Question

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

Short answer

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

Step-by-step solution

Step 1.  Given information 

In this question $\sin \theta =\frac{\sqrt{3}}{2},\cos \theta =\frac{1}{2}$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

Here$$\begin{gathered} \sin \theta =\frac{\sqrt{3}}{2}socsc\theta =\frac{1}{\sin \theta } \\ =\frac{1}{{\displaystyle \frac{\sqrt{3}}{2}}} \\ =\frac{2}{\sqrt{3}} \\ \cos \theta =\frac{1}{2}thensec\theta =\frac{1}{\cos \theta } \\ =\frac{1}{{\displaystyle \frac{1}{2}}} \\ =2 \end{gathered}$$$$\begin{gathered} \sin \theta =\frac{\sqrt{3}}{2}socsc\theta =\frac{1}{\sin \theta } \\ =\frac{1}{{\displaystyle \frac{\sqrt{3}}{2}}} \\ =\frac{2}{\sqrt{3}} \\ \cos \theta =\frac{1}{2}thensec\theta =\frac{1}{\cos \theta } \\ =\frac{1}{{\displaystyle \frac{1}{2}}} \\ =2 \end{gathered}$$

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

Since

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