en:proof-of-the-law-of-cosines

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en:proof-of-the-law-of-cosines [2017/07/14 10:46] federico [Proof] |
en:proof-of-the-law-of-cosines [2017/07/14 10:47] (current) federico [Proof] |
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Using trigonometry, the sides of the green triangle are | Using trigonometry, the sides of the green triangle are | ||

- | * Longest side: a | + | * 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 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 | + | * 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|}} |

en/proof-of-the-law-of-cosines.txt · Last modified: 2017/07/14 10:47 by federico