Trigonometry (10th Edition)

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

Chapter 6 - Review Exercises - Page 285: 53

Answer

$\theta = 180^{\circ}+360^{\circ}~n$, where $n$ is a non-negative integer.

Work Step by Step

$tan~\theta-sec~\theta = 1$ $\frac{sin~\theta}{cos~\theta}-\frac{1}{cos~\theta} = 1$ $sin~\theta-1 = cos~\theta$ $sin^2~\theta-2~sin~\theta +1 = cos^2~\theta$ $sin^2~\theta-2~sin~\theta +1 = 1-sin^2~\theta$ $2~sin^2~\theta = 2~sin~\theta$ $sin^2~\theta = sin~\theta$ Therefore, $sin~\theta = 0$ or $sin~\theta = 1$ When $sin~\theta = 0$: $\theta = 0, 180^{\circ}$ When we check these possible solutions in the original equation, only $\theta = 180^{\circ}$ is a valid solution. When $sin~\theta = 1$: $\theta = 90^{\circ}$ When we check this possible solution in the original equation, $sec~\theta$ is undefined. Therefore, $\theta = 90^{\circ}$ is not a valid solution. The only valid solution is $\theta = 180^{\circ}$ In general, the solutions have the form: $\theta = 180^{\circ}+360^{\circ}~n$, where $n$ is a non-negative integer.
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