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

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

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

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

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

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

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

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

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

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

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

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

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

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

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