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The Law of Cosines

If you are trying to understand trigonometry you should definitely read this. This theorem is a commonly used trick for some problems that require pretty genius and creative geometric problems.

In this article, I will be proving the law of cosines. If you know a little about triangles and angles you can do it yourself! So, before reading the proof, you had better try to prove it.

The Law of Cosines

Let’s start with the theorem. In any type of triangle, the following equation holds.

What does cosα mean?

As you can understand from the name of the theorem it has something to do with cosines. But what is that “cosα”? Cosines is a trigonometric function that is equal to the ratio of the side adjacent to an acute angle (in a right-angled triangle) to the hypotenuse.

This drawing will help you understand what cosα means.

Proof

Now, the proof is coming. Make sure you have dealt with the problem before you read the proof.

Let h be the length of the height of the triangle from vertex B. Let y be the length of the right side of the height. By definition, cosα = y/c. Then we can write a² = h² + (b-y)² from Pythagoras theorem. Also, we know c² = h² + y² from Pythagoras theorem. If we write c² -y² instead of h, we will get a² = c² + b² -2by. We want to use cosα somewhere to get c² + b² -2bc(cosα). If we write -2bc.y/c instead of -2by, we will get -2bc(cosα). This gives us the equality which we were trying to prove.

Same proof occurs if there is an obtuse triangle. I need to note that,

cosα=-cos(180-α)

and this comes from the unit circle.

Resources,

1 The Law of Cosines. WolframMathWorld. Web. 28.04.2020 (Link)

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Ceren Şahin

Ceren Şahin

Co-founder at Betamat, YGA volunteer, table tennis passionate, web developer and a curious coder.

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