1. Splash Screen

2. Five-Minute Check (over Lesson 13–6) 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. 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. A B C D
5-Minute Check 2

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

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

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

8. You examined periodic functions. (Lesson 13–6)
Describe and graph the sine, cosine, and tangent functions.
Describe and graph other trigonometric functions.
Then/Now

9. amplitude
frequency
Vocabulary

10. Concept

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

12. A B C D 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° A B C D
Example 1

13. A. Graph the function y = sin 3.
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

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

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

16. Graph Sine and Cosine Functions
Answer:
Example 2

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

18. B. Graph the function y = 4 cos 2.D. A B C D
Example 2

19. 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.
Answer: So, the period is or about second.
Example 3

20. Write the relationship between the period and b.
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
Multiply each side by |b|. =
Divide each side by 0.025; b is positive =
Example 3

21. y = a sin b Write the general equation for the sine function.
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

22. 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 second
B second
C second
D. 0.1 second A B C D
Example 3

23. 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 A B C D
Example 3

24. Concept

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

26. Use y = tan , but only draw one cycle every 360°.
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

27. A B C D 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. A B C D
Example 4

28. Concept

29. Find the period of y = 3 csc. Then graph the function.
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

30. Graph Other Trigonometric Functions
Answer:
Example 5

31. 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° A B C D
Example 5

32. End of the Lesson