Q.39 - Precalculus Enhanced with Graphing Utilities

Question

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

Step-by-Step Solution

Answer

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

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  

Step 2 . Finding the value of trigonometric functions c s c θ   a n d   s e c θ

In this question 

$$\begin{gathered} \sin \theta =\frac{1}{2} so csc\theta =\frac{1}{\sin \theta } \\ =\frac{1}{{\displaystyle \frac{1}{2}}} \\ =2 \\ \cos \theta =\frac{\sqrt{3}}{2} then sec\theta =\frac{1}{\cos \theta } \\ =\frac{1}{{\displaystyle \frac{\sqrt{3}}{2}}} \\ =\frac{2}{\sqrt{3}} \end{gathered}$$

Step 3. Finding the value of the trigonometric function tan θ   a n d   c o t θ

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