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Precalculus Enhanced with Graphing Utilities

Sullivan

$Math Studyset 91影视 Explanations$ Math

6th Edition 路 1200 pages

ISBN: 9780321795465

Chapter 6: Q. 23 (page 435)

In Problems 16鈥�23, find the exact value of each of the remaining trigonometric functions.
$c o t 胃 = - 2, \frac{蟺}{2} < 胃 < 蟺$ .

Short Answer

Expert verified

The exact values of the remaining trigonometric functions of $c o t 胃 = - 2$ are,

$\sin 胃 = \frac{\sqrt{5}}{5} \cos 胃 = - \frac{2 \sqrt{5}}{5} \tan 胃 = - \frac{1}{2} c s c 胃 = \sqrt{5} s e c 胃 = - \frac{\sqrt{5}}{2}$

Step by step solution

01

Step 1 Given expression is,

$c o t 胃 = - 2$ and $\frac{蟺}{2} < 胃 < 2 蟺$ .

So, $胃$ lies in the second quadrant where $c o t 胃$ is negative.

Let $伪$ be the reference angle for $胃$ then $c o t 胃 = \frac{a}{b} = c o t 伪$ .

So, $c o t 伪 = \frac{2}{1} = \frac{A d j a c e n t}{O p p o s i t e}$

Use the values $a = 2, b = 1$ to construct a triangle.

From the figure, we can obtain $r = \sqrt{5}$ .

02

Step 2 Now find the exact values of the remaining trigonometric functions.

The exact values of the remaining trigonometric functions of the reference angle $伪$ are,

role="math" localid="1646419068024" $\sin 伪 = \frac{b}{r} = \frac{1}{\sqrt{5}} = \frac{\sqrt{5}}{5} \cos 伪 = \frac{a}{r} = \frac{2}{\sqrt{5}} = \frac{2 \sqrt{5}}{5} \tan 伪 = \frac{b}{a} = \frac{1}{2} c s c 伪 = \frac{r}{b} = \frac{\sqrt{5}}{1} = \sqrt{5} s e c 伪 = \frac{r}{a} = \frac{\sqrt{5}}{2}$

03

Step 3 The exact values of the remaining trigonometric functions of the angle θ are,

$\sin 胃 = \frac{\sqrt{5}}{5} \cos 胃 = - \frac{2 \sqrt{5}}{5} \tan 胃 = - \frac{1}{2} c s c 胃 = \sqrt{5} s e c 胃 = - \frac{\sqrt{5}}{2}$

These are the exact values of the remaining trigonometric functions.

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