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Law of Cosines: a2 = b2 + c2 - 2bccos(θ)
If divide the triangle into 2 right angle triangles:
image-2
Using trigonometry, the sides of the green triangle are
Using the Pythagorean Theorem:
a2 = (b sin θ)2 + (c - b cos(θ))2
= b2 sin 2 θ + c2 - 2cbcos(θ) + b 2 cos 2 θ
= b2 (sin 2 θ + cos 2 θ) + c2 - 2cbcos(θ)
Knowing that (sin 2 θ + cos 2 θ = 1) 1)
= b2 * 1 + c2 - 2cbcos(θ)
a2 = b2 + c2 - 2cbcos(θ)
Discussion