Question

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

Short 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-by-step solution

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

. Finding the value of trigonometric functions cscθ  and secθ

According to the question
$$\begin{gathered} \sin \theta =-\frac{1}{3}socsc\theta =\frac{1}{\sin \theta } \\ =-\frac{1}{{\displaystyle \frac{1}{3}}} \\ =-3 \\ \cos \theta =\frac{2\sqrt{2}}{3}thensec\theta =\frac{1}{\cos \theta } \\ =\frac{1}{{\displaystyle \frac{2\sqrt{2}}{3}}} \\ =\frac{3}{2\sqrt{2}} \end{gathered}$$

Step 3. Finding the value of the trigonometric function tanθ  and cotθ 

Since

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