Question

Use the Double-Angle Identity to find the exact value for cos 2x , given `sin x= sqrt2/ 4`?

Answer 1

The first step to answering this question is figuring out which Identity to use, taking into account the information we are given in the question.

For `Cos2x`, we have:

`Cos2x = cos^2x - sin^2x`
`Cos2x = 2Cos^2x - 1`
and `Cos2x = 1 - 2Sin^2x`

As we are looking for `Cos`, we really don't want an identity with `cos` in it, therefore we can choose `1 - 2Sin^2x`.

We know three things at this point:

`Sinx = (sqrt2)/4`
`Cos2x = 1 - 2Sin^2x`
and
`Sin^2x` is the same as `(sinx)^2`

We can use the above to find `Cos2x`:

Use the identity we chose:
`Cos2x = 1 - 2Sin^2x`

Change the notation to make it easier to manipulate:
`Cos2x = 1 - 2(Sinx)^2`

Substitute `Sinx` for the `sqrt2/4`:
`Cos2x = 1 - 2(sqrt2/4)^2`

Square both the numerator and denominator of the fraction:
`Cos2x = 1 - 2(2/16)`

Expand (break the brackets):
`Cos2x = 1 - 4/16`

Simplify:
`Cos2x = 1 - 1/4`

Solve:
`Cos2x = 3/4`