Question

how do I use a sketch to find the exact value of the problem in the image?

Answer 1

Solution:

We want to solve the following expression:

$\cos (\sin ^{-1}(\frac{4}{5}))$

Let us denote by epsilon the argument of cosine function:

$\epsilon=\sin ^{-1}(\frac{4}{5})$

now, applying the sine function to both sides of the equation, we get:

$\sin (\epsilon)=\frac{4}{5}=\frac{\text{ opposite side}}{hypotenuse}$

this equation can be represented in a right triangle like this:

now, we want to find:

$\cos (\sin ^{-1}(\frac{4}{5}))=\cos (\epsilon)=\frac{\text{adjacent side}}{hypotenuse}=\frac{x}{5}$

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:

$x=\sqrt[]{5^2-4^2}\text{ =3}$

thus, we can conclude that:

$\cos (\sin ^{-1}(\frac{4}{5}))=\cos (\epsilon)=\frac{\text{adjacent side}}{hypotenuse}=\frac{x}{5}=\frac{3}{5}$

So that, the correct answer is:

$\cos (\sin ^{-1}(\frac{4}{5}))=\frac{3}{5}$