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6. Sketch the graph of y= Cos (0.5x + 20m) from x = -2π to x = 2π

6. Sketch the graph of y= Cos (0.5x + 20m) from x = -2π to x = 2π

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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**Problem 6: Sketch the Graph of y = Cos(0.5x + 20π)** **Instructions:** - Sketch the graph of the function \( y = \cos(0.5x + 20\pi) \). - Consider the interval from \( x = -2\pi \) to \( x = 2\pi \). **Graph Explanation:** The image contains a Cartesian plane featuring an x-axis labeled with increments of \( \pi \), ranging from \( -2\pi \) to \( 2\pi \). The y-axis is oriented upright, intersecting the x-axis at the origin (0,0). Both axes are marked, but the sketch of the cosine function is not displayed. **Key Considerations:** - **Amplitude:** The amplitude of the cosine function is 1. - **Period:** The coefficient of x is 0.5, affecting the period of the function. The period \( T \) is calculated as \( T = \frac{2\pi}{0.5} = 4\pi \). - **Phase Shift:** The phase shift is determined by the constant term inside the cosine function, \( 20\pi \), which results in no visible shift as it is a multiple of \( 2\pi \). - **Sketching Steps:** 1. Identify critical points where \( y = 1, 0, -1 \). 2. Recognize that due to the period adjustment, one full cycle of the cosine curve spans from 0 to \( 4\pi \). 3. Plot the corresponding y-values at significant x-values within the specified range. By sketching this function, students can visualize how changes in coefficients and constants inside trigonometric functions modify their graphs.
Transcribed Image Text:**Problem 6: Sketch the Graph of y = Cos(0.5x + 20π)** **Instructions:** - Sketch the graph of the function \( y = \cos(0.5x + 20\pi) \). - Consider the interval from \( x = -2\pi \) to \( x = 2\pi \). **Graph Explanation:** The image contains a Cartesian plane featuring an x-axis labeled with increments of \( \pi \), ranging from \( -2\pi \) to \( 2\pi \). The y-axis is oriented upright, intersecting the x-axis at the origin (0,0). Both axes are marked, but the sketch of the cosine function is not displayed. **Key Considerations:** - **Amplitude:** The amplitude of the cosine function is 1. - **Period:** The coefficient of x is 0.5, affecting the period of the function. The period \( T \) is calculated as \( T = \frac{2\pi}{0.5} = 4\pi \). - **Phase Shift:** The phase shift is determined by the constant term inside the cosine function, \( 20\pi \), which results in no visible shift as it is a multiple of \( 2\pi \). - **Sketching Steps:** 1. Identify critical points where \( y = 1, 0, -1 \). 2. Recognize that due to the period adjustment, one full cycle of the cosine curve spans from 0 to \( 4\pi \). 3. Plot the corresponding y-values at significant x-values within the specified range. By sketching this function, students can visualize how changes in coefficients and constants inside trigonometric functions modify their graphs.
Expert Solution
Introduction

Cosine graph:

   The cosine graph is up and down graph i.e. 

            Advanced Math homework question answer, step 1, image 1

   The cosine graph is always between -1 and 1.

 

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