Law of Cosines
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In triangle problems dealing with 2 sides and 2 angles we have seen that the Law of Sines is used to find the missing item.  There are many problems, however, that deal with all three sides and only one angle of the triangle.  For these problems we have another method of solution called the Law of Cosines.

With the diagram labeled at the left,
the Law of Cosines is as follows: 

Notice that <C and side c are at opposite ends of the formula.  Also, notice the resemblance (in the beginning of the formula) to the Pythagorean Theorem.

We can write the Law of Cosines for each angle around the triangle.  Notice in each statement how the pattern of the letters remains the same.

Law of Cosines


The Law of Cosines can be used to find a missing side for a triangle, or a missing angle.  Let's take a look .

Example 1:  In , side b = 12, side c = 20 and m<A = 45.  Find side a to the nearest integer.

Since the only known angle is A, we use the version of the Law of Cosines dealing with angle A.

This problem involves all three sides but only one angle of the triangle.  This fits the profile for the Law of Cosines.


Example 2:   Find the largest angle, to the nearest tenth of a degree, of a triangle whose sides are 9, 12 and 18.

In a triangle, the largest angle is opposite the largest side.  We need to find <B.

Use the Law of Cosines:


Example 3:   In a parallelogram, the adjacent sides measure 40 cm and 22 cm.  If the larger angle of the parallelogram measure 116, find the length of the larger diagonal, to the nearest integer.

In , we have 2 sides, 1 angle, and we are finding the 3rd side.  Use the Law of Cosines:
Remember that cosine of an obtuse angle is negative, which changes the sign of the last term!!