Question

What is `cos (2 arcsin (3/5))`?

Answer 1

`7/25`

Explanation:

First consider that : `epsilon=arcsin(3/5)`

`epsilon` simply represents an angle.

This means that we are looking for `color(red)cos(2epsilon)!`

If `epsilon=arcsin(3/5)` then,

`=>sin(epsilon)=3/5`

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

`=>cos(2epsilon)=1-2*(3/5)^2=(25-18)/25=color(blue)(7/25)`

Answer 2

We have:

`y = cos(2arcsin(3/5))`

I will do something similar to Antoine's method, but expand on it.
Let `arcsin(3/5) = theta`

`y = cos(2theta)`

`theta = arcsin(3/5)`
`sintheta = 3/5`

Using the identity `cos(theta+theta) = cos^2theta - sin^2theta`, we then have:

`cos(2theta) = (1-sin^2theta) - sin^2theta = 1-2sin^2theta`
(I didn't remember the result, so I just derived it)

`= 1-2{sin[arcsin(3/5)]}^2`

`= 1-2(3/5)^2`

`= 25/25 - 2(9/25)`

`= 25/25 - 18/25 = color(blue)(7/25)`