Question

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

Short answer

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

Step-by-step solution

Step 1.  Given information 

Here $\sin \theta =\frac{1}{2},\cos \theta \frac{\sqrt{3}}{2}$is 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

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

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

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