Concept

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

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

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

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

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

Graph Sine and Cosine Functions Answer: Example 2

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

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

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 0.025 second. Example 3

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

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

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

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

Concept

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

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

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

Concept

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

Graph Other Trigonometric Functions Answer: Example 5

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