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Law of Sines

Law of Sines. Objective: To solve triangles that are not right triangles. Law of Sines. We have been solving for sides and angles of right triangles with a variety of methods. We will now look at solving triangles that are not right triangles. These triangles are called oblique triangles.

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Law of Sines

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  1. Law of Sines Objective: To solve triangles that are not right triangles

  2. Law of Sines • We have been solving for sides and angles of right triangles with a variety of methods. We will now look at solving triangles that are not right triangles. These triangles are called oblique triangles.

  3. Law of Sines • We have been solving for sides and angles of right triangles using a variety of methods. We will now look at solving triangles that are not right triangles. These triangles are called oblique triangles. • To solve an oblique triangle with the Law of Sines, you need to know the measure of at least one side and the opposite angle. This breaks down into the following cases.

  4. Law of Sines • To use the Law of Sines, you need to have: • Two angles and any side (AAS or ASA) • Two sides and an angle opposite one of them (SSA)

  5. Law of Sines • Given triangle ABC with sides a, b and c, then:

  6. Example 1(AAS) • For the given triangle, find the remaining angle and sides.

  7. Example 1(AAS) • For the given triangle, find the remaining angle and sides. • We know that <B=28.70 • We know that <C=102.30 • We can find that <A=490

  8. Example 1(AAS) • For the given triangle, find the remaining angle and sides.

  9. Example 1(AAS) • For the given triangle, find the remaining angle and sides.

  10. Example 1(AAS) • For the given triangle, find the remaining angle and sides.

  11. You Try • Given triangle ABC, find the missing angle and sides.

  12. You Try • Given triangle ABC, find the missing angle and sides. • <A = 350

  13. You Try • Given triangle ABC, find the missing angle and sides. • <A = 350

  14. Example 2(ASA) • A pole tilts toward the sun at an 80 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 430. How tall is the pole?

  15. Example 2(ASA) • A pole tilts toward the sun at an 80 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 430. How tall is the pole? • We know that <A=430 • We know that <B=980 • We can find that <C=390

  16. Example 2(ASA) • A pole tilts toward the sun at an 80 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 430. How tall is the pole?

  17. Example 2(ASA) • A pole tilts toward the sun at an 80 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 430. How tall is the pole?

  18. Example 2(ASA) • A pole tilts toward the sun at an 80 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 430. How tall is the pole?

  19. You Try • Given triangle ABC, find the missing angle and sides. • <A = 700 • <B = 440 • c = 12 ft

  20. You Try • Given triangle ABC, find the missing angle and sides. • <A = 700 • <B = 440 • c = 12 ft • <C = 660

  21. You Try • Given triangle ABC, find the missing angle and sides. • <A = 700 • <B = 440 • c = 12 ft • <C = 660

  22. Example 3 • We know from Geometry that SSA does not make a unique triangle. When given SSA, one of three situations may occur. • One unique triangle • No triangle • Two different triangles

  23. Example 3 • For the given triangle, find the missing angles and side.

  24. Example 3 • For the given triangle, find the missing angles and side.

  25. Example 3 • For the given triangle, find the missing angles and side.

  26. Example 3 • Remember, there are two answers to • <B = 21.40 or <B = 158.60. • The answer of 158.60 won’t work since this angle added to the given angle of 420 would be greater than 1800, and we know that doesn’t make sense for a triangle, therefore there is only one solution to this problem.

  27. Example 4 • For the given triangle, find the missing angles and side.

  28. Example 4 • For the given triangle, find the missing angles and side. • There is no solution to this.

  29. Example 5 • For the given triangle, find the missing angles and side. • a = 12m • b = 31m • <A = 20.50

  30. Example 5 • For the given triangle, find the missing angles and side. • A = 12m • B = 31m • <A = 20.50

  31. Example 5 • For the given triangle, find the missing angles and side. • A = 12m • B = 31m • <A = 20.50

  32. Example 5 • Remember, there are two answers to . • <B = 64.80 or 115.20. • Since both answers work with an angle of 20.50, there are two triangles possible for this problem.

  33. Example 5 • For the given triangle, find the missing angles and side. • A = 12m • B = 31m • <A = 20.50

  34. Example 5 • For the given triangle, find the missing angles and side. • A = 12m • B = 31m • <A = 20.50

  35. Example 5 • Here are the two triangles together.

  36. You Try • Page 598 • 20

  37. You Try • Page 598 • 20 There is no triangle

  38. You Try • Page 598 • 22

  39. You Try • Page 598 • 22

  40. You Try • Page 598 • 22

  41. You Try • Page 598 • 22 • If <B = 36.80, it can also be 143.20. This with the given angle of 760 is more than 1800, so there is only one triangle.

  42. You Try • Page 598 • 6

  43. You Try • Page 598 • 6

  44. Example 5 • Remember, there are two answers to . • <B = 74.20 or 105.80. • Since both answers work with an angle of 600, there are two triangles possible for this problem.

  45. You Try • Page 598 • 6

  46. Area of an Oblique Triangle • In the past, we needed the height of a triangle in order to find the area. We will now use an equation to find area that doesn’t need the height. There are three forms of this equation. They are:

  47. Area of an Oblique Triangle • In the past, we needed the height of a triangle in order to find the area. We will now use an equation to find area that doesn’t need the height. There are three forms of this equation. They are:

  48. Area of an Oblique Triangle • In the past, we needed the height of a triangle in order to find the area. We will now use an equation to find area that doesn’t need the height. There are three forms of this equation. They are: • In words, this equation says that area is equal to: ½(side)(side)(sin of the included angle)

  49. Example 6 • Find the area of a triangular lot having two sides of length 90 meters and 52 meters and an included angle of 1020.

  50. Example 6 • Find the area of a triangular lot having two sides of length 90 meters and 52 meters and an included angle of 1020.

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