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SHORT ANSWER QUESTIONS: Using radians, find the amplitude and period of each function. Then graph. Show your work 2πt 13) y = 4sin + 14) y = 3cos 3cos (40+)+2 3 T T 3 5m R KIN. 3x 2π 5x 3x T 4 R 7z2z

SHORT ANSWER QUESTIONS: Using radians, find the amplitude and period of each function. Then graph. Show your work 2πt 13) y = 4sin + 14) y = 3cos 3cos (40+)+2 3 T T 3 5m R KIN. 3x 2π 5x 3x T 4 R 7z2z

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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--- ### Trigonometric Functions: Amplitude and Period #### Objective: Using the graphs provided, determine the amplitude and period of each trigonometric function. Represent the function graphically and show detailed work. **Short Answer Questions:** 1. **Function 1: \( y = 4\sin\left( \frac{\theta + 2\pi}{3} \right) \)** - **Amplitude Calculation:** - The amplitude of the function \( y = a \sin(b\theta + c) \) is given by the coefficient \( a \). - For \( y = 4 \sin\left( \frac{\theta + 2\pi}{3} \right) \), the amplitude is \( |4| = 4 \). - **Period Calculation:** - The period of \( y = \sin(b\theta) \) is given by \( \frac{2\pi}{|b|} \). - Here, \( b = \frac{1}{3} \), so the period is \( \frac{2\pi}{\frac{1}{3}} = 6\pi \). **Graph Explanation:** - The x-axis is labeled in multiples of \( \pi \) (i.e., \( -3\pi, -2\pi, -\pi, \pi, 2\pi, 3\pi \)). - The y-axis ranges from -4 to 4, indicating the amplitude range. - The sinusoidal wave oscillates between 4 and -4 along the y-axis over the period of \( 6\pi \). 2. **Function 2: \( y = 3\cos\left(4\theta + \frac{\pi}{3}\right) + 2 \)** - **Amplitude Calculation:** - The amplitude of the function \( y = a \cos(b\theta + c) \) is given by the coefficient \( a \). - For \( y = 3\cos\left(4\theta + \frac{\pi}{3}\right) + 2 \), the amplitude is \( |3| = 3 \). - **Period Calculation:** - The period of \( y = \cos(b\theta) \) is given by \( \frac{2\pi}{|b|}
Transcribed Image Text:--- ### Trigonometric Functions: Amplitude and Period #### Objective: Using the graphs provided, determine the amplitude and period of each trigonometric function. Represent the function graphically and show detailed work. **Short Answer Questions:** 1. **Function 1: \( y = 4\sin\left( \frac{\theta + 2\pi}{3} \right) \)** - **Amplitude Calculation:** - The amplitude of the function \( y = a \sin(b\theta + c) \) is given by the coefficient \( a \). - For \( y = 4 \sin\left( \frac{\theta + 2\pi}{3} \right) \), the amplitude is \( |4| = 4 \). - **Period Calculation:** - The period of \( y = \sin(b\theta) \) is given by \( \frac{2\pi}{|b|} \). - Here, \( b = \frac{1}{3} \), so the period is \( \frac{2\pi}{\frac{1}{3}} = 6\pi \). **Graph Explanation:** - The x-axis is labeled in multiples of \( \pi \) (i.e., \( -3\pi, -2\pi, -\pi, \pi, 2\pi, 3\pi \)). - The y-axis ranges from -4 to 4, indicating the amplitude range. - The sinusoidal wave oscillates between 4 and -4 along the y-axis over the period of \( 6\pi \). 2. **Function 2: \( y = 3\cos\left(4\theta + \frac{\pi}{3}\right) + 2 \)** - **Amplitude Calculation:** - The amplitude of the function \( y = a \cos(b\theta + c) \) is given by the coefficient \( a \). - For \( y = 3\cos\left(4\theta + \frac{\pi}{3}\right) + 2 \), the amplitude is \( |3| = 3 \). - **Period Calculation:** - The period of \( y = \cos(b\theta) \) is given by \( \frac{2\pi}{|b|}
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