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Q. 4

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Found in: Page 485

### Precalculus Enhanced with Graphing Utilities

Book edition 6th
Author(s) Sullivan
Pages 1200 pages
ISBN 9780321795465

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# if $\mathrm{sin}\alpha =-\frac{4}{5}$, $\mathrm{\pi }<\mathrm{\alpha }<\frac{3\mathrm{\pi }}{2}$, then $\mathrm{cos}\alpha =$_____

$\mathrm{cos}\alpha =-\frac{3}{5}$

See the step by step solution

## Step 1. Given information

Given is:

$\mathrm{sin}\alpha =-\frac{4}{5}$

We have to find the value of $\mathrm{cos}\alpha$.

## Step 2. Calculate the adjacent side

We consider the right-angled triangle which opposite side is equal to $-4$ and its hypotenuse equals to $5$ and let $\theta$ be the angle formed by the intersection of the adjacent and the hypotenuse sides. We get

$\mathrm{sin}\alpha =-\frac{4}{5}:\frac{opposite}{hypotenuse}$

Calculate the adjacent side,

$adjacent=\sqrt{hypotenus{e}^{2}-opposit{e}^{2}}\phantom{\rule{0ex}{0ex}}=\sqrt{25-16}\phantom{\rule{0ex}{0ex}}=\sqrt{9}\phantom{\rule{0ex}{0ex}}=3$

## Step 3. Find cos α

As $\mathrm{cos}\alpha =\frac{adjacent}{hypotenuse}$

and $\mathrm{cos}\alpha$ is negative in the second quadrant.

Therefore,

$\mathrm{cos}\alpha =-\frac{3}{5}$

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