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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. 18 (page 435)

In Problems 16鈥�23, find the exact value of each of the remaining trigonometric functions.
$s e c 胃 = - \frac{5}{4}, \tan 胃 < 0$

Short Answer

Expert verified

The exact value of the remaining trigonometric functions of $s e c 胃 = - \frac{5}{4}$ are,

$\sin 胃 = - \frac{3}{5} \cos 胃 = \frac{4}{5} \tan 胃 = - \frac{3}{4} c s c 胃 = - \frac{5}{3} c o t 胃 = - \frac{4}{3}$

Step by step solution

01

Step 1 Given expression is,

$s e c 胃 = - \frac{5}{4}$ , $\tan 胃 < 0$ .

Since $胃$ lies in the second quadrant, $t a n 胃, s e c 胃$ are negative.

Let $伪$ be the reference angle for $胃$ then, $s e c 胃 = \frac{r}{a} = s e c 伪$ .

So, $s e c 伪 = \frac{5}{4} = \frac{H y p o t e n u s e}{A d j a c e n t}$

02

Step 2 use the values r=5,a=4 to construct a triangle.

From the figure, we obtain $b = 3$ .

03

Step 3 Now calculate the remaining trigonometric functions.

The remaining trigonometric functions of the reference angle $伪$ are calculated as follows,

$\sin 伪 = \frac{b}{r} = \frac{3}{5} \cos 伪 = \frac{a}{r} = \frac{4}{5} \tan 伪 = \frac{b}{a} = \frac{4}{3} c s c 伪 = \frac{r}{b} = \frac{5}{3} c o t 伪 = \frac{a}{b} = \frac{4}{3}$

04

Step 4 Now calculate the remaining trigonometric functions of the angle θ.

$\sin 胃 = - \frac{3}{5} \cos 胃 = \frac{4}{5} \tan 胃 = - \frac{3}{4} c s c 胃 = - \frac{5}{3} c o t 胃 = - \frac{4}{3}$

These are the exact values of the remaining trigonometric functions.

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Most popular questions from this chapter

Convert each angle in degrees to radians. Express your answer as a multiple of $蟺$ .

$270 掳$

For a circle of radius r, a central angle of u degrees subtends an arc whose length s is $s = \frac{蟺}{180} r 胃$ . Discuss whether this is a true or false statement. Give reasons to defend your position.

Find the amplitude, period, and phase shift of each function. Graph each function. Be sure to label key points. Show at least two periods.

$y = 2 \cos (3 x + \frac{蟺}{2})$

Find the exact values of the six trigonometric functions of the given angle. If any are not defined, say "not defined." Do not use a calculator.

$\frac{2 蟺}{3}$

Convert each angle to a decimal in degrees. Round your answer to two decimal places.

$40 掳 10' 25''$

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