Question #13d24

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Question #13d24

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Answer 1 · 2016-11-17T18:48:54

see below

Explanation:

$\sqrt{\frac{1 + cos A}{2}} = cos (\frac{A}{2})$ $sqrt((1+cosA)/2)=cos (A/2 )$

We will prove this by using the double argument formula for cosine.

Recall that one of the formula for $cos 2 A$ $cos 2A$ is $cos 2 A = 2 {cos}^{2} A - 1$ $cos2A=2cos^2A-1$

So if we isolate cos A then we have

$\frac{cos 2 A + 1}{2} = {cos}^{2} A$ $(cos 2A+1)/2=cos^2 A$

$\pm \sqrt{\frac{1 + cos 2 A}{2}} = cos A$ $+-sqrt ( (1+cos2A)/2)=cos A$

So from this formula we can see that you double angle A inside the square root. Since A in this case is $\frac{1}{2} A$ $1/2A$ then we have $2 A = 2 (\frac{1}{2} A) = A$ $2A=2(1/2A)=A$ ,

$\pm \sqrt{\frac{1 + cos 2 (\frac{1}{2} A)}{2}} = cos (\frac{1}{2} A)$ $+-sqrt ( (1+cos2(1/2A))/2)=cos (1/2A)$

$:. +-sqrt((1+cosA)/2)=cos (A/2 )$

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Question #13d24

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