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

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Answer 1 · 2015-07-21T22:51:47

$7/25$

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 · 2015-07-22T05:12:34

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)

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

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

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

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

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