Question

if $\sin \theta =\frac{4}{5}$and $\theta$is in quadrant 2, then$\cos \theta =$______

Short answer

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

Step-by-step solution

Step 1. Given information

We have been given the value of $\sin \theta$which is $\frac{4}{5}$and $\theta$is in second quadrant.

We have to find the value of$\cos \theta$

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 & be the angle formed by the intersection of the adjacent and the hypotenuse sides. We get

role="math" localid="1646455498886" $\sin \theta =\frac{4}{5}:\frac{opposite}{hypotenuse}$

Calculate the adjacent side,

role="math" localid="1646455604923" $$\begin{gathered} adjacent=\sqrt{hypotenus{e}^2-opposit{e}^2} \\ =\sqrt{25-16} \\ =\sqrt{9} \\ =3 \end{gathered}$$

Step 3. Find the value of cos θ

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

and $\cos \left(\theta \right)$is negative in the second quadrant.

Thus,

localid="1646455781307" $\cos \theta =-\frac{3}{5}$