Find the exact value for cos 2x , given sin x = \frac{\sqrt{2}}{4} ?

Step 1: Note down the given data and recall the double angle identities
\sin x = \frac{\sqrt{2}}{4}
\cos x = ?
Double identities:
\cos 2\theta = \cos^2 \theta - \sin^2\theta
\cos 2\theta = 2 \cos^2 - 1 \theta
\cos 2 \theta = 1 - 2\sin^2\theta
Step 2: Choose the appropriate formula to calculate \cos 2x
We know the \sin \theta so take
\cos 2x = 1 - 2\sin^2 x
\cos 2x = 1 - 2(\frac{\sqrt{2}}{4})^2 \because \sin x = \frac{\sqrt{2}}{4}
\cos 2x = 1 - 2(\frac{2}{16})
\cos 2x = 1 - \frac{1}{4}
\cos 2x = \frac{3}{4}