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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. 45 (page 435)

Find the exact values of the six trigonometric functions of $t$ if $P = (- \frac{1}{3}, \frac{2 \sqrt{2}}{3})$ is the point on the unit circle that corresponds to $t$ .

Short Answer

Expert verified

The value is

$\tan t = - 2 \sqrt{2}, \csc t = \frac{3 \sqrt{2}}{4} \sec t = - 3 \cot t = - \frac{\sqrt{2}}{4}$

Step by step solution

01

Given information

Given the point $(- \frac{1}{3}, \frac{2 \sqrt{3}}{3})$

02

Identify the point and use the formula

A point on circle is of the form $P (\sin 胃, \cos 胃)$ ,so

$\sin t = \frac{2 \sqrt{2}}{3} \cos t = - \frac{1}{3}$

$\tan t = \frac{y}{x} \tan t = \frac{\frac{2 \sqrt{2}}{3}}{- \frac{1}{3}} \tan t = - 2 \sqrt{2}$

03

Calculating other ratios

Calculating, we get

$\csc t = \frac{1}{y} c s c t = \frac{\frac{1}{2 \sqrt{2}}}{3} c s c t = \frac{3}{2 \sqrt{2}} 路 \frac{\sqrt{2}}{\sqrt{2}} c s c t = \frac{3 \sqrt{2}}{4}$

$\sec t = \frac{1}{x} s e c t = \frac{1}{- \frac{1}{3}} s e c t = - 3$

04

Calculating other ratios

Calculating, we get

$\cot t = \frac{x}{y} \cot t = \frac{- \frac{1}{3}}{\frac{2 \sqrt{2}}{3}} \cot t = - \frac{1}{2 \sqrt{2}} 路 \frac{\sqrt{2}}{\sqrt{2}} \cot t = - \frac{\sqrt{2}}{4}$

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Most popular questions from this chapter

In Problems 13鈥�20, $P (x, y)$ is the point on the unit circle that corresponds to a real number $t$ . Find the exact values of the six trigonometric functions of $t$ .

$(- \frac{2}{5}, \frac{\sqrt{21}}{5})$ .

In problems 13-20, $P (x, y)$ is the point on the unit circle that corresponds to a real number $t$ . Find the exact values of the six trigonometric functions of $t$ .

$(- \frac{1}{5}, \frac{2 \sqrt{6}}{5})$ .

In Problems 13鈥�20, $P (x, y)$ is the point on the unit circle that corresponds to a real number $t$ . Find the exact values of the six trigonometric functions of $t$ .

$(\frac{1}{2}, - \frac{\sqrt{3}}{2})$ .

Find the exact value of each expression. Do not use a calculator.

$2 \sin \frac{蟺}{3} - 3 \tan \frac{蟺}{6}$

Use a calculator to find the approximate value of each expression rounded to two decimal places: $\csc \frac{5 蟺}{13}$

See all solutions

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