Q.41 - Precalculus Enhanced with Graphing Utilities

Question

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

Step-by-Step Solution

Answer

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

Step 1. Given information

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

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

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

Since

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