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Triangles: the Ambiguous Case By: Rachel Atmadja, 2002 Source: Glencoe Adv. Mathematical Concepts Pg 324 #16

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The Ambiguous Case Find all solutions for the triangle below: B = 37 a = 32 b = 27 C B A 32 27 37

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The Ambiguous Case Find all solutions for the triangle below: B = 37 a = 32 b = 27 C B A 32 27 37 The ‘magic’ number: 32(sin 37 )= 19.3

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The Ambiguous Case Find all solutions for the triangle below: B = 37 a = 32 b = 27 C B A 32 27 37 The ‘magic’ number: 32(sin 37 )= 19.3 19.3 < 27 < 32

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The Ambiguous Case Find all solutions for the triangle below: B = 37 a = 32 b = 27 C B A 32 27 37 The ‘magic’ number: 32(sin 37 )= 19.3 19.3 < 27 < 32 Therefore, there are 2 solutions

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The Ambiguous Case Find all solutions for the triangle below: B = 37 a = 32 b = 27 C B A 32 27 37 27 Use law of Sines to find A: Sin 37 Sin A 27 32 =

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The Ambiguous Case Find all solutions for the triangle below: B = 37 a = 32 b = 27 C B A 32 27 37 27 Sin 37 Sin A 27 32 A = 45.5 = 45.5 Use law of Sines to find A:

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The Ambiguous Case Find all solutions for the triangle below: B = 37 a = 32 b = 27 C B A 32 27 37 27 Sin 37 Sin A 27 32 A = 45.5 C = 180 – 37 – 45.5 = 97.5 = 45.5 Now find C:

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The Ambiguous Case Find all solutions for the triangle below: B = 37 a = 32 b = 27 C B A 32 27 37 27 Sin 37 Sin A Sin 97.5 27 32 c A = 45.5 C = 97.5 = 45.5 Now use the Law of Sines to find side c =

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The Ambiguous Case Find all solutions for the triangle below: B = 37 a = 32 b = 27 C B A 32 27 37 27 Sin 37 Sin A Sin 97.5 27 32 c A = 45.5 C = 97.5 c = 44.5 = 45.5 Now use the Law of Sines to find side c =

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The Ambiguous Case Find all solutions for the triangle below: B = 37 a = 32 b = 27 C B A 32 27 37 27 A = 45.5 C = 97.5 c = 44.5 45.5 Now to find the 2 nd set of solution, let’s being by finding A2: A

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The Ambiguous Case Find all solutions for the triangle below: B = 37 a = 32 b = 27 C B A 32 27 37 27 A = 45.5 A2 = 134.5 C = 97.5 c = 44.5 45.5 Notice that A and A2 are supplementary angles. To find A2, simply subtract A from 180 A

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The Ambiguous Case Find all solutions for the triangle below: B = 37 a = 32 b = 27 C B A 32 27 37 27 A = 45.5 A2 = 134.5 C = 97.5C2 = 8.5 c = 44.5 45.5 Now find C2. You can find C2 simply by subtracting angles B and A2 from 180. A

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The Ambiguous Case Find all solutions for the triangle below: B = 37 a = 32 b = 27 C B A 32 27 37 27 A = 45.5 A2 = 134.5 C = 97.5C2 = 8.5 c = 44.5 45.5 Use the Law of Sines to find side c2: A

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The Ambiguous Case Find all solutions for the triangle below: B = 37 a = 32 b = 27 C B A 32 27 37 27 Sin 37 Sin 8.5 27 c2 A = 45.5 A2 = 134.5 C = 97.5C2 = 8.5 c = 44.5 c2 = 6.6 45.5 Use the Law of Sines to find side c2: A =

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13.5 Law of Cosines Objectives: 1.Solve problems by using the Law of Cosines 2.Determine whether a triangle can be solved by first using the Law of Sines.

13.5 Law of Cosines Objectives: 1.Solve problems by using the Law of Cosines 2.Determine whether a triangle can be solved by first using the Law of Sines.

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