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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. 14 (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 胃 = \frac{b}{c} = \frac{3}{4} c s c 胃 = \frac{c}{b} = \frac{4}{3} \cos 胃 = \frac{a}{c} = \frac{\sqrt{7}}{4} s e c 胃 = \frac{c}{a} = \frac{4}{\sqrt{7}} \tan 胃 = \frac{b}{a} = \frac{3}{\sqrt{7}} c o t 胃 = \frac{a}{b} = \frac{\sqrt{7}}{3}

Step by step solution

01

Step 1. Given Information

From the figure, we have,

hypotenuse, $c = 4$ and opposite, $b = 3$ .

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 out the adjacent first.

By Pythagorean Theorem,

$c^{2} = a^{2} + b^{2} a^{2} = 4^{2} - 3^{2} a^{2} = 16 - 9 a^{2} = 7 a = \sqrt{7}$

Now, the six trigonometric functions are:

$\sin 胃 = \frac{b}{c} = \frac{3}{4} c s c 胃 = \frac{c}{b} = \frac{4}{3} \cos 胃 = \frac{a}{c} = \frac{\sqrt{7}}{4} s e c 胃 = \frac{c}{a} = \frac{4}{\sqrt{7}} \tan 胃 = \frac{b}{a} = \frac{3}{\sqrt{7}} c o t 胃 = \frac{a}{b} = \frac{\sqrt{7}}{3}$

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