Triangles: the Ambiguous Case

Triangles: the Ambiguous Case Source: Glencoe Adv. Mathematical Concepts Pg 324 #16 By: Rachel Atmadja, 2002 The Ambiguous Case Find all solutions for the triangle below: B = 37 a = 32 b = 27 C 32 B 37 27 A The Ambiguous Case Find all solutions for the triangle below: B = 37 a = 32

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

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

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

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

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

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

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

Sin A A = 45.5 32 27 27 C = 97.5 c = 44.5 B 37 45.5 A 32 = Sin 97.5 c The Ambiguous Case Find all solutions for the triangle below: B = 37 a = 32

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

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

a = 32 Now find C2. You can find C2 simply by subtracting angles B and A2 from 180. b = 27 C 27 32 27 A = 45.5 A2 = 134.5 C = 97.5 C2 = 8.5 c = 44.5 B 37 45.5 A A

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

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

C2 = 8.5 45.5 A c2 A = 45.5 c = 44.5 B Sin 8.5 A c2 = 6.6