Question

$\sin \theta ,\cos \theta$ are given. Find the exact value of each of the four remaining trigonometric functions $\sin \theta =-\frac{2\sqrt{5}}{5}and\cos \theta =-\frac{\sqrt{5}}{5}$

Short answer

The value of$csc\theta =-\frac{5}{2\sqrt{5}},sec\theta =-\frac{5}{\sqrt{5}},\tan \theta =2andcot\theta =\frac{1}{2}$

Step-by-step solution

Step 1. Given information

$\sin \theta =-\frac{2\sqrt{5}}{5}and\cos \theta =-\frac{\sqrt{5}}{5}$are given.

We have to find out the exact value of each of the four remaining trigonometric functions.

. Finding the value of trigonometric functions cscθ and secθ 

According to the question

$$\begin{gathered} \sin \theta =-\frac{2\sqrt{5}}{5}socsc\theta =\frac{1}{\sin \theta } \\ =-\frac{1}{{\displaystyle \frac{2\sqrt{5}}{5}}} \\ =-\frac{5}{2\sqrt{5}} \\ \cos \theta =-\frac{\sqrt{5}}{5}thensec\theta =\frac{1}{\cos \theta } \\ =-\frac{1}{{\displaystyle \frac{\sqrt{5}}{5}}} \\ =-\frac{5}{\sqrt{5}} \end{gathered}$$

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

We know that$$\begin{gathered} \tan \theta =\frac{\sin \theta }{\cos \theta } \\ =\frac{{\displaystyle \frac{-2\sqrt{5}}{5}}}{{\displaystyle \frac{-\sqrt{5}}{5}}}\sin ce\sin \theta =\frac{-2\sqrt{5}}{5}and\cos \theta =\frac{-\sqrt{5}}{5} \\ =2 \\ andcot\theta =\frac{1}{\tan \theta } \\ =\frac{1}{{\displaystyle 2}} \end{gathered}$$