In any triangle ABC ,prove that
a cos A + b cos B + c cos C = 2a sin B sin C
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22
Let (C)(C)
be a unit circle, and M∈(C)M∈(C)
. Also, we will denote ∠IOM∠IOM
as θθ
(see the diagram). From the unit circle definition, the coordinates of the point MM
are (cosθ,sinθ)(cosθ,sinθ)
. And so, OC¯¯¯¯¯¯¯¯OC¯
is cosθcosθ
and OS¯¯¯¯¯¯¯OS¯
is sinθsinθ
. Therefore, OM=OC¯¯¯¯¯¯¯¯2+OS¯¯¯¯¯¯¯2−−−−−−−−−√=cos2θ+sin2θ−−−−−−−−−−−√OM=OC¯2+OS¯2=cos2θ+sin2θ
. Since MM
lies in the unit circle, OMOM
is the radius of that circle, and by definition, this radius is equal to 11
. It immediately follows that: cos2θ+sin2θ=1cos2θ+sin2θ=1
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