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Trigonometric Functions: Amplitude, Period, and Graphing

Learn how to find the amplitude and period of sine and cosine functions and how to graph them. Explore real-world examples and discover the properties of other trigonometric functions.

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Trigonometric Functions: Amplitude, Period, and Graphing

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  1. Splash Screen

  2. Five-Minute Check (over Lesson 12–6) CCSS Then/Now New Vocabulary Key Concept: Sine and Cosine Functions Example 1: Find Amplitude and Period Example 2: Graph Sine and Cosine Functions Example 3: Real-World Example: Model Periodic Situations Key Concept: Tangent Function Example 4: Graph Tangent Functions Key Concept: Cosecant, Secant, and Cotangent Functions Example 5: Graph Other Trigonometric Functions Lesson Menu

  3. The terminal side of angle  in standard position intersects the unit circle at . Find sin  and cos  . A. B. C. D. 5-Minute Check 1

  4. The terminal side of angle  in standard position intersects the unit circle at . Find sin  and cos  . A. B. C. D. 5-Minute Check 1

  5. The terminal side of angle  in standard position intersects the unit circle at . Find sin  and cos  . A. B. C. D. 5-Minute Check 2

  6. The terminal side of angle  in standard position intersects the unit circle at . Find sin  and cos  . A. B. C. D. 5-Minute Check 2

  7. A. B. C. D. Find the exact value of sin 225°. 5-Minute Check 3

  8. A. B. C. D. Find the exact value of sin 225°. 5-Minute Check 3

  9. A.2 B. C. D. Determine the period of the function graphed at the right. 5-Minute Check 4

  10. A.2 B. C. D. Determine the period of the function graphed at the right. 5-Minute Check 4

  11. If x is a positive acute angle and cos x = what is the exact value of sin x? A. B. C. D. 5-Minute Check 5

  12. If x is a positive acute angle and cos x = what is the exact value of sin x? A. B. C. D. 5-Minute Check 5

  13. Content Standards F.IF.7.e Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude. F.TF.5 Choose trigonometric functions to model periodic phenomena with specified amplitude, frequency, and midline. Mathematical Practices 1 Make sense of problems and persevere in solving them. CCSS

  14. You examined periodic functions. • Describe and graph the sine, cosine, and tangent functions. • Describe and graph other trigonometric functions. Then/Now

  15. amplitude • frequency Vocabulary

  16. Concept

  17. |a| = |1| The coefficient of is 1. Find Amplitude and Period Find the amplitude and period of . First, find the amplitude. Next, find the period. = 1080° Answer: Example 1

  18. |a| = |1| The coefficient of is 1. Find Amplitude and Period Find the amplitude and period of . First, find the amplitude. Next, find the period. = 1080° Answer: amplitude: 1; period: 1080° Example 1

  19. Find the amplitude and period for A. amplitude: 1; period: 720° B. amplitude: 1; period: 180° C. amplitude: 2; period: 1080° D. amplitude: 2; period: 360° Example 1

  20. Find the amplitude and period for A. amplitude: 1; period: 720° B. amplitude: 1; period: 180° C. amplitude: 2; period: 1080° D. amplitude: 2; period: 360° Example 1

  21. Graph Sine and Cosine Functions A. Graph the function y = sin 3. Find the amplitude, the period, and the x-intercepts: a = 1 and b = 3. amplitude: |a| = |1| or 1 → The maximum is 1 and the minimum is –1. One cycle has a length of 120°. Example 2

  22. Graph Sine and Cosine Functions x-intercepts: (0, 0) Answer: Example 2

  23. Graph Sine and Cosine Functions x-intercepts: (0, 0) Answer: Example 2

  24. B. Graph the function The graph is compressed vertically. The maximum is and the minumum is – . Graph Sine and Cosine Functions One cycle has a length of 360°. Example 2

  25. Graph Sine and Cosine Functions Answer: Example 2

  26. Graph Sine and Cosine Functions Answer: Example 2

  27. A. B. C. D. A. Graph the function y = sin 2. Example 2

  28. A. B. C. D. A. Graph the function y = sin 2. Example 2

  29. A. • B. • C. • D. B. Graph the function y = 3 cos 2. Example 2

  30. A. • B. • C. • D. B. Graph the function y = 3 cos 2. Example 2

  31. Model Periodic Situations A. SOUND Humans can hear sounds with a frequency of 40 Hz. Find the period of the function that models the sound waves. A sound with the frequency of 40 Hz, has 40 cycles per second. The period is the time it takes for one cycle. Answer: Example 3

