Question

What is `cos (arcsin (5/13))`?

Answer 1

`12/13`

Explanation:

First consider that : `epsilon=arcsin(5/13)`

`epsilon` simply represents an angle.

This means that we are looking for `color(red)cos(epsilon)!`

If `epsilon=arcsin(5/13)` then,

`=>sin(epsilon)=5/13`

To find `cos(epsilon)` We use the identity : `cos^2(epsilon)=1-sin^2(epsilon)`

`=>cos(epsilon)=sqrt(1-sin^2(epsilon)`

`=>cos(epsilon)=sqrt(1-(5/13)^2)=sqrt((169-25)/169)=sqrt(144/169)=color(blue)(12/13)`

Answer 2

`12/13`

Explanation:

First, see `arcsin(5/13)`. This represents the ANGLE where `sin=5/13`.

That is represented by this triangle:

Now that we have the triangle that `arcsin(5/13)` is describing, we want to figure out `costheta`. The cosine will be equal to the adjacent side divided by the hypotenuse, `15`.

Use the Pythagorean Theorem to determine that the adjacent side's length is `12`, so `cos(arcsin(5/13))=12/13`.