6-3: Law of Cosines

Use the Law of Cosines to solve triangles when: 1.2 sides and the included angle are known (SAS) 2.all 3 sides are known (SSS)

1.SAS SOLUTION: Step 1: Use the Law of Cosines to find side length b. b 2 = a 2 + c 2 – 2ac cos B b 2 = – 2(11)(14) cos 34° b 2  61.7 b 2  61.7  7.85 Law of cosines Substitute for a, c, and B. Simplify. Take positive square root. Solve ∆ABC a = 11, c = 14, &  B = 34°

Step 2: Use the Law of Sines to find the measure of  A sin A a sin B b = sin A 11 = sin 34° 7.85 sin A = 11 sin 34° 7.85  A  sin –  51.68° Law of Sines Substitute for a, b, and B. Multiply each side by 11 and simplify. Use inverse sine.  C = 180° – 34° – 51.68° = 94.32° In ABC, b  7.85,  A  51.68°, &  C  94.32° ANSWER Step 3: Subtract  A &  B from 180 ° to find  C

Solve ∆ABC a = 12, b = 27, and c = 20 Step 1: Use the Law of Cosines to solve for  B. b 2 = a 2 + c 2 – 2ac cos B 27 2 = – 2(12)(20) cos B 27 2 = – 2(12)(20) = cos B –  cos B  B  cos –1 (– )  112.7° Law of cosines Substitute. Solve for cos B. Simplify. Use inverse cosine. SOLUTION: 2.SSS

sin A a = sin B b sin A 12 sin 112.7° 27 = sin A= 12 sin 112.7° 27   A  sin –  24.2° Law of sines Substitute for a, b, and B. Multiply each side by 12 & simplify. Use inverse sine.  C = 180° – 24.2° – 112.7° = 43.1° In ∆ABC, A  24.2°, B  112.7°, & C  43.1° Step 2: Use the Law of Sines to find A ANSWER: Step 3: Subtract  A &  B from 180 ° to find  C

SOLUTION b 2 = a 2 + c 2 – 2ac cos B b 2 = – 2(8)(10) cos 48° b 2  57 b 2  57  7.55 Law of Cosines Substitute for a, c, and B Simplify Take positive square root Solve ∆ABC a = 8, c = 10, & B = 48° Step 1: Use the Law of Cosines to find side length b. 1.SAS

sin A a sin B b = sin A 8 = sin 48° 7.55 sin A = 8 sin 48° 7.55  A  sin –  51.6° Law of Sines Substitute for a, b, and B Multiply each side by 8 & simplify Use inverse sine  C = 180° – 48° – 51.6° = 80.4° In ∆ABC, b  7.55, A  51.6°, & C  80.4° ANSWER: Step 2: Use the Law of Sines to find the measure of  A. Step 3: Subtract  A &  B from 180 ° to find  C

Scientists can use a set of footprints to calculate an organism’s step angle, which is a measure of walking efficiency. The closer the step angle is to 180°, the more efficiently the organism walked. The diagram at the right shows a set of footprints for a dinosaur. Find the step angle B. SOLUTION b 2 = a 2 + c 2 – 2ac cos B = – 2(155)(197) cos B = – 2(155)(197) = cos B –  cos B B  cos –1 (– )  127.3° Use inverse cosine. Simplify. Solve for cos B. Substitute. Law of cosines The step angle B is about 127.3° ANSWER

W HEN SOLVING A TRIANGLE : 1.Use the Trigonometric Functions (sine, cosine & tangent) and their inverses when: you have a RIGHT triangle. 2.Use the Law of Sines when: 2 Angles and 1 Side (AAS or ASA) 2 Sides and 1 Angle (SSA) 3.Use the Law of Cosines when: 2 Sides and the included Angle (SAS) 3 Sides (SSS) SUMMARY