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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 8: Q. 11 (page 516)

Find the exact value of the six trigonometric functions of the angle $胃$ in each figure.

Short Answer

Expert verified

The required six trigonometric functions are:

$\sin 蠎 = \frac{3}{\sqrt{13}} \cos 胃 = \frac{2}{\sqrt{13}} \tan 胃 = \frac{3}{2} c s c 胃 = \frac{\sqrt{13}}{3} s e c 胃 = \frac{\sqrt{13}}{2} c o t 胃 = \frac{2}{3}$

Step by step solution

01

Step 1. Given Information

From the figure, we have

$a d j a c e n t, a = 2$ and $o p p o s i t e, b = 3$

02

Step 2. Formula Used

Pythagorean Theorem: $a^{2} + b^{2} = c^{2}$

where, a, b and c are adjacent, opposite and hypotenuse respectively.

Trigonometric Functions:

$\sin 胃 = \frac{b}{c} c s c 胃 = \frac{c}{b} \cos 胃 = \frac{a}{c} s e c 胃 \frac{c}{a} \tan 胃 = \frac{b}{a} c o t 胃 = \frac{a}{b}$

03

Step 3. Calculation

We have, $a d j a c e n t, a = 2$ and $o p p o s i t e, b = 3$

We have to find hypotenuse, c

Using Pythagorean Theorem,

$c^{2} = a^{2} + b^{2}$

$c^{2} = 2^{2} + 3^{2}$

$c^{2} = 4 + 9$

$c^{2} = 13$

$c = \sqrt{13}$

Now, the value of the trigonometric functions are:

$\sin 胃 = \frac{b}{c} = \frac{3}{\sqrt{13}}$ $\cos 胃 = \frac{a}{c} = \frac{2}{\sqrt{13}}$

$\tan 胃 = \frac{b}{a} = \frac{3}{2}$ $c s c 胃 = \frac{c}{b} = \frac{\sqrt{13}}{3}$

$s e c 胃 \frac{c}{a} = \frac{\sqrt{13}}{2}$ $c o t 胃 \frac{a}{b} = \frac{2}{3}$

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