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Section 11.2 Law of Cosines

Section 11.2 Law of Cosines. Law of Sines. Can you use the Law of Sines ?. C. a. 6. No. 70 . B. 8. Law of Sines. Can you use the Law of Sines ?. C. 7. 6. No. A. B. 8. Law of Cosines. The Law of Cosines can be used when SAS or SSS information is given about the triangle.

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Section 11.2 Law of Cosines

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  1. Section 11.2Law of Cosines

  2. Law of Sines Can you use the Law of Sines? C a 6 No 70 B 8

  3. Law of Sines Can you use the Law of Sines? C 7 6 No A B 8

  4. Law of Cosines • The Law of Cosines can be used when SAS or SSS information is given about the triangle.

  5. Law of Cosines • SAS C a 3 52 B 8

  6. Law of Cosines • SSS C 13 6 B A 8

  7. Law of Cosines • Three forms: • a2 = b2 + c2 – 2bccosA • b2 = a2 + c2 – 2accosB • c2 = a2 + b2 – 2abcosC

  8. Solving SAS Triangles • Solve for the side opposite the given angle first. • Solve for the angle opposite the shorter side using the Law of Sines or the Law of Cosines.

  9. Solving SAS Triangles • Find the last angle by subtracting from 180.

  10. Law of Cosines • What is c? • c = 26.7 78 16 25 B A c

  11. Law of Cosines • What is mA? (Round to the nearest tenth.) • A = 35.9 78 16 25 B A c

  12. Law of Cosines • What is mB? (Round to the nearest tenth.) • B = 66.1 78 16 25 B A c

  13. Solving SSS Triangles • Solve for the angle opposite the shortest side. • Solve for the angle opposite the medium side using the Law of Sines or the Law of Cosines.

  14. Solving SSS Triangles • Subtract these angles from 180 to find the last angle.

  15. Law of Cosines • What is mB? (Round to the nearest tenth.) • B = 22.3 A 8 4 B C 10

  16. Law of Cosines • What is mC? (Round to the nearest tenth.) • C = 49.4 A 8 4 B C 10

  17. Law of Cosines • What is mA? (Round to the nearest tenth.) • A = 108.3 A 8 4 B C 10

  18. Which Law? • Sines • Cosines C 3 b B 16 6

  19. Which Law? • Sines • Cosines C a 12 B 50 20

  20. Which Law? • Sines • Cosines C 10 7 B A 12

  21. Which Law? • Sines • Cosines C a b 16 6 50

  22. AAA Triangles • Why can’t AAA ever be solved? • Infinitely many similar triangles exist. • Example: 30-60-90 triangle or 45-45-90 triangle

  23. Law of Cosines What happens if you apply the Law of Cosines to a right triangle? • c2 = a2 + b2 – 2ab cosC • cos 90 = 0 • c2 = a2 + b2 – 0 • Pythagorean Theorem!

  24. Section 11.2 • pp. 451-452

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