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TRIGONOMETRYSINE LAW: THE AMBIGUOUS CASE
Updated: 2016/12/08File: 20160829-125919-rs2.2B-SineLawWorkflow
Sigtunaskolan Humanistiska Läroverket (SSHL)
Overview
1. ASS Trigangles
2. The Workflow
ASS TRIGANGLES
Sine Law: Given Information
hb a
cA B
C
Sine Lawsin A
a�
sin Bb
�sin C
c
Sine Law: The Ambiguous Case
Essential QuestionHow do we determine the number of solutions of an ASStriangle using the sine law?
THE WORKFLOW
Sine Law: The Ambiguous Case
]A > 90◦
a > bba
A
C
B
ba
A
C
B
a ≥ b
b a
cA
C
B
find h a � h
hb
acA
C
B
a < h
hb a
A
C
B
hb a
A
C
BB
OneSolution
OneSolution
No Solution No Solution
OneSolution
TwoSolutions
Yes
NoYes
No
Yes
No
Yes
No
No
Yes
Sine Law: The Ambiguous Case
]A > 90◦
a > b
ba
A
C
B
ba
A
C
B
a ≥ b
b a
cA
C
B
find h a � h
hb
acA
C
B
a < h
hb a
A
C
B
hb a
A
C
BB
OneSolution
OneSolution
No Solution No Solution
OneSolution
TwoSolutions
Yes
NoYes
No
Yes
No
Yes
No
No
Yes
Sine Law: The Ambiguous Case
]A > 90◦
a > bba
A
C
B
ba
A
C
B
a ≥ b
b a
cA
C
B
find h a � h
hb
acA
C
B
a < h
hb a
A
C
B
hb a
A
C
BB
OneSolution
OneSolution
No Solution No Solution
OneSolution
TwoSolutions
Yes
No
Yes
No
Yes
No
Yes
No
No
Yes
Sine Law: The Ambiguous Case
]A > 90◦
a > bba
A
C
B
ba
A
C
B
a ≥ b
b a
cA
C
B
find h a � h
hb
acA
C
B
a < h
hb a
A
C
B
hb a
A
C
BB
OneSolution
OneSolution
No Solution No Solution
OneSolution
TwoSolutions
Yes
No
Yes
No
Yes
No
Yes
No
No
Yes
Sine Law: The Ambiguous Case
]A > 90◦
a > bba
A
C
B
ba
A
C
B
a ≥ b
b a
cA
C
B
find h a � h
hb
acA
C
B
a < h
hb a
A
C
B
hb a
A
C
BB
OneSolution
OneSolution
No Solution No Solution
OneSolution
TwoSolutions
Yes
NoYes
No
Yes
No
Yes
No
No
Yes
Sine Law: The Ambiguous Case
]A > 90◦
a > bba
A
C
B
ba
A
C
B
a ≥ b
b a
cA
C
B
find h a � h
hb
acA
C
B
a < h
hb a
A
C
B
hb a
A
C
BB
OneSolution
OneSolution
No Solution
No Solution
OneSolution
TwoSolutions
Yes
NoYes
No
Yes
No
Yes
No
No
Yes
Sine Law: The Ambiguous Case
]A > 90◦
a > bba
A
C
B
ba
A
C
B
a ≥ b
b a
cA
C
B
find h a � h
hb
acA
C
B
a < h
hb a
A
C
B
hb a
A
C
BB
OneSolution
OneSolution
No Solution
No Solution
OneSolution
TwoSolutions
Yes
NoYes
No
Yes
No
Yes
No
No
Yes
Sine Law: The Ambiguous Case
]A > 90◦
a > bba
A
C
B
ba
A
C
B
a ≥ b
b a
cA
C
B
find h a � h
hb
acA
C
B
a < h
hb a
A
C
B
hb a
A
C
BB
OneSolution
OneSolution
No Solution
No Solution
OneSolution
TwoSolutions
Yes
NoYes
No
Yes
No
Yes
No
No
Yes
Sine Law: The Ambiguous Case
]A > 90◦
a > bba
A
C
B
ba
A
C
B
a ≥ b
b a
cA
C
B
find h a � h
hb
acA
C
B
a < h
hb a
A
C
