 when x = CHECKThe graph of has zeros at on the interval [ 0, 2π).  Answer: Solve a Trigonometric Equation Example 7

  40. Find the solutions of on the interval [0, 2π). A. B. C. D. Example 7

  41. End of the Lesson

More Related