Trigonometry 7th Edition

Published by Cengage Learning
ISBN 10: 1111826854
ISBN 13: 978-1-11182-685-7

Chapter 1 - Section 1.3 - Definition I: Trigonometric Functions - 1.3 Problem Set - Page 32: 37


With $\csc{\theta} = \dfrac{r}{y}$ and $r \ge |y|$, then $|\csc{\theta}|\ge1$ for any angle in standard position since the value of $|\frac{r}{y}|$ will never be less than $1$. Refer to the step-by-step part for the explanation.

Work Step by Step

Definition I states that $\csc{\theta}= \dfrac{r}{y}, y\ne0$. Figure 1 shows that $r$ is the hypotenuse (longest side) of the right triangle, while $x$ and $y$ are the legs. Since $r$ is longer than or equal to $|y|$, then $|\frac{r}{y}|$ will be never smaller than $1$ Thus,for any angle in standard position, $|\csc{\theta}|$ will always be greater than or equal to $1$.
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