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

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

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Answer 1 · 2015-04-24T22:31:59

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:
$Cos2x = 3/4$

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

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