Q.48 - Precalculus Enhanced with Graphing Utilities

Question

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

$\cos \theta =\frac{4}{5} , {270}^o<\theta <{360}^o$

Step-by-Step Solution

Answer

 The exact values are 

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

Step 1. Given information

 $\cos \theta =\frac{4}{5} ,{270}^0<\theta <{360}^0$is given.

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

Step 2. Find the value of sin &#952;

 

$$\begin{gathered} We know that {\sin }^2\theta +{\cos }^2\theta =1 , here \cos \theta =\frac{4}{5} \\ \sin \theta =\pm \sqrt{1-{\cos }^2\theta } , here \theta lies on {270}^o<\theta <{360}^o so \cos \theta is on fourth quadrant all other except \cos \theta and sec\theta are will be negative. \\ hence \sin \theta =-\sqrt{1-co{s}^2\theta } \\ =-\sqrt{1-\frac{4^2}{5^2}} \\ =-\sqrt{1-\frac{16}{25}} \\ =-\frac{3}{5} \end{gathered}$$

step 3. Find the values of s e c &#952; , &#160; c s c &#952;

 According to the question 

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

Step 4. Find the values of tan &#952; &#160; a n d &#160; c o t &#952;

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