Proof of The Law of Cosines

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

Divide the triangle into 2 right angle triangles:

Using trigonometry, the sides of the green triangle are

• Longest side: a
• Side on the left: b . sin θ ; because the sine of θ is opposite divided the hypotenuse, and the opposite is the length of this side. Then, the length of this side is the hypotenuse multiplied by the sine of θ
• Side on the right: *c - b cos(θ) ; we must subtract from the length “c”, the part of c that is occupied by the red triangle. We know that, in the red triangle, cos θ is the adjacent side divided the hypotenuse. The adjacent is the “base” of the red triangle, and it is cos θ multiplied by the hypotenuse. The hypotenuse is b. Then, the base of the red triangle is (cos θ)b. So, the length of the “base” of the green triangle is 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)