Question

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

Short 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-by-step solution

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

. Finding the value of trigonometric functions cscθ and secθ 

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

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

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