B
hb a
A
C
BB
OneSolution
OneSolution
No Solution
No Solution
OneSolution
TwoSolutions
Yes
NoYes
No
Yes
No
Yes
No
No
Yes
Sine Law: The Ambiguous Case
]A > 90◦
a > bba
A
C
B
ba
A
C
B
a ≥ b
b a
cA
C
B
find h a � h
hb
acA
C
B
a < h
hb a
A
C
B
hb a
A
C
BB
OneSolution
OneSolution
No Solution
No Solution
OneSolution
TwoSolutions
Yes
NoYes
No
Yes
No
Yes
No
No
Yes
Sine Law: The Ambiguous Case
]A > 90◦
a > bba
A
C
B
ba
A
C
B
a ≥ b
b a
cA
C
B
find h
a � h
hb
acA
C
B
a < h
hb a
A
C
B
hb a
A
C
BB
OneSolution
OneSolution
No Solution
No Solution
OneSolution
TwoSolutions
Yes
NoYes
No
Yes
No
Yes
No
No
Yes
Sine Law: The Ambiguous Case
]A > 90◦
a > bba
A
C
B
ba
A
C
B
a ≥ b
b a
cA
C
B
find h a � h
hb
acA
C
B
a < h
hb a
A
C
B
hb a
A
C
BB
OneSolution
OneSolution
No Solution
No Solution
OneSolution
TwoSolutions
Yes
NoYes
No
Yes
No
Yes
No
No
Yes
Sine Law: The Ambiguous Case
]A > 90◦
a > bba
A
C
B
ba
A
C
B
a ≥ b
b a
cA
C
B
find h a � h
hb
acA
C
B
a < h
hb a
A
C
B
hb a
A
C
BB
OneSolution
OneSolution
No Solution
No Solution
OneSolution
TwoSolutions
Yes
NoYes
No
Yes
No
Yes
No
No
Yes
Sine Law: The Ambiguous Case
]A > 90◦
a > bba
A
C
B
ba
A
C
B
a ≥ b
b a
cA
C
B
find h a � h
hb
acA
C
B
a < h
hb a
A
C
B
hb a
A
C
BB
OneSolution
OneSolution
No Solution
No Solution
OneSolution
TwoSolutions
Yes
NoYes
No
Yes
No
Yes
No
No
Yes
Sine Law: The Ambiguous Case
]A > 90◦
a > bba
A
C
B
ba
A
C
B
a ≥ b
b a
cA
C
B
find h a � h
hb
acA
C
B
a < h
hb a
A
C
B
hb a
A
C
BB
OneSolution
OneSolution
No Solution
No Solution
OneSolution
TwoSolutions
Yes
NoYes
No
Yes
No
Yes
No
No
Yes
Sine Law: The Ambiguous Case
]A > 90◦
a > bba
A
C
B
ba
A
C
B
a ≥ b
b a
cA
C
B
find h a � h
hb
acA
C
B
a < h
hb a
A
C
B
hb a
A
C
BB
OneSolution
OneSolution
No Solution
No Solution
OneSolution
TwoSolutions
Yes
NoYes
No
Yes
No
Yes
No
No
Yes
Sine Law: The Ambiguous Case
]A > 90◦
a > bba
A
C
B
ba
A
C
B
a ≥ b
b a
cA
C
B
find h a � h
hb
acA
C
B
a < h
hb a
A
C
B
hb a
A
C
BB
OneSolution
OneSolution
No Solution
No Solution
OneSolution
TwoSolutions
Yes
NoYes
No
Yes
No
Yes
No
No
Yes
Sine Law: The Ambiguous Case
]A > 90◦
a > bba
A
C
B
ba
A
C
B
a ≥ b
b a
cA
C
B
find h a � h
hb
acA
C
B
a < h
hb a
A
C
B
hb a
A
C
BB
OneSolution
OneSolution
No Solution
No Solution
OneSolution
TwoSolutions
Yes
NoYes
No
Yes
No
Yes
No
No
Yes
Sine Law: The Ambiguous Case
]A > 90◦
a > bba
A
C
B
ba
A
C
B
a ≥ b
b a
cA
C
B
find h a � h
hb
acA
C
B
a < h
hb a
A
C
B
hb a
A
C
BB
OneSolution
OneSolution
No Solution No Solution
OneSolution
TwoSolutions
Yes
NoYes
No
Yes
No
Yes
No
No
Yes
