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19 May 2011

19 May 2011. Algebra 2. Law of Cosines 5/19. a 2 = b 2 + c 2 – 2bc* cos (A) b 2 = a 2 + c 2 – 2ac* cos (B) c 2 = a 2 + b 2 – 2ab* cos (C). Three forms of L.o.C . Law of Cosines 5/19. Law of Sines is easier, but Law of Cosines can be used more often.

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19 May 2011

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  1. 19 May 2011 Algebra 2

  2. Law of Cosines 5/19 a2 = b2 + c2 – 2bc*cos(A) b2 = a2 + c2 – 2ac*cos(B) c2 = a2 + b2 – 2ab*cos(C) Three forms of L.o.C.

  3. Law of Cosines 5/19 Law of Sines is easier, but Law of Cosines can be used more often. When to use L.o.C.

  4. Law of Cosines 5/19 Law of Sines is easier, but Law of Cosines can be used more often b = ? When to use L.o.C.

  5. Law of Cosines 5/19 Law of Sines is easier, but Law of Cosines can be used more often b = ? When to use L.o.C.

  6. Law of Cosines 5/19 Law of Sines is easier, but Law of Cosines can be used more often b = ? Law of Sines doesn’t help us! When to use L.o.C.

  7. Law of Cosines 5/19 Law of Sines is easier, but Law of Cosines can be used more often b = ? Law of Cosines to the rescue!!! When to use L.o.C.

  8. Law of Cosines 5/19 Law of Sines is easier, but Law of Cosines can be used more often b = ? b2 = a2 + c2 – 2ac*cos(B) When to use L.o.C.

  9. Law of Cosines 5/19 Law of Sines is easier, but Law of Cosines can be used more often b = ? b2 = a2 + c2 – 2ac*cos(B) b2 = 72 + 52 – 2*7*5*cos(36) When to use L.o.C.

  10. Law of Cosines 5/19 Law of Sines is easier, but Law of Cosines can be used more often b = ? b2 = a2 + c2 – 2ac*cos(B) b2 = 72 + 52 – 2*5*7*cos(36) b2 = 17.4 When to use L.o.C.

  11. Law of Cosines 5/19 Law of Sines is easier, but Law of Cosines can be used more often b = ? b2 = a2 + c2 – 2ac*cos(B) b2 = 72 + 52 – 2*5*7*cos(36) b2 = 17.4 b = 4.2 When to use L.o.C.

  12. Law of Cosines 5/19 Example 2: find c C 15 26o 12.5 A c B Which form of the L.o.C. should we use? When to use L.o.C.

  13. Law of Cosines 5/19 Example 2: find c C 15 26o 12.5 A c B c2 = a2 + b2 – 2ab*cos(C) When to use L.o.C.

  14. Law of Cosines 5/19 Example 2: find c C 15 26o 12.5 A c B c2 = a2 + b2 – 2ab*cos(C) c2 = 12.52 + 152 – 2*12.5*15*cos(26) When to use L.o.C.

  15. Law of Cosines 5/19 Example 2: find c C 15 26o 12.5 A c B c2 = a2 + b2 – 2ab*cos(C) c2 = 12.52 + 152 – 2*12.5*15*cos(26) c2 = 44.20 When to use L.o.C.

  16. Law of Cosines 5/19 Example 2: find c C 15 26o 12.5 A c B c2 = a2 + b2 – 2ab*cos(C) c2 = 12.52 + 152 – 2*12.5*15*cos(26) c2 = 44.2 c = 6.6 When to use L.o.C.

  17. Student Practice: • Complete #1,6,9,10 on pages 843,844 • #1 find b • #6 find b • #9 find c • #10 find b

  18. Law of Cosines 5/19 Example 3: find <B C 18 9 A 10 B Use L.o.C. formula that has <B Finding an angle

  19. Law of Cosines 5/19 Example 3: find <B C 18 9 A 10 B b2 = a2 + c2 – 2ac*cos(B) Finding an angle

  20. Law of Cosines 5/19 Example 3: find <B C 18 9 A 10 B b2 = a2 + c2 – 2ac*cos(B) 182 = 92 + 102 – 2*9*10*cos(B) Finding an angle

