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en:proof-of-the-law-of-cosines [2017/07/14 10:36] federico |
en:proof-of-the-law-of-cosines [2017/07/14 10:46] federico [Proof] |
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| Using trigonometry, the sides of the green triangle are | Using trigonometry, the sides of the green triangle are | ||
| - | * a | + | * Longest side: a |
| - | * (b sin θ) | + | * 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 |
| - | * (c - b cos(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 |
| {{:en:law-of-cosines3-fs8.png?nolink|}} | {{:en:law-of-cosines3-fs8.png?nolink|}} | ||