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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. 12 (page 516)

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

Short Answer

Expert verified

The required trigonometric functions are:

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

Step by step solution

01

Step 1. Given Information

We have, $a d j a c e n t, a = 3$ 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, $a, b a n d c$ 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 hypotenuse, c first.

So using Pythagorean Theorem,

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

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

$9 + 9 = c^{2}$

$c = \sqrt{18} c = 3 \sqrt{2}$

Now, the six trigonometric functions are:

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

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