cosine problem

**jarny** · September 12th 2007, 01:18 PM

Can someone explain to me how cos(theta/2) can be recongfigured into The square root of (1/2+ cos(theta)/2). Thanks.

**Jhevon** · September 12th 2007, 01:29 PM

Originally Posted by jarny

Can someone explain to me how cos(theta/2) can be recongfigured into The square root of (1/2+ cos(theta)/2). Thanks.

We can use the double angle formula to get it:

$\cos \theta = \cos \left(2 \cdot \frac {\theta}{2} \right)$

$= \cos^2 \left( \frac {\theta}{2} \right) - \sin^2 \left( \frac {\theta}{2} \right)$

$= \cos^2 \left( \frac {\theta}{2} \right) - \left[ 1 - \cos^2 \left( \frac {\theta}{2} \right) \right]$

$= 2 \cos^2 \left( \frac {\theta}{2} \right) - 1$

So we have: $\cos \theta = 2 \cos^2 \left( \frac {\theta}{2} \right) - 1$

solving for $\cos \left( \frac {\theta}{2} \right)$ we obtain the desired result

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