Section 8.2 The Law of Cosines Copyright ©2013, 2009, 2006, 2001 Pearson Education, Inc.

Objectives Use the law of cosines to solve triangles. Determine whether the law of sines or the law of cosines should be applied to solve a triangle.

Law of Cosines The Law of Cosines In any triangle ABC, B a c Thus, in any triangle, the square of a side is the sum of the squares of the other two sides, minus twice the product of those sides and the cosine of the included angle. When the included angle is 90º, the law of cosines reduces to the Pythagorean theorem. A B C a b c

When to use the Law of Cosines The Law of Cosines is used to solve triangles given two sides and the included angle (SAS) or given three sides (SSS).

Example In !ABC, a = 32, c = 48, and B = 125.2º. Solve the triangle. Solution: Draw and label a triangle.

Example (cont) Use the law of cosines to find the third side, b. We need to find the other two angle measures. We can use either the law of sines or law of cosines. Using the law of cosines avoids the possibility of the ambiguous case. So use the law of cosines.

Example (cont) Find angle A. Now find angle C.

Example Solve !RST, r = 3.5, s = 4.7, and t = 2.8. Solution: Draw and label a triangle.

Example (cont) Similarly, find angle R.

Example (cont) Now find angle T. T ≈ 180º – (95.86º + 47.80º) ≈ 36.34º

Example Knife makers know that the bevel of the blade (the angle formed at the cutting edge of the blade) determines the cutting characteristics of the knife. A small bevel like that of a straight razor makes for a keen edge, but is impractical for heavy-duty cutting because the edge dulls quickly and is prone to chipping. A large bevel is suitable for heavy-duty work like chopping wood. Survival knives, being universal in application, are a compromise between small and large bevels. The diagram illustrates the blade of a hand-made Randall Model 18 survival knife. What is its bevel?

Example (cont) Solution: Use the law of cosines to find angle A. The bevel is approximately 14.36º.

Example In triangle ABC, three measures are given. Determine which law to use when solving the triangle. You need not solve the triangle. a. a = 14, b = 23, c = 10 SSS Law of Cosines b. a = 207, B = 43.8, C = 57.6 ASA Law of Sines

Example (cont) c. A = 112, C = 37, a = 84.7 AAS Law of Sines d. B = 101, a = 960, c = 1042 SAS Law of Cosines e. b = 17.26, a = 27.29, A = 39 SSA Law of Sines f. A = 61, B = 39, C = 80 AAA cannot be solved