Question

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

$\cos \theta =\frac{3}{5},\theta isonfourthquadrant$

Short answer

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

Step-by-step solution

Step 1. Given information 

Here $\cos \theta =\frac{3}{5},\theta isonfourthquadrantisgiven.$

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

Step 2.Description of  finding the value of  sinθ

In this question$$\begin{gathered} \cos \theta =\frac{3}{5} \\ Weknowthat{\sin }^2\theta +{\cos }^2\theta =1 \\ \sin \theta =\pm \sqrt{1-{\cos }^2\theta }and\sin ce\cos \theta isonfourthquadrantallothertrignometricratioswillbenegative. \\ hence\sin \theta =-\sqrt{1-{\cos }^2\theta } \\ =-\sqrt{1-\frac{3^2}{5^2}} \\ =-\sqrt{1-\frac{9}{25}} \\ =-\frac{4}{5} \end{gathered}$$

Step  3 . Finding the value of cscθ and secθ 

According to the question

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

Step 4. Finding the value of tanθ and cot θ     

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