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 6.2 Trigonometric Equations I - 6.2 Exercises - Page 273: 56


The solution set is $$\{60^\circ+180^\circ n, n\in Z\}$$

Work Step by Step

$$\tan\theta\csc\theta-\sqrt3\csc\theta=0$$ 1) Solve the equation over the interval $[0^\circ,360^\circ)$ $$\tan\theta\csc\theta-\sqrt3\csc\theta=0$$ $$\csc\theta(\tan\theta-\sqrt3)=0$$ $$\csc\theta=0\hspace{1cm}\text{or}\hspace{1cm}\tan\theta=\sqrt3$$ - For $\csc\theta=0$ The range of a cosectant function is $(-\infty,-1]\cup[1,\infty)$. Since $0\notin(-\infty,-1]\cup[1,\infty)$, there cannot be any values of $\theta\in[0^\circ, 360^\circ)$ that have $\csc\theta=0$. - For $\tan\theta=\sqrt3$ Over the interval $[0^\circ, 360^\circ)$, there are 2 values of $\theta$ where $\tan\theta=\sqrt3$, which are $60^\circ$ and $240^\circ$. Therefore, overall, $$\theta=\{60^\circ, 240^\circ\}$$ 2) Solve the equation for all solutions - The integer multiples of the period of the tangent function is $180^\circ$. - We apply it to each solution found in step 1, which would lead to $\theta=60^\circ+180^\circ n$ and $\theta=240^\circ+180^\circ n$ $(n\in Z)$ - However, both $60^\circ+180^\circ n$ and $240^\circ+180^\circ n$ refer to the same set of points, so we only need to include one in the solution set. The solution set is $$\{60^\circ+180^\circ n, n\in Z\}$$
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