www.iTutoring.com -­‐ NOTES The Ambiguous Case for the Law of Sines Name _________________________________ Date ___________________ Period _______ To solve a triangle:
Determine the length of all three sides and the measure of all three angles.
Need to know the measure of one side and...
two angles
two sides
The Law of Sines
The Law of
Cosines
A
A
S
A
A
ASA
S
S
S
SAA
one side and one angle
The Ambiguous
Case
S
S
The Law of
Cosines
A
S
S
A
S
SSS
SSA
SAS
Given two sides and an angle opposite one side (SSA Triangle)...
this gives us the ambiguous case for a triangle.
if a < h or a < b·sin !
No triangle can be formed
b
h
b
b·sin ! = h
h
sin ! =
a
!
Given two sides and an angle opposite one side (SSA Triangle)...
this gives us the ambiguous case for a triangle.
if a = h or a = b·sin !
One right triangle can be formed
b
h
b
b·sin ! = h
sin ! =
h a
!
Given two sides and an angle opposite one side (SSA Triangle)...
this gives us the ambiguous case for a triangle.
if a > h or a > b·sin !, and a < b,
Two triangles can be formed
b
h
sin ! =
b
b·sin ! = h
a h
!
One Obtuse
a
One Acute
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The Ambiguous Case for the Law of Sines Pg. 2 Given two sides and an angle opposite one side (SSA Triangle)...
this gives us the ambiguous case for a triangle.
if a > h or a > b·sin !, and a > b,
One triangle can be formed
b
h
b
b·sin ! = h
a
h
sin ! =
!
Given two sides and an angle opposite one side (SSA Triangle)...
this gives us the ambiguous case for a triangle.
a < h or a < b·sin !
a = h or a = b·sin !
No triangle
One Right Triangle
b
!
a
b
!
a > h or a > b·sin !,
and a < b,
Two triangles
b
a
a
a
!
a > h or a > b·sin !,
and a > b,
One triangle
b
a
!
One Obtuse One Acute
© iTutoring.com
The Ambiguous Case for the Law of Sines Pg. 3 Solve the following triangles
a = 4, b = 5, ! = 58°
5
4
58°
Solve the following triangles
a = 6, b = 12, ! = 30°
12
12
6
6
30°
30°
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The Ambiguous Case for the Law of Sines Pg. 4 Solve the following triangles
a = 5, b = 3, ! = 100°
5
3
5
100°
3
100°
Solve the following triangles
!1
3
a = 2, b = 3, ! = 40°
3
2
2
"1
40°
c1
40°
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The Ambiguous Case for the Law of Sines Pg. 5 Solve the following triangles
3
!2
3
a = 2, b = 3, ! = 40°
"2
40°
c2
2
2
2
40°
Given two sides and an angle opposite one side (SSA Triangle)...
this gives us the ambiguous case for a triangle.
a < h or a < b·sin !
a = h or a = b·sin !
No triangle
One Right Triangle
b
!
a
b
!
a > h or a > b·sin !,
and a < b,
Two triangles
b
a
a
a
!
a > h or a > b·sin !,
and a > b,
One triangle
b
a
!
One Obtuse One Acute
© iTutoring.com
The Ambiguous Case for the Law of Sines Pg. 6