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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. 13 (page 516)

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

Short Answer

Expert verified

The required trigonometric functions are:

$\sin 胃 = \frac{b}{c} = \frac{2 \sqrt{3}}{4} = \frac{\sqrt{3}}{2} c s c 胃 = \frac{c}{b} = \frac{4}{2 \sqrt{3}} = \frac{2}{\sqrt{3}} \cos 胃 = \frac{a}{c} = \frac{2}{4} = \frac{1}{2} s e c 胃 = \frac{c}{a} = \frac{4}{2} = 2 \tan 胃 = \frac{b}{a} = \frac{2 \sqrt{3}}{2} = \sqrt{3} c o t 胃 = \frac{a}{b} = \frac{2}{2 \sqrt{3}} = \frac{1}{\sqrt{3}}$

Step by step solution

01

Step 1. Given Information

From the figure, we have, $h y p o t e n u s e, c = 4$ and localid="1647160803805" $a d j a c e n t . a = 2$

02

Step 2. Formula Used

Pythagorean Theorem:

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

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 need to find the adjacent first,

So using Pythagorean Theorem,

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

$b^{2} = 4^{2} - 2^{2} b^{2} = 16 - 4 b^{2} = 12$

$b = 2 \sqrt{3}$

Now, trigonometric functions are:

$\sin 胃 = \frac{b}{c} = \frac{2 \sqrt{3}}{4} = \frac{\sqrt{3}}{2} c s c 胃 = \frac{c}{b} = \frac{4}{2 \sqrt{3}} = \frac{2}{\sqrt{3}} \cos 胃 = \frac{a}{c} = \frac{2}{4} = \frac{1}{2} s e c 胃 = \frac{c}{a} = \frac{4}{2} = 2 \tan 胃 = \frac{b}{a} = \frac{2 \sqrt{3}}{2} = \sqrt{3} c o t 胃 = \frac{a}{b} = \frac{2}{2 \sqrt{3}} = \frac{1}{\sqrt{3}}$

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