  32. Answer:So, the period is or about 0.025 second. Model Periodic Situations A. SOUND Humans can hear sounds with a frequency of 40 Hz. Find the period of the function that models the sound waves. A sound with the frequency of 40 Hz, has 40 cycles per second. The period is the time it takes for one cycle. Example 3

  33. Multiply each side by |b|. = Divide each side by 0.025; b is positive. = Model Periodic Situations B. SOUND Humans can hear sounds with a frequency of 40 Hz. Let the amplitude equal 1 unit. Write a sine equation to represent the sound wave y as a function of time t. Then graph the equation. Write the relationship between the period and b. Substitution Example 3

  34. Model Periodic Situations y = a sin b Write the general equation for the sine function. y = 1 sin 80t a = 1, b = 80, and  = t y = sin 80t Simplify. Answer: Example 3

  35. Model Periodic Situations y = a sin b Write the general equation for the sine function. y = 1 sin 80t a = 1, b = 80, and  = t y = sin 80t Simplify. Answer: Example 3

  36. A. INSTRUMENTS The bass tuba can produce sounds with as low a frequency as 50 Hz. Find the period of the function that models the sound waves. A. 0.025 second B. 0.02 second C. 0.05 second D. 0.1 second Example 3

  37. A. INSTRUMENTS The bass tuba can produce sounds with as low a frequency as 50 Hz. Find the period of the function that models the sound waves. A. 0.025 second B. 0.02 second C. 0.05 second D. 0.1 second Example 3

  38. B. INSTRUMENTS The bass tuba can produce sounds with as low a frequency as 50 Hz. Let the amplitude equal 1 unit. Determine the correct sine equation to represent the sound wave y as a function of time t. A.y = sin 60t B.y = sin 80t C.y = sin 100t D.y = sin 120t Example 3

  39. B. INSTRUMENTS The bass tuba can produce sounds with as low a frequency as 50 Hz. Let the amplitude equal 1 unit. Determine the correct sine equation to represent the sound wave y as a function of time t. A.y = sin 60t B.y = sin 80t C.y = sin 100t D.y = sin 120t Example 3

  40. Concept

  41. Find the period of . Then graph the function. Graph Tangent Functions Example 4

  42. Graph Tangent Functions Sketch asymptotes at –1 ● 180° or –180°, 1 ● 180° or 180°, 3 ● 180° or 540°, and so on. Use y = tan , but only draw one cycle every 360°. Answer: Example 4

  43. Graph Tangent Functions Sketch asymptotes at –1 ● 180° or –180°, 1 ● 180° or 180°, 3 ● 180° or 540°, and so on. Use y = tan , but only draw one cycle every 360°. Answer: Example 4

  44. Find the period of y = tan 3. Then determine the asymptotes. A. period = 60°, asymptotes at 0°, 60°, 120°, and so on. B. period = 90°, asymptotes at –30°, 60°, 150°, and so on. C. period = 60°, asymptotes at –30°, 30°, 90°, and so on. D. period = 120°, asymptotes at –30°, 150°, 270°, and so on. Example 4

  45. Find the period of y = tan 3. Then determine the asymptotes. A. period = 60°, asymptotes at 0°, 60°, 120°, and so on. B. period = 90°, asymptotes at –30°, 60°, 150°, and so on. C. period = 60°, asymptotes at –30°, 30°, 90°, and so on. D. period = 120°, asymptotes at –30°, 150°, 270°, and so on. Example 4

  46. Concept

  47. Graph Other Trigonometric Functions Find the period of y = 3 csc. Then graph the function. Since 3 csc θ is a reciprocal of 3 sin θ, the graphs have the same period, 360°. The vertical asymptotes occur at the points where 3 sin θ = 0. So, the asymptotes are at θ = 0° and θ = 180° and 360°, and so on. Sketch y = 3 sin and use it to graph y = 3 csc . Example 5

  48. Graph Other Trigonometric Functions Answer: Example 5

  49. Graph Other Trigonometric Functions Answer: Example 5

  50. Find the period of y = 4 sec 2. Then determine the asymptotes. A. period = 90°, asymptotes at 30° and 120° B. period = 180°, asymptotes at 90° and 270° C. period = 180°, asymptotes at 45°, 135°, 225°, and 315° D. period = 360°, asymptotes at 90°, 180°, 270° and 360° Example 5

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