The

The Law of Cosines gives us another relationship between the sides and of a . The sides and angles are configured as in the previous section on the Law of Sines, with side a opposite 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 of a given side is equal to the sum of the 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 . 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  232 68cos55  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 . 212  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 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  226 19 cos110 1375  c  37 feet. For the shorter side, a, a2  262 192  226 19 cos70  699  a  26 feet.