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Albert Einstein (1879 – 1955)

There are only two ways to live your life. One is as though nothing is a miracle. The other is as though everything is a miracle. Albert Einstein (1879 – 1955) a German-born theoretical physicist who developed the theory of general relativity, effecting a revolution in physics. Chapter 6.

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Albert Einstein (1879 – 1955)

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  1. There are only two ways to live your life. One is as though nothing is a miracle. The other is as though everything is a miracle. Albert Einstein (1879 – 1955) a German-born theoretical physicist who developed the theory of general relativity, effecting a revolution in physics.

  2. Chapter 6 Additional Topics in Trigonometry

  3. Day I. Law of Sines (6.1) Part 1

  4. 6.1 GOAL 1 How to use the Law of Sines to solve oblique triangles.

  5. Why should you learn it? You can use the Law of Sines to solve real-life problems, for example to determine the distance from a ranger station to a forest fire

  6. Oblique Triangle -- a triangle that has no right angles

  7. To Solve you need

  8. 1-Today 2 angles and any side V S A A A A S

  9. 2-Next Time 2 sides and an angle opposite one of them SSA A S S

  10. 3-Later 3 sides S S S

  11. 4-Later 2 sides and their included angle S A S

  12. Standard 13.1 Students know the Law of Sines.

  13. Law of Sines If ABC is a triangle with sides a, b, and c, then a b c sin A sin B sin C = =

  14. Law of Sines Alternate Form sin A sin B sin C a b c = =

  15. Standard 13.2 Students can apply the Law of Sines to solve problems.

  16. Example 1 Finding a Measurement

  17. Find, to the nearest meter, the distance across Perch Lake from point A to point B. The length of AC, or b, equals 110 m, and measures of the angles of the triangle are as shown.

  18. A c b h C B a h c h = c sin B sin B = OR h b h = b sin C OR sin C =

  19. A b c h C B a c sin B = b sin C c b sin C sin B =

  20. A 110 m h 40 67 C B

  21. 110 c b sin C sin B = 40 67 Solve.

  22. sin 67sin 40 c 110 = 110sin 67 = csin 40 110sin 67 sin 40 c = c  158 meters

  23. Example 2 Given Two Angles and One Side - AAS

  24. Using the given information, solve the triangle. C 3.5 b 35 25 B A c

  25. What does it mean “to solve a triangle”? Find all unknowns

  26. sin 25sin 35 3.5 b = bsin 25 = 3.5sin 35 3.5sin 35 sin 25 b = b  4.8

  27. Using the given information, solve the triangle. C  4.8 3.5 35 25 B A c

  28. How do we find C? C = 180 – (25 + 35) C = 120

  29. Using the given information, solve the triangle. C  4.8 3.5 120 35 25 B A c

  30. Can I use the Pythagorean Theorem to find c? Why or why not? NOT A RIGHT TRIANGLE!

  31. sin 25sin 120 3.5 c = csin 25 = 3.5sin 120 3.5sin 120 sin 25 c = c  7.2

  32. Using the given information, solve the triangle. C  4.8 3.5 120 35 25 B A  7.2

  33. Your Turn

  34. Using the given information, solve the triangle. C a b 135 10 A B 45

  35. sin 135sin 10 45 b = bsin 135 = 45sin 10 45sin 10 sin 135 b = b  11.0

  36. Using the given information, solve the triangle. C a 11.0 135 10 A B 45

  37. A = 180 – (135 + 10) A = 35

  38. Using the given information, solve the triangle. C a 11.0 135 10 35 A B 45

  39. sin 135sin 35 45 a = asin 135 = 45sin 35 45sin 35 sin 135 a = a  36.5

  40. Using the given information, solve the triangle. C 36.5 11.0 135 10 35 A B 45

  41. Example 3 Finding a Measurement

  42. A pole tilts away from the sun at an 8° angle from the vertical, and it casts a 22 foot shadow. The angle of elevation from the tip of the shadow to the top of the pole is 43°. How tall is the pole?

  43. What do we need to find in order to use AAS or ASA? C 8 p 43 A B 22’ CBA

  44. CBA = 90 - 8 C 55 = 82 p 8 82 43 A B 22’ BCA = 180 - (82 + 43) = 55

  45. sin 55sin 43 22 p = psin 55 = 22sin 43 22sin 43 sin 55 p = p  18.3 ft

  46. What was the psychiatrist’s reply when a patient exclaimed, “I’m a teepee. I’m a wigwam!”?

  47. “Relax… You’re too tense!”

  48. Example 4 Given Two Angles and One Side - ASA

  49. Using the given information, solve the triangle. A = 102.4, C = 16.7, and b = 21.6

  50. A 21.6 102.4 c 16.7 60.9 B C a B=180 – (102.4 + 16.7) = 60.9

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