Wikipedia:Reference desk/Archives/Mathematics/2012 November 26

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November 26

Can anyone tell me what the full sine rule (including 2r) is? thanks.

I'm doing British mathematical olympiad round 1, and the one thing in the booklet of reccomended things you ought to know, but I don't actually know if it mentions the full sine rule-which I knew last summer but now I've forgotten. I presume that r is radius, but radus of what? — Preceding unsigned comment added by 86.176.99.71 (talk) 18:38, 26 November 2012 (UTC)

Does Law of sines#Relation to the circumcircle give you what you are looking for? Looie496 (talk) 20:29, 26 November 2012 (UTC)

thanks it does indeed. that was relatively simple after all-if only they could say so in as many words... — Preceding unsigned comment added by 86.176.99.71 (talk) 20:24, 27 November 2012 (UTC)

Draw a circumcircle on your triangle, then slide the vertex A along the circle's arc to make B a right angle. Based on an inscribed angle theorem the A angle keeps its measure unchanged (blue and green on the picture), so does the BC chord. Now it's clear that sin(A) = a/(2R), isn't it? --CiaPan (talk) 06:32, 28 November 2012 (UTC)

Thanks from me too, CiaPan! That's a very nice demonstration, new to me even though I've used the result. Duoduoduo (talk) 15:32, 28 November 2012 (UTC)

Glad to be helpful. However, despite how nice demonstration it is, it's not a valid proof. :( Would be better to ask to shift A so that AC becomes the circle's diameter, then use the Thales' theorem to obtain 'B is a right angle'.
Furthermore the procedure would not work for obtuse triangle with A > 90°, unless we replace A with a new vertex D on the other side of BC and prove
    1. D = 180° − A
then
    2. sin D = sin A
CiaPan (talk) 06:30, 29 November 2012 (UTC)