Question

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

Short answer

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

Step-by-step solution

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.

. Finding the value of trigonometric functions cscθ,secθ 

Here

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

Step 3.. Finding the value of trigonometric function tanθ,cotθ

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