Q.42 - Precalculus Enhanced with Graphing Utilities

Question

$\sin \theta , \cos \theta$ are given .Find the exact value of each of the four remaining trigonometric functions  $\sin \theta = \frac{2\sqrt{2}}{3} , \cos \theta =-\frac{1}{3}$

Step-by-Step Solution

Answer

The value of $csc\theta =\frac{3}{2\sqrt{2}},sec\theta =-\frac{1}{3}, \tan \theta =--2\sqrt{2}, cot\theta =-\frac{1}{2\sqrt{2}}$

Step 1. Given information

Here $\sin \theta = \frac{2\sqrt{2}}{3} , \cos \theta =-\frac{1}{3}$are given.

We need to find the exact value of each of the four remaining trigonometric functions   

Step 2 . Finding the value of trigonometric functions c s c θ   a n d   s e c θ

 In this question $$\begin{gathered} \sin \theta =\frac{2\sqrt{2}}{3} so csc\theta =\frac{1}{\sin \theta } \\ =\frac{1}{{\displaystyle \frac{2\sqrt{2}}{3}}} \\ =\frac{3}{2\sqrt{2}} \\ \cos \theta =\frac{-1}{3} then sec\theta =\frac{1}{\cos \theta } \\ =\frac{1}{{\displaystyle \frac{-1}{3}}} \\ =-3 \end{gathered}$$

Step 3. Finding the value of the trigonometric function tan θ     a n d   c o t θ

Here $$\begin{gathered} \tan \theta =\frac{\sin \theta }{\cos \theta } \\ =\frac{{\displaystyle \frac{2\sqrt{2}}{3}}}{{\displaystyle -\frac{1}{3}}} \sin ce \sin \theta =\frac{2\sqrt{2}}{3} and \cos \theta =-\frac{1}{3} \\ =-2\sqrt{2} \\ and cot\theta =\frac{1}{\tan \theta } \\ =-\frac{1}{{\displaystyle 2\sqrt{2}}} \end{gathered}$$