  21. Law of Cosines 5/19 Example 3: find <B C 18 9 A 10 B b2 = a2 + c2 – 2ac*cos(B) 182 = 92 + 102 – 2*9*10*cos(B) 182 – 92 – 102= -2*9*10*cos(B) Finding an angle

  22. Law of Cosines 5/19 Example 3: find <B C 18 9 A 10 B b2 = a2 + c2 – 2ac*cos(B) 182 = 92 + 102 – 2*9*10*cos(B) 182 – 92 – 102= -2*9*10*cos(B) = cos(B) Finding an angle

  23. Law of Cosines 5/19 Example 3: find <B C 18 9 A 10 B b2 = a2 + c2 – 2ac*cos(B) 182 = 92 + 102 – 2*9*10*cos(B) 182 – 92 – 102= -2*9*10*cos(B) = cos(B) cos(B) = -.794 Finding an angle

  24. Law of Cosines 5/19 Example 3: find <B C 18 9 A 10 B b2 = a2 + c2 – 2ac*cos(B) 182 = 92 + 102 – 2*9*10*cos(B) 182 – 92 – 102= -2*9*10*cos(B) = cos(B) <B = 142o Finding an angle

  25. Student Practice • Complete # 2,11,12,19 on pages 843-844 • Always find the angle opposite the longest side

  26. Law of Cosines 5/19 You will be asked to “solve” a triangle. This means find all missing sides and angles. Easiest way to do this is to use the Law of Cosines once, then use the Law of Sines. Using L.o.C.

  27. Law of Cosines 5/19 Solve the triangle Step 1: Use Law of Cosines to find the largest angle (A) Using L.o.C.

  28. Law of Cosines 5/19 Solve the triangle Step 1: Use Law of Cosines to find the largest angle (A) “Mr. Meeks which formula do I use, there are three?” Using L.o.C.

  29. Law of Cosines 5/19 Solve the triangle Step 1: Use Law of Cosines to find the largest angle (A) Use the one that will give you < A. Using L.o.C.

  30. Law of Cosines 5/19 Solve the triangle a2 = b2 + c2 – 2bc*cos(A) Using L.o.C.

  31. Law of Cosines 5/19 Solve the triangle a2 = b2 + c2 – 2bc*cos(A) 162 = 92 + 102 – 2*9*10*cos(A) Using L.o.C.

  32. Law of Cosines 5/19 Solve the triangle a2 = b2 + c2 – 2bc*cos(A) 162 = 92 + 102 – 2*9*10*cos(A) 162 - 92 - 102 = – 2*9*10*cos(A) Using L.o.C.

  33. Law of Cosines 5/19 Solve the triangle a2 = b2 + c2 – 2bc*cos(A) 162 = 92 + 102 – 2*9*10*cos(A) 162 - 92 - 102 = – 2*9*10*cos(A) = cos(A) Using L.o.C.

  34. Law of Cosines 5/19 Solve the triangle a2 = b2 + c2 – 2bc*cos(A) 162 = 92 + 102 – 2*9*10*cos(A) 162 - 92 - 102 = – 2*9*10*cos(A) = cos(A) - 0.4167 = cos(A) Using L.o.C.

  35. Law of Cosines 5/19 Solve the triangle a2 = b2 + c2 – 2bc*cos(A) 162 = 92 + 102 – 2*9*10*cos(A) 162 - 92 - 102 = – 2*9*10*cos(A) = cos(A) 115o = A Using L.o.C.

  36. Law of Cosines 5/19 Solve the triangle Step 2: Use Law of Sines to find < B Using L.o.C.

  37. Law of Cosines 5/19 Solve the triangle Step 2: Use Law of Sines to find < B Using L.o.C.

  38. Law of Cosines 5/19 Solve the triangle Step 2: Use Law of Sines to find < B sin(B) = 0.5098 Using L.o.C.

  39. Law of Cosines 5/19 Solve the triangle Step 2: Use Law of Sines to find < B sin(B) = 0.5098 B = 31 Using L.o.C.

  40. Law of Cosines 5/19 Solve the triangle Step 3: Subtract to find C 180 – 115 – 31 = 34 C = 34o Using L.o.C.

  41. VERY IMPORTANT I just gave you an EXAMPLE of how to solve this. Do NOT try to use steps 1-3 for every problem. IT WILL NOT WORK. Always think to yourself, “What do I need in order to be able to use the Law of Sines?”

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