Q. 36 - Precalculus Enhanced with Graphing Utilities

Question

$\sin \theta ,\cos \theta$ are given. Find the exact value of each of the four remaining trigonometric functions.$\sin \theta = \frac{4}{5} ,\cos \theta =-\frac{3}{5}$

Step-by-Step Solution

Answer

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

Step 1. Given information

Here $\sin \theta = \frac{4}{5} and \cos \theta =-\frac{3}{5}$are given.

We have to find out the exact value of each of the four remaining trigonometric functions. 

Step 2 . Finding the value of trigonometric functions c s c θ , s e c θ  

Here

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

Step 3.. Finding the value of trigonometric function tan θ , c o t θ

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