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en:proof-of-the-law-of-cosines [2017/04/11 10:44] federico created |
en:proof-of-the-law-of-cosines [2017/07/14 10:36] federico |
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| ====== Proof of The Law of Cosines ====== | ====== Proof of The Law of Cosines ====== | ||
| - | **Law of Cosines**: a<sup>2</sup> = b<sup>2</sup> + c<sup>2</sup> - 2 bc cos Θ | + | {{:en:law-of-cosines1-fs8.png?nolink|}} |
| + | |||
| + | **Law of Cosines**: a<sup>2</sup> = b<sup>2</sup> + c<sup>2</sup> - 2bccos(θ) | ||
| ===== Proof ===== | ===== Proof ===== | ||
| - | a<sup>2</sup> = (b sin Θ)<sup>2</sup> + (c - b cos(Θ))<sup>2</sup> | + | Divide the triangle into 2 right angle triangles: |
| + | |||
| + | {{:en:law-of-cosines2-fs8.png?nolink|}} | ||
| + | |||
| + | Using trigonometry, the sides of the green triangle are | ||
| + | * a | ||
| + | * (b sin θ) | ||
| + | * (c - b cos(θ)) | ||
| + | |||
| + | {{:en:law-of-cosines3-fs8.png?nolink|}} | ||
| + | |||
| + | Using the Pythagorean Theorem: | ||
| + | |||
| + | a<sup>2</sup> = (b sin θ)<sup>2</sup> + (c - b cos(θ))<sup>2</sup> | ||
| + | |||
| + | = b<sup>2</sup> sin <sup>2</sup> θ + c<sup>2</sup> - 2cbcos(θ) + b <sup>2</sup> cos <sup>2</sup> θ | ||
| - | = b<sup>2</sup> sin <sup>2</sup> Θ + c<sup>2</sup> - 2cbcos(Θ) + b <sup>2</sup> cos <sup>2</sup> Θ | + | = b<sup>2</sup> (sin <sup>2</sup> θ + cos <sup>2</sup> θ) + c<sup>2</sup> - 2cbcos(θ) |
| - | = b<sup>2</sup> (sin <sup>2</sup> Θ + cos <sup>2</sup> Θ + c<sup>2</sup> - 2cbcos(Θ) | + | Knowing that (sin <sup>2</sup> θ + cos <sup>2</sup> θ = 1) ((see https://en.wikibooks.org/wiki/Trigonometry/Sine_Squared_plus_Cosine_Squared)) |
| - | = b<sup>2</sup> * 1 + c<sup>2</sup> - 2cbcos(Θ) | + | = b<sup>2</sup> * 1 + c<sup>2</sup> - 2cbcos(θ) |
| - | = b<sup>2</sup> + c<sup>2</sup> - 2cbcos(Θ) | + | **a<sup>2</sup> = b<sup>2</sup> + c<sup>2</sup> - 2cbcos(θ)** |