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9.6 Apply the Law of Cosines

9.6 Apply the Law of Cosines. In which cases can the law of cosines be used to solve a triangle? What is Heron’s Area Formula? What is the semiperimeter ?. Law of Cosines.

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9.6 Apply the Law of Cosines

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  1. 9.6 Apply the Law of Cosines In which cases can the law of cosines be used to solve a triangle? What is Heron’s Area Formula? What is the semiperimeter?

  2. Law of Cosines Use the law of cosines to solve triangles when two sides and the included angle are known (SAS), or when all three side are known (SSS).

  3. Solve ABCwith a = 11,c = 14, and B = 34°. b2 61.7 b2 61.7 7.85 SOLUTION Use the law of cosines to find side length b. b2 = a2 + c2 – 2ac cosB Law of cosines b2 = 112 + 142 – 2(11)(14) cos 34° Substitute for a, c, and B. Simplify. Take positive square root.

  4. = sin 34° sinA = 7.85 11 sin A sin B 0.7836 sin A = a b A sin–1 0.7836 51.6° 11 sin 34° The third angle Cof the triangle isC180° – 34° – 51.6° = 94.4°. 7.85 InABC,b 7.85,A 51.68,andC 94.48. ANSWER Use the law of sines to find the measure of angle A. Law of sines Substitute for a, b, and B. Multiply each side by 11 and Simplify. Use inverse sine.

  5. SolveABC witha = 12,b = 27, and c = 20. First find the angle opposite the longest side, AC. Use the law of cosines to solve for B. 272 = 122 + 202 cosB = – 2(12)(20) – 0.3854cosB B cos –1 (– 0.3854) 112.7° SOLUTION b2 = a2 + c2 – 2ac cosB Law of cosines 272 = 122 + 202 – 2(12)(20) cosB Substitute. Solve for cosB. Simplify. Use inverse cosine.

  6. sin A sin B = a b sin A = 12 0.4100 sin A = sin 112.7° 12 sin 112.7° 27 The third angle Cof the triangle isC180° – 24.2° – 112.7° = 43.1°. 27 A sin–1 0.4100 24.2° InABC,A 24.2,B 112.7,andC 43.1. ANSWER Now use the law of sines to find A. Law of sines Substitute for a, b, and B. Multiply each side by 12 and simplify. Use inverse sine.

  7. 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 to180°, the more efficiently the organism walked. 3162 = 1552 + 1972 cosB = – 2(155)(197) – 0.6062 cos B B cos –1 (– 0.6062) 127.3° ANSWER The step angle Bis about 127.3°. Science The diagram at the right shows a set of footprints for a dinosaur. Find the step angle B. b2 = a2 + c2 – 2ac cosB Law of cosines SOLUTION 3162 = 1552 + 1972 – 2(155)(197) cosB Substitute. Solve for cos B. Simplify. Use inverse cosine.

  8. Find the area of ABC. b2 57 b2 57 7.55 1. a = 8,c = 10,B = 48° SOLUTION Use the law of cosines to find side length b. b2 = a2 + c2 – 2ac cosB Law of cosines b2 = 82 + 102 – 2(8)(10) cos 48° Substitute for a,c, and B. Simplify. Take positive square root.

  9. = sin 48° sinA = 7.55 8 sin A sin B 0.7874 sin A = b a 8 sin 48° The third angle Cof the triangle isC180° – 48° – 51.6° = 80.4°. 7.55 A sin –1 0.7836 51.6° ANSWER InABC,b 7.55,A 51.6°,andC 80.4°. Use the law of sines to find the measure of angle A. Law of sines Substitute for a, b, and B. Multiply each side by 8 and simplify. Use inverse sine.

  10. Find the area of ABC. Bcos–1(– 0.0834) 85.7° First find the angle opposite the longest side, AC. Use the law of cosines to solve for B. a = 14,b = 16,c = 9 2. 162 = 142 + 92 cosB = – 2(14)(9) – 0.0834 cosB SOLUTION b2 = a2 + c2 – 2ac cosB Law of cosines 162 = 142 + 92 – 2(14)(9) cosB Substitute. Solve for cos B. Simplify. Use inverse cosine.

  11. sin A sin B a b The third angle Cof the triangle isC180° – 85.2° – 60.7° = 34.1°. sin A 14 InABC,A 60.7°,B 85.2°,andC 34.1°. ANSWER = sin 85.2° = 16 A sin–1 0.8719 60.7° 14sin 85.2° 0.8719 sin A = 16 Use the law of sines to find the measure of angle A. Law of sines Substitute for a,b, and B. Multiply each side by 14 and simplify. Use inverse sine.

  12. Heron’s Area Formula Heron (Hero) of Alexandria, the Greek mathematician (10 - 70 A.D.) is credited with using the law of cosines to find this formula for the area of a triangle.

  13. STEP 2 Use Heron’s formula to find the area of ABC. The intersection of three streets forms a piece of land called a traffic triangle. Find the area of the traffic triangle shown. Area = 18,300 = 380 (380 – 170) (380 – 240) (380 – 350) s (s – a)(s – b)(s – c) STEP 1 Find the semiperimeter s. 1 1 s = (a + b + c ) = (170 + 240 + 350) 2 2 Urban Planning SOLUTION = 380 The area of the traffic triangle is about 18,300 square yards.

  14. STEP 2 Use Heron’s formula to find the area of ABC. Find the area of ABC. Area 4. = 18.3 = 12 (12 – 8) (12 – 11) (12 – 5) s (s – a)(s – b)(s – c) STEP 1 Find the semiperimeter s. 1 1 s = (a + b + c ) = (5 + 8 + 11) 2 2 SOLUTION = 12 The area is about 18.3 square units.

  15. STEP 2 Use Heron’s formula to find the area of ABC. Find the area of ABC. Area = 5. 13.4 = 10 (10– 4) (10 – 9) (10 – 7) s (s – a)(s – b)(s – c) STEP 1 Find the semiperimeter s. 1 1 2 2 s = (a + b + c ) = (4 + 9+ 7) SOLUTION = 10 The area is about 13.4 square units.

  16. STEP 2 Use Heron’s formula to find the area of ABC. Find the area of ABC. Area 6. = 80.6 = 25 (25– 15) (25 – 23) (25 – 12) s (s – a)(s – b)(s – c) STEP 1 Find the semiperimeter s. 1 1 2 2 s = (a + b + c ) = (15 + 23+ 127) SOLUTION = 25 The area is about 80.6 square units.

  17. 9.6 Assignment Page 596, 3-39 every 3rd problem No work is the same as a missing problem

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