What Is the Law of Cosines?

Triangle with the law of cosines

The law of cosines is the general form of the Pythagorean theorem, which means it can be used on any triangle. You can see that when A = 90, you get 2bc cos A = 0, and you’re left with the Pythagorean theorem.

Formula

The Law of Cosines

If b and c are two sides that span an angle A, then

a2 = b2 + c2 2bc cos A (1) cos A = b2 + c2 a2 2bc (2)

Rule

Uses

You can use the law of cosines to

  • Find a side in a triangle when you know the two other sides and one angle in the triangle—Formula (1).

  • Find an angle in a triangle when you know all three sides of the triangle—Formula (2).

Example 1

You have a quadrilateral ABCD, where AB = 12, AD = 9 and A = 120. Find the diagonal BD.

The first thing to do is to make an auxiliary figure to help you visualize the problem.

Example of using the law of cosines 1

You can see that this fits with the law of cosines in (1). Insert the numbers to find BD:

BD2 = AB2 + AD2 2 AB AD cos 120 = 122 + 92 2 12 9 cos 120 = 144 + 81 216 (0.5) = 333 BD = 333 18.25

Thus BD 18.25.

Example 2

You have a triangle ABC, where AB = 7, AC = 5 and BC = 10. Find A.

The first thing to do is to make an auxiliary figure, to help you visualize the problem.

Example of using the law of cosines 2

You can see that this fits with the law of cosines in (2). Insert the numbers to find A.

cos A = AB2 + AC2 BC2 2 AB AC = 72 + 52 102 2 7 5 = 49 + 25 100 70 = 26 70 A = cos 1 ( 13 35 ) 111.8

So the angle A is about 111.8.

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