Trigonometry (11th Edition) Clone

Published by Pearson
ISBN 10: 978-0-13-421743-7
ISBN 13: 978-0-13421-743-7

Chapter 4 - Graphs of the Circular Functions - Section 4.1 Graphs of the Sine and Cosine Functions - 4.1 Exercises - Page 152: 62

Answer

In general, $\cos{(bx)} \ne b\cdot \cos{x}$ because they have different amplitudes and periods. Refer to the graph in the step-by-step part below.

Work Step by Step

RECALL: The function $y=a \cdot \cos{(bx)}$ has : amplitude = $|a|$ period = $\frac{2\pi}{b}$ Thus: The function $y=\cos{(3x)}$ has an amplitude of $|1|=1$ and a period of $\frac{2\pi}{3}$. The function $y=3\cos{x}$ has an amplitude of $|3|=3$ and a period of $\frac{2\pi}{1}=2\pi$. From the information above, it is obvious that the two functions are different from each other because they have different amplitudes and periods. Thus, it cannot be said that in general, $\cos{(bx)}=b \cdot \cos{x}$. Use a graphing utility to graph the two functions. (Refer to the attached image below for the graph, the green graph is $y=3\cos{x}$ while the red graph is $y=\cos{(3x)}$.) Use a graphing utility to graph the given functions. (Refer to the graph below.) Notice that the graphs are different.
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