Trigonometry (10th Edition)

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

Chapter 6 - Inverse Circular Functions and Trigonometric Equations - Section 6.3 Trigonometric Equations II - 6.3 Exercises - Page 273: 5

Answer

Step 3 (perform the division) is wrong, as we cannot simplify $\frac{\tan2\theta}{2}$ into $\tan\theta$, leading to wrong results in step 4.

Work Step by Step

Solve $\tan2\theta=2$ over the interval $[0,2\pi)$ We examine each step: 1) Original equation: $$\tan2\theta=2$$ This is obviously correct. 2) Divide by 2 $$\frac{\tan2\theta}{2}=\frac{2}{2}$$ This is probably weird but still correct. 3) Perform the division $$\tan\theta=1$$ Now this step is wrong. Divide the whole $\tan2\theta$ by $2$ would not make it into $\tan\theta$. Instead, it would stay the same as $\frac{\tan2\theta}{2}$. Only $\tan(\frac{2\theta}{2})$ could change to $\tan\theta$ 4) Definition of inverse tangent $$\theta=\frac{\pi}{4}\hspace{1cm}\text{or}\hspace{1cm}\theta=\frac{5\pi}{4}$$ Step 3 was carried out wrongly, leading to wrong results in step 4, though the application of inverse tangent definition is correct.
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