Half-angle Formula and Law of Cosine---Prove

greatwhiteshark

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Half-angle Formula and Law of Cosine Prove:

C = gamma, sqrt = square root

Show that cos(C/2) = sqrts(s-c)/ab

where s = 1/2(a + b + c).
 
Hello, greatwhiteshark!

Half-angle Formula and Law of Cosine
Prove: cos(C/2) = sqrt[s(s - c)/ab] . . . where s = (a + b + c)2
We have the half-angle formula: . 2 cos<sup>2</sup>(C/2) - 1 . = . cos C
. . . . . From the Law of Cosines: . cos C . = . (a<sup>2</sup> + b<sup>2</sup> - c<sup>2</sup>)/2ab

. . . . . . . . . . . . . . . . . . . . . . . . . . . a<sup>2</sup> + b<sup>2</sup> - c<sup>2</sup>
So we have: . 2cos<sup>2</sup>(C/2) - 1 . = . ---------------
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2ab

. . . . . . . . . . . . . . . . . . . a<sup>2</sup> + b<sup>2</sup> - c<sup>2</sup> . . . . . . . a<sup>2</sup> + 2ab + b<sup>2</sup> - c<sup>2</sup> . . . .(a + b)<sup>2</sup> - c<sup>2</sup>
Then: . 2cos<sup>2</sup>(C/2) . = . --------------- + 1 . = . ----------------------- . = . ---------------- . .[1]
. . . . . . . . . . . . . . . . . . . . . . 2ab . . . . . . . . . . . . . . . . 2ab . . . . . . . . . . . . .2ab


Since .s = (a + b + c)/2, then: . a + b .= .2s - c

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . (2s - c)<sup>2</sup> - c<sup>2</sup> . . . . .4s(s - c)
Substitute into [1]: . 2cos<sup>2</sup>(C/2) . = . ---------------- . = . -----------
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2ab . . . . . . . . . .2ab

. . . . . . . . . . . . . . . . . . . s(s - c)
Hence: . cos<sup>2</sup>(C/2) . = . ---------
. . . . . . . . . . . . . . . . . . . . .ab

. . . . . . . . . . . . . . . . . . . . . . . . [ s(s - c) ]
Therefore: . cos(C/2) . = . sqrt| --------- |
. . . . . . . . . . . . . . . . . . . . . . . . [ . .ab . .]

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This is <u>not</u> a demonstration of how clever I think I am.

This is Teaching experience ... to show what can be done.
. . If it is not seen that way, too bad . . .

With every problem, there is a proper point at which to stop.
. . Sometimes it is with a brief and terse hint.
. . With others, it may be a more elaborate explanation.
It is, and always will be, a judgment call
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With this problem, the intricacies were such that I found no good place to stop
. . and say, "Okay, you finish it!"

If anyone of you think I've overexplained it
. . (mistaking an "or" for an "and", for example),
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Just don't think that I'm interested in it.
 
OK, I'll bite just a little. Why would anyone have a problem with what CLEARLY are the must useful and lucid posts on this board?

I make sense most of the time. You make sense EVERY time. Folks are having trouble with that? Wow.
 
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