1. Concept

2. 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

3. 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

4. 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

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

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

7. Graph Sine and Cosine Functions
Answer:
Example 2

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

9. B. Graph the function y = 3 cos 2.D.
Example 2

10. 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

11. 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

12. 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

13. 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
Example 3

14. 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

15. Concept

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

17. 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

18. 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

19. Concept

20. 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

21. Graph Other Trigonometric Functions
Answer:
Example 5

22. 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