Trigonometry (10th Edition)

Published by Pearson
ISBN 10: 0321671775
ISBN 13: 978-0-32167-177-6

Chapter 5 - Trigonometric Identities - Section 5.2 Verifying Trigonometric Identities - 5.2 Exercises - Page 204: 92

Answer

$$\cos x=-\sqrt{1-\sin^2x}$$ This is a true statement only when the angle $x$ lies in quadrant II or III.

Work Step by Step

$$\cos x=-\sqrt{1-\sin^2x}$$ As from a Pythagorean Identity: $$\cos^2x=1-\sin^2x$$ That means $$\sqrt{\cos^2x}=\sqrt{1-\sin^2x}$$ $$|\cos x|=\sqrt{1-\sin^2x}$$ In other words, $$\cos x=\pm\sqrt{1-\sin^2x}$$ So we see now that there are 2 cases of $\cos x$, whether it is going to be $\cos x=\sqrt{1-\sin^2x}$ when $\cos x\ge0$ or $\cos x=-\sqrt{1-\sin^2x}$ when $\cos x\lt0$. In the requirement of the exercise given, it asks for the situation when $$\cos x=-\sqrt{1-\sin^2x}$$which happens when $$\cos x\lt0$$ That happens when the angle $x$ lies in quadrant II or III.
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