Question

$\sin \theta ,\cos \theta$are given.Find the exact value of each of the four remaining trigonometric functions localid="1646650767506" role="math" $\sin \theta =\frac{2\sqrt{5}}{5}and\cos \theta =\frac{\sqrt{5}}{5}$

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

In this 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 trigonometric function tanθ and cotθ 

We know that

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