University Calculus: Early Transcendentals (3rd Edition)

Published by Pearson
ISBN 10: 0321999584
ISBN 13: 978-0-32199-958-0

Chapter 1 - Section 1.3 - Trigonometric Functions - Exercises - Page 27: 7


$$\cos x=-\frac{4}{5}$$ $$\tan x=-\frac{3}{4}$$

Work Step by Step

$$\sin x = \frac{3}{5}, x\in\Big[\frac{\pi}{2},\pi\Big]$$ 1) As $x\in\Big[\frac{\pi}{2},\pi\Big]$, it means angle $x$ is in the second quadrant, so $\cos x\lt0$ and $\tan x\lt0$. 2) To find $\cos x$, we employ the formula: $$\cos^2 x+\sin^2x=1$$ $$\cos^2x=1-\sin^2x=1-\Big(\frac{3}{5}\Big)^2=1-\frac{9}{25}=\frac{16}{25}$$ $$|\cos x|=\frac{4}{5}$$ Since $\cos x\lt0$, $$\cos x=-\frac{4}{5}$$ 3) To find $\tan x$, we employ the formula: $$\tan x=\frac{\sin x}{\cos x}=\frac{\frac{3}{5}}{-\frac{4}{5}}=-\frac{3}{4}$$
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