A new trig identity
by John from John D. Cook on (#6ASEW)
This evening I ran across a trig identity I hadn't seen before. I doubt it's new to the world, but it's new to me.
Let A, B, and C be the angles of an arbitrary triangle. Then
sin^2 A + sin^2 B + sin^2 C = 2 + 2 cos A cos B cos C.
This looks a little like the Pythagorean theorem, but the Pythagorean theorem involves the sides of a triangle, not the angles. (I expect there's an interesting generalization of the identity above to spherical geometry where sides are angles.)
This identity also looks a little like the law of cosines, but the law of cosines mixes sides and angles, and this identity only involves angles.
Source: Lemma 4.4.3 in A Panoply of Polygons by Claudi Alsina and Roger B. Nelsen.
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