Q.38 - Precalculus Enhanced with Graphing Utilities

Question

$\sin \theta , \cos \theta$ are given. Find the exact value of each of the four remaining trigonometric functions  $\sin \theta =-\frac{2\sqrt{5}}{5} and \cos \theta =-\frac{\sqrt{5}}{5}$

Step-by-Step Solution

Answer

 The value of $csc\theta =-\frac{5}{2\sqrt{5}} ,sec\theta =-\frac{5}{\sqrt{5}},\tan \theta =2 and cot\theta =\frac{1}{2}$

Step 1. Given information

$\sin \theta =-\frac{2\sqrt{5}}{5} and \cos \theta =-\frac{\sqrt{5}}{5}$ 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   a n d   s e c θ  

 According to the question 

$$\begin{gathered} \sin \theta =-\frac{2\sqrt{5}}{5} so csc\theta =\frac{1}{\sin \theta } \\ =-\frac{1}{{\displaystyle \frac{2\sqrt{5}}{5}}} \\ =-\frac{5}{2\sqrt{5}} \\ \cos \theta =-\frac{\sqrt{5}}{5} then sec\theta =\frac{1}{\cos \theta } \\ =-\frac{1}{{\displaystyle \frac{\sqrt{5}}{5}}} \\ =-\frac{5}{\sqrt{5}} \end{gathered}$$

Step 3. Finding the value of the trigonometric function tan θ   a n d   c o t θ

We know that $$\begin{gathered} \tan \theta =\frac{\sin \theta }{\cos \theta } \\ =\frac{{\displaystyle \frac{-2\sqrt{5}}{5}}}{{\displaystyle \frac{-\sqrt{5}}{5}}} \sin ce \sin \theta =\frac{-2\sqrt{5}}{5} and \cos \theta =\frac{-\sqrt{5}}{5} \\ =2 \\ and cot\theta =\frac{1}{\tan \theta } \\ =\frac{1}{{\displaystyle 2}} \end{gathered}$$