Question

if$\sin \alpha =-\frac{4}{5}$,$\mathrm{\pi }<\mathrm{\alpha }<\frac{3\mathrm{\pi }}{2}$, then$\cos \alpha =$_____

Short answer

$\cos \alpha =-\frac{3}{5}$

Step-by-step solution

Step 1. Given information

Given is:

$\sin \alpha =-\frac{4}{5}$

We have to find the value of$\cos \alpha$.

Step 2. Calculate the adjacent side

We consider the right-angled triangle which opposite side is equal to $-4$and its hypotenuse equals to $5$and let $\theta$be the angle formed by the intersection of the adjacent and the hypotenuse sides. We get

$\sin \alpha =-\frac{4}{5}:\frac{opposite}{hypotenuse}$

Calculate the adjacent side,

$$\begin{gathered} adjacent=\sqrt{hypotenus{e}^2-opposit{e}^2} \\ =\sqrt{25-16} \\ =\sqrt{9} \\ =3 \end{gathered}$$

Step 3. Find cos α

As $\cos \alpha =\frac{adjacent}{hypotenuse}$

and $\cos \alpha$is negative in the second quadrant.

Therefore,

$\cos \alpha =-\frac{3}{5}$