Chapter 6.3, Problem 42AYU

In problems 35-42, $\sin \theta$ and $\cos \theta$ are given. Find the exact value of the four remaining trigonometric functions.

$\sin \theta \text{=}\frac{2\sqrt{2}}{3}$, $\cos \theta \text{=}-\frac{1}{3}$

To determine

To find: The exact value of each of the four remaining trigonometric functions.

Answer to Problem 42AYU

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

Explanation of Solution

Given:

$\text{sin}\theta =\frac{2\sqrt{2}}{3},\text{cos}\theta =-\frac{1}{3}$

Calculation:

$\text{tan}\theta =\frac{\text{sin}\theta }{\text{cos}\theta }=\frac{\frac{2\sqrt{2}}{3}}{-\frac{1}{3}}=-2\sqrt{2}$

$\text{csc}\theta =\frac{1}{\text{sin}\theta }=\frac{1}{\frac{2\sqrt{2}}{3}}=\frac{3}{2\sqrt{2}}=\frac{3\sqrt{2} }{4}$

$\text{sec}\theta =\frac{1}{\text{cos}\theta }=\frac{1}{-\frac{1}{3}}=-3$

$\text{cot}\theta =\frac{1}{\text{tan}\theta }=\frac{1}{-2\sqrt{2}}=-\frac{\sqrt{2}}{4}$