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: 33


The solution set to this problem is $$\{45^\circ,225^\circ\}$$

Work Step by Step

$$\csc^2\theta-2\cot\theta=0$$ over interval $[0^\circ,360^\circ)$ 1) Solve the equation: $$\csc^2\theta-2\cot\theta=0$$ Here we have two different trigonometric functions: cosecant and cotangent. It would be better if we have only one type, since only by then can we solve trigonometric functions. Seeing that we have $\csc^2\theta$, we can recall the identity: $\csc^2\theta=1+\cot^2\theta$. In fact, we would replace $\csc^2\theta$ with $1+\cot^2\theta$ $$1+\cot^2\theta-2\cot\theta=0$$ $$(\cot\theta-1)^2=0$$ $$\cot\theta-1=0$$ $$\cot\theta=1$$ 2) Apply the inverse function: Over the interval $[0^\circ,360^\circ)$, $\cot\theta\gt0$ means that $\theta$ angle lies either in quadrant I or quadrant III. In quadrant I, angle $45^\circ$ and in quadrant III, angle $225^\circ$ would have cosecant equal $1$. Therefore, $$\theta\in\{45^\circ,225^\circ\}$$ In other words, the solution set to this problem is $$\{45^\circ,225^\circ\}$$
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