Calculus (3rd Edition)

Published by W. H. Freeman
ISBN 10: 1464125260
ISBN 13: 978-1-46412-526-3

Chapter 1 - Precalculus Review - 1.4 Trigonometric Functions - Exercises - Page 30: 21


$\sin\theta=\frac{2}{\sqrt {53}}$, $\sec \theta=\frac{\sqrt {53}}{7}$ and $\cot \theta=\frac{7}{2}$

Work Step by Step

$\cot \theta=\frac{1}{\tan\theta}=\frac{7}{2}$ $\sec^{2}\theta =1+\tan^{2}\theta=1+\frac{4}{49}=\frac{53}{49}$ $\implies \sec \theta=\frac{\sqrt {53}}{7}$ ($\sec \theta$ is positive as $0\leq\theta\lt\frac{\pi}{2}$) $\csc^{2}\theta=1+\cot^{2}\theta=1+\frac{49}{4}=\frac{53}{4}$ $\implies \csc\theta= \frac{\sqrt {53}}{2}$ $\sin\theta= \frac{1}{\csc\theta}=\frac{2}{\sqrt {53}}$
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