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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Disclaimer
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
. . and we
all take full responsibility for the decision.
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),
. . you are certainly entitled to your opinion.
Just don't think that I'm interested in it.