Precalculus (10th Edition)

Published by Pearson
ISBN 10: 0-32197-907-9
ISBN 13: 978-0-32197-907-0

Chapter 10 - Analytic Geometry - Chapter Review - Cumulative Review - Page 701: 10

Answer

$x\not=\dfrac{3\pi}{4}+n\pi, n$ is an integer

Work Step by Step

We are given the function: $f(x)=\dfrac{3}{\sin x+\cos x}$ In order to find the function's domain, we must eliminate the values of $x$ which are solutions of the equation: $\sin x+\cos x=0$ $(\sin x+\cos x)^2=0$ $\sin^2 x+2\sin x\cos x+\cos^2 x=0$ $(\sin^2 x+\cos^2 x)+\sin (2x)=0$ $1+\sin (2x)=0$ $\sin (2x)=-1$ $2x=\dfrac{3\pi}{2}+2n\pi$ $x=\dfrac{3\pi}{4}+n\pi$ The domain of the function is the set of all real numbers except those in the form $\dfrac{3\pi}{4}+n\pi$, where $n$ is any integer.
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