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Proof of The Law of Cosines

Law of Cosines: a2 = b2 + c2 - 2bccos(θ)

Proof

If divide the triangle into 2 right angle triangles:

image-2

Using trigonometry, the sides of the green triangle are

  • a
  • (b sin θ)
  • (c - b cos(θ))

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)

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en/proof-of-the-law-of-cosines.1492017236.txt.gz · Last modified: 2017/04/12 13:13 by federico