The Law of Cosines
The Law of Cosines gives us another relationship between the sides and angles of a triangle. The sides and angles are configured as in the previous section on the Law of Sines, with side a opposite angle A and so on.
Law of Cosines
a 2 b 2 c 2 2bc cos A b 2 a 2 c 2 2ac cos B c 2 a 2 b 2 2abcosC
We can summarize the various forms of the Law of Cosines as follows: The square of a given side is equal to the sum of the squares of the other two sides minus twice the product of those two sides and the angle opposite the given side. Also note that, in the equation c2 a2 b2 2abcosC , if C is a right angle, the Law of Cosines reduces to the Pythagorean Theorem. We use the Law of Cosines to solve a triangle if we are given two sides and the angle between those two sides (SAS) or three sides (SSS).
EXAMPLE 1: Suppose that a 32m, b 68 m, and C 55. Solve triangle ABC.
Solution: We will start by using the Law of Cosines to find side c. c2 322 682 232 68cos55 3151.8 c 56 m. We can now use the Law of Sines to find a second angle. When we use the Law of Sines to find an angle, choose the smallest angle, because we know that angle is acute, and we want to avoid ambiguities. Since side a is shorter than side c, we know that angle A will be smaller than angle B. sin A sin55 32sin55 sin A 0.4681 A 28. 32 56 56 Finally, B 180 55 28 97 .
We can solve each of the forms of the Law of Cosines for the angle to obtain an alternate version of the Law of Cosines which is useful for finding angles.
Law of Cosines
b 2 c 2 a 2 cos A 2bc a 2 c 2 b 2 cos B 2ac a 2 b 2 c 2 cos C 2ab
EXAMPLE 2: Suppose that a 12, b 27, and c 34 . Solve triangle ABC.
Solution: We will use the Law of Cosines to find the first angle, and then we will be able to use the Law of Sines to find another angle. To avoid any ambiguities in the angle found by Law of Sines, we use the Law of Cosines to find the largest angle first (that way, the remaining smaller angles must be acute). Here, the longest side is side c, so we use the Law of Cosines to find angle C: 122 27 2 342 283 283 cosC . Thus, C cos 1 116 . 212 27 648 648 Now we can use the Law of Sines to find either A or B. We will find the smallest angle: sin A sin116 12sin116 sin A .3172 A sin 1 .3172 18. 12 34 34 Finally, B 180 116 18 46 .
EXAMPLE 3: The diagonals of a parallelogram are 38 feet and 52 feet and intersect at a 70 angle. Find the lengths of the sides of the parallelogram.
Solution: In the parallelogram above, the longer side, c, satisfies the equation c2 262 192 226 19 cos110 1375 c 37 feet. For the shorter side, a, a2 262 192 226 19 cos70 699 a 26 feet.