Q.40 - Precalculus Enhanced with Graphing Utilities

Question

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

Step-by-Step Solution

Answer

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

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  

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

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

Step 3. Finding the value of the trigonometric function tan θ   a n d   s e c θ

 Since 

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