Q.44 - Precalculus Enhanced with Graphing Utilities

Question

Find the exact value of each of the remaining trigonometric functions of $\theta .$

$\cos \theta =\frac{3}{5} ,\theta is on fourth quadrant$

Step-by-Step Solution

Answer

The value of $\sin \theta =-\frac{4}{5} , sec\theta =\frac{5}{3},csc\theta =-\frac{5}{4} , \tan \theta =-\frac{4}{3} and cot \theta =-\frac{3}{4}$

Step 1. Given information

 Here $\cos \theta =\frac{3}{5} , \theta is on fourth quadrant is given.$

 we have to find exact value of each of the remaining trigonometric functions  

Step 2.Description of finding the value of sin θ

 In this question $$\begin{gathered} \cos \theta =\frac{3}{5 } \\ We know that {\sin }^2\theta +{\cos }^2\theta =1 \\ \sin \theta =\pm \sqrt{1-{\cos }^2\theta } and \sin ce \cos \theta is on fourth quadrant all other trignometric ratios will be negative. \\ hence \sin \theta =-\sqrt{1-{\cos }^2\theta } \\ =-\sqrt{1-\frac{3^2}{5^2}} \\ =-\sqrt{1-\frac{9}{25}} \\ =-\frac{4}{5} \end{gathered}$$

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

  According to the question

 $$\begin{gathered} \cos \theta =\frac{3}{5} then sec\theta =\frac{1}{\cos \theta } \\ =\frac{1}{{\displaystyle \frac{3}{5}}} \\ =\frac{5}{3} \\ csc\theta =\frac{1}{\sin \theta } \\ =\frac{1}{-{\displaystyle \frac{4}{5}}} \\ =-\frac{5}{4} \end{gathered}$$

Step 4. Finding the value of tan θ   a n d   c o t   θ          

Since $$\begin{gathered} \sin \theta =-\frac{4}{5} and \cos \theta =\frac{3}{5} \\ \tan \theta =\frac{\sin \theta }{\cos \theta } \\ =\frac{-\frac{4}{5}}{\frac{3}{5}} \\ =\frac{-4}{3} \\ cot\theta =\frac{1}{\tan \theta } \\ =\frac{1}{-{\displaystyle \frac{4}{3}}} \\ =-\frac{3}{4} \end{gathered}$$