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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. 19 (page 435)

In Problems 16鈥�23, find the exact value of each of the remaining trigonometric functions.
$\sin 胃 = \frac{12}{13}, 胃$ lies in the quadrant ll.

Short Answer

Expert verified

The exact values of the remaining trigonometric functions of $\sin 胃 = \frac{12}{13}$ are,

$\cos 胃 = - \frac{5}{13} \tan 胃 = - \frac{12}{5} c s c 胃 = \frac{13}{12} s e c 胃 = - \frac{13}{5} c o t 胃 = - \frac{5}{12}$

Step by step solution

01

Step 1 Given expression is,

$\sin 胃 = \frac{12}{13}$ and $胃$ lies in the second quadrant.

So, $\sin 胃, c s c 胃$ are positive in the second quadrant and the remaining trigonometric functions are negative.

Let $伪$ be the reference angle for $胃$ .

$\sin 胃 = \sin 伪$ .

So, $\sin 胃 = \frac{12}{13} = \frac{o p p o s i t e}{h y p o t e n u s e}$

Use $b = 12, r = 13$ to construct a triangle.

From the figure, we obtain $a = 5$ .

02

Step 2 Now calculate the remaining trigonometric functions.

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

$\cos 伪 = \frac{a}{r} = \frac{5}{13} \tan 伪 = \frac{b}{a} = \frac{12}{5} c s c 伪 = \frac{r}{b} = \frac{13}{12} s e c 伪 = \frac{r}{a} = \frac{13}{5} c o t 伪 = \frac{a}{b} = \frac{5}{12}$

03

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

$\cos 胃 = - \frac{5}{13} \tan 胃 = - \frac{12}{5} c s c 胃 = \frac{13}{12} s e c 胃 = - \frac{13}{5} c o t 胃 = - \frac{5}{12}$

These are the remaining trigonometric functions.

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