Solution:
We want to solve the following expression:
[tex]\cos (\sin ^{-1}(\frac{4}{5}))[/tex]
Let us denote by epsilon the argument of cosine function:
[tex]\epsilon=\sin ^{-1}(\frac{4}{5})[/tex]
now, applying the sine function to both sides of the equation, we get:
[tex]\sin (\epsilon)=\frac{4}{5}=\frac{\text{ opposite side}}{hypotenuse}[/tex]
this equation can be represented in a right triangle like this:
now, we want to find:
[tex]\cos (\sin ^{-1}(\frac{4}{5}))=\cos (\epsilon)=\frac{\text{adjacent side}}{hypotenuse}=\frac{x}{5}[/tex]
Note that we just need to find x to solve this problem. Then, to find x, we can apply the pythagorean theorem:
According to the right triangle, we get:
[tex]x=\sqrt[]{5^2-4^2}\text{ =3}[/tex]
thus, we can conclude that:
[tex]\cos (\sin ^{-1}(\frac{4}{5}))=\cos (\epsilon)=\frac{\text{adjacent side}}{hypotenuse}=\frac{x}{5}=\frac{3}{5}[/tex]
So that, the correct answer is:
[tex]\cos (\sin ^{-1}(\frac{4}{5}))=\frac{3}{5}[/tex]
