Precalculus: Mathematics for Calculus, 7th Edition

Published by Brooks Cole
ISBN 10: 1305071751
ISBN 13: 978-1-30507-175-9

Chapter 5 - Section 5.4 - More Trigonometric Graphs - 5.4 Exercises - Page 439: 64

Answer

If $f$ and $g$ are periodic with period $p$, then $\frac{f}{g}$ is also periodic but its period could be smaller than $p$

Work Step by Step

$f(x) = f(x+p)$ and $g(x) = g(x+p)$ by the definition of periodic functions. Let's consider $\frac{f}{g}$: $\frac{f(x)}{g(x)} = \frac{f(x+p)}{g(x+p)}$ We can see that $\frac{f}{g}$ is also periodic. The period could be $p$, or $p$ could be subdivided into two or more smaller periods. To see an example, we can consider $f(x) = sin~x$ and $g(x) = cos~x$ $\frac{f(x)}{g(x)} = \frac{sin~x}{cos~x} = tan~x$ The period of both $sin~x$ and $cos~x$ is $2\pi$ However, the period of $tan~x$ is $\pi$ Therefore, if $f$ and $g$ are periodic with period $p$, then $\frac{f}{g}$ is also periodic but its period could be smaller than $p$
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