Unit 7 Day 4 – Law of
Sines – Ambiguous Case
We will apply the Law of Sines to SSA
triangles
and
I will solve SSA triangles with obtuse
angles.
January 5, 2017
Law of Sines – Ambiguous Case
• If you are given two angles and one side
(ASA or AAS), the Law of Sines gives ONE
solution for a missing side.
• If you are given two sides and one angle—
SSA (where you must find an angle), there
could be one, two or no triangles
• This is called Ambiguous Case.
• Ambiguous - open to two or more
interpretations.
• Before you can solve the triangle, you have
to figure out how many triangles you have!
Given an angle, a side opposite the angle
and a side adjacent to the angle. Solve
the triangle.
Step 1: Is the given angle obtuse or
acute?
For obtuse angles, there will be either 0
or 1 triangle.
adj
A
opp
Situation #1
Angle is obtuse
opp. ≤ adj
No Triangle!
opp
adj
A
Situation #2
Angle is obtuse
One Triangle!
opp. > adj.
opp
adj
A
Example –Two sides and an angle are given.
Determine whether the given information results in
one triangle, two triangles, or no triangle at all. Solve
any triangle(s) that result.
a = 2, c = 1,  = 100
opp
adj
A
Given  is obtuse, so 0 or 1 Δ.
opp adj
1 < 2
No triangle is possible.
b
c=1
C
100
a=2
B
Example –Two sides and an angle are given.
Determine whether the given information results in
one triangle, two triangles, or no triangle at all. Solve
any triangle(s) that result.
a = 1, c = 2,  = 100
adj
opp
Given  is obtuse, so 0 or 1 Δ. b
  29.5
Opp (2) > adj (1)
A
c=2
One triangle exists.
sin  sin 

a
c
sin  sin100

1
2
sin   .4924
  29.5
sin  sin 

b
c
sin 50.5 sin100

b
2
2 sin 50.5  b sin100
C
100
a=1
50.5
B
2 sin 50.5
b
sin100
b  1.56
Law of Sines – Ambiguous Case – Given  is
Obtuse (flipbook)
Given an angle, a side opposite the angle and a
side adjacent to the angle of a potential
triangle:
Angle is obtuse, opp ≤ adj
Angle is obtuse, opp > adj
opp
adj
opp
adj
A
NO TRIANGLE!
A
ONE TRIANGLE!