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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] Both sides previous revision Previous revision 2017/07/14 10:47 federico [Proof] 2017/07/14 10:46 federico [Proof] 2017/07/14 10:36 federico 2017/07/14 10:34 federico 2017/04/12 13:20 federico [Proof] 2017/04/12 13:16 federico [Proof] 2017/04/12 13:15 federico [Proof] 2017/04/12 13:13 federico [Proof] 2017/04/11 11:54 federico [Proof of The Law of Cosines] 2017/04/11 10:46 federico 2017/04/11 10:46 federico 2017/04/11 10:44 federico created 2017/07/14 10:47 federico [Proof] 2017/07/14 10:46 federico [Proof] 2017/07/14 10:36 federico 2017/07/14 10:34 federico 2017/04/12 13:20 federico [Proof] 2017/04/12 13:16 federico [Proof] 2017/04/12 13:15 federico [Proof] 2017/04/12 13:13 federico [Proof] 2017/04/11 11:54 federico [Proof of The Law of Cosines] 2017/04/11 10:46 federico 2017/04/11 10:46 federico 2017/04/11 10:44 federico created Line 12: Line 12: 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|}}