Law of Cosines
The Law of Cosines is used to find the remaining parts of an oblique (non-right) triangle when either the lengths of two sides and the measure of the included angle is known (SAS) or the lengths of the three sides (SSS) are known. In either of these cases, it is impossible to use the Law of Sines because we cannot set up a solvable proportion.
The Law of Cosines states:
This resembles the Pythagorean Theorem except for the third term and if is a right angle the third term equals because the cosine of is and we get the Pythagorean Theorem. So, the Pythagorean Theorem is a special case of the Law of Cosines.
The Law of Cosines can also be stated as
Example 1: Two Sides and the Included Angle-SAS
Given and . Find the remaining side and angles.
To find the remaining angles, it is easiest to now use the Law of Sines.
Note that angle is opposite to the longest side and the triangle is not a right triangle. So, when you take the inverse you need to consider the obtuse angle whose sine is .
Example 2: Three Sides-SSS
Given and . Find the measures of the angles.
It is best to find the angle opposite the longest side first. In this case, that is side .
Since is negative, we know that is an obtuse angle.
Since is an obtuse angle and a triangle has at most one obtuse angle, we know that angle and angle are both acute.
To find the other two angles, it is simplest to use the Law of Sines.