Trigonometry (11th Edition) Clone

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

Chapter 6 - Inverse Circular Functions and Trigonometric Equations - Section 6.1 Inverse Circular Functions - 6.1 Exercises - Page 264: 27


The value of $y$ here is $$y=\frac{3\pi}{4}$$

Work Step by Step

$$y=\cot^{-1} (-1)$$ First, we see that the domain of inverse cotangent function is $(-\infty,\infty)$. Therefore, in fact when we deal with inverse cotangent function, we do not need to do this checking step. The range of inverse cotangent function is $(0,\pi)$. In other words, $y\in(0,\pi)$. We can rewrite $y=\cot^{-1}(-1)$ into $\cot y=-1$ We know that $$\cot\frac{\pi}{4}=1$$ which means $$-\cot\frac{\pi}{4}=-1$$ $$\cot(-\frac{\pi}{4})=-1$$ (If you need a proof here, take $\cot(-X)=\frac{\cos(-X)}{\sin(-X)}$. We know that $\cos(-X)=\cos X$ and $\sin(-X)=-\sin X$. That means $\cot(-X)=\frac{\cos X}{-\sin X}=-\cot X$). However, $-\frac{\pi}{4}$ does not belong to the range $(0,\pi)$. So we must take another similar value but stays in quadrant 1 or 2, which belongs to the range. Since $\cot (X+\pi) =\cot x$, $\frac{3\pi}{4}$ is such a value. Therefore, the exact value of $y$ here is $$y=\frac{3\pi}{4}$$
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