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en:proof-of-the-law-of-cosines [2017/04/12 13:20]
federico [Proof]
en:proof-of-the-law-of-cosines [2017/07/14 10:47] (current)
federico [Proof]
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 ===== Proof ===== ===== Proof =====
  
-If divide ​the triangle into 2 right angle triangles:+Divide ​the triangle into 2 right angle triangles:
  
 {{:​en:​law-of-cosines2-fs8.png?​nolink|}} {{:​en:​law-of-cosines2-fs8.png?​nolink|}}
- 
-{{:​en:​law-of-cosines3-fs8.png?​nolink|}} 
  
 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: **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|}}
  
 Using the Pythagorean Theorem: Using the Pythagorean Theorem:
en/proof-of-the-law-of-cosines.1492017626.txt.gz · Last modified: 2017/04/12 13:20 by federico