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Document related concepts
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Transcript 
LAW OF SINES:
THE AMBIGUOUS CASE
MENTAL DRILL
Identify if the given oblique triangle can be
solved using the Law of Sines or the Law of
Cosines
1. X = 210, Z = 650 and y = 34.7
Law of Sines
2. s = 73.1, r = 93.67 and T =
0
65
Law of Cosines
3. a = 78.3, b = 23.5 and c = 36.8
/ctr
Law of Cosines
AMBIGUOUS
•
•
•
/ctr
Open to various
interpretations
Having double meaning
Difficult to classify,
distinguish, or
comprehend
RECALL:
• Opposite sides of
angles of a triangle
• Interior Angles of a
Triangle Theorem
• Triangle
Inequality Theorem
/ctr
RECALL:
•
Oblique Triangles
Triangles that do
not have right
angles
(acute or obtuse
triangles)
/ctr
RECALL:
•
LAW OF SINE
sin A sin B sin C


a
b
c
– 1  sin   1
/ctr
RECALL:
• Sine values of
supplementary
angles are equal.
Example:
Sin 80o = 0.9848
Sin 100o = 0.9848
/ctr
Law of Sines:
The Ambiguous Case
Given:
lengths of two sides
and the angle opposite
one of them (S-S-A)
/ctr
Possible Outcomes
Case 1: If A is acute and a < b
C
a. If a < b sinA
a
b
C
a
h = b sin A
A
c
b
B
A
h
c
B
NO SOLUTION
Possible Outcomes
Case 1: If A is acute and a < b
C
b. If a = b sinA
a
b
C
h = b sin A
A
c
b
B
A
h=a
c
B
1 SOLUTION
Possible Outcomes
Case 1: If A is acute and a < b
C
b
b. If a > b sinA
a
C
h = b sin A
A
c
b
B
a

180 - 
A
c
h
a

B
2 SOLUTIONS
B
Possible Outcomes
Case 2: If A is obtuse and a > b
C
a
b
A
c
B
ONE SOLUTION
Possible Outcomes
Case 2: If A is obtuse and a ≤ b
C
a
b
A
c
B
NO SOLUTION
EXAMPLE 1
Given: ABC where
a = 22 inches
a>b
b = 12 inches
mA > mB
SINGLE–SOLUTION CASE
(acute)
mA =
Find m B, m C, and c.
o
42
sin A = sin B
a
b
Sin B  0.36498
mB = 21.41o or 21o
Sine values of supplementary
angles are equal.
The supplement of B is B2.
 mB2=159o
mC = 180o – (42o + 21o)
o
mC = 117
sin A = sin C
a
c
c = 29.29 inches
SINGLE–SOLUTION CASE
EXAMPLE 2
Given: ABC where
c = 15 inches
b = 25 inches
c ? b sin C
c<b
15 < 25 sin 85o
NO SOLUTION CASE
(acute)
mC =
Find m B, m C, and c.
o
85
sin A = sin B
a
b
Sin B  1.66032
mB = ?
Sin B > 1 NOT POSSIBLE !
Recall: – 1  sin   1
NO SOLUTION CASE
/ctr
EXAMPLE 3
Given: ABC where
b = 15.2 inches
a = 20 inches
b<a
NO SOLUTION CASE
(obtuse)
mB =
Find m B, m C, and c.
o
110
sin A = sin B
a
b
Sin B  1.23644
mB = ?
Sin B > 1 NOT POSSIBLE !
Recall: – 1  sin   1
NO SOLUTION CASE
/ctr
EXAMPLE 4
Given: ABC where
a = 24 inches
b = 36 inches
a ? b sin A
a<b
24 > 36 sin 25o
TWO – SOLUTION CASE
o
mA = 25 (acute)
Find m B, m C, and c.
sin A = sin B
a
b
Sin B  0.63393
mB = 39.34o or 39o
The supplement of B is B2. 
mB2 = 141o
mC1 = 180o – (25o + 39o)
mC1 = 116o
mC2 = 180o – (25o+141o)
mC2 = 14o
sin A = sin C
a
c1
c1 = 51.04 inches
sin A = sin C
a
c2
c = 13.74 inches
/ctr
EXAMPLE 3
Final Answers:
mB1 = 39o mB2 = 141o
mC1 = 116o mC2 = 14o
c1 = 51.04 in.C2= 13.74 in.
TWO – SOLUTION CASE
/ctr
SEATWORK: (notebook)
Answer in pairs.
Find m B, m C, and c, if
they exist.
1) a = 9.1, b = 12, mA = 35o
2) a = 25, b = 46, mA = 37o
3) a = 15, b = 10, mA = 66o
/ctr
Answers:
1)Case 1:
mB=49o,mC=96o,c=15.78
Case 2:
mB=131o,mC=14o,c=3.84
2)No possible solution.
3)mB=38o,mC=76o,c=15.93
/ctr
LAW OF SINES:
THE AMBIGUOUS CASE
MENTAL DRILL
Identify if the given oblique triangle can be
solved using the Law of Sines or the Law of
Cosines
1. X = 210, Z = 650 and y = 34.7
Law of Sines
2. s = 73.1, r = 93.67 and T =
0
65
Law of Cosines
3. a = 78.3, b = 23.5 and c = 36.8
/ctr
Law of Cosines
RECALL:
• Opposite sides of
angles of a triangle
• Interior Angles of a
Triangle Theorem
• Triangle
Inequality Theorem
/ctr
RECALL:
•
Oblique Triangles
Triangles that do
not have right
angles
(acute or obtuse
triangles)
/ctr
RECALL:
•
LAW OF SINE
sin A sin B sin C


a
b
c
– 1  sin   1
/ctr
RECALL:
• Sine values of
supplementary
angles are equal.
Example:
Sin 80o = 0.9848
Sin 100o = 0.9848
/ctr
Law of Sines:
The Ambiguous Case
Given:
lengths of two sides
and the angle opposite
one of them (S-S-A)
/ctr
Possible Outcomes
Case 1: If A is acute and a < b
C
a. If a < b sinA
a
b
C
a
h = b sin A
A
c
b
B
A
h
c
B
NO SOLUTION
Possible Outcomes
Case 1: If A is acute and a < b
C
b. If a = b sinA
a
b
C
h = b sin A
A
c
b
B
A
h=a
c
B
1 SOLUTION
Possible Outcomes
Case 1: If A is acute and a < b
C
b
b. If a > b sinA
a
C
h = b sin A
A
c
b
B
a

180 - 
A
c
h
a

B
2 SOLUTIONS
B
Possible Outcomes
Case 2: If A is obtuse and a > b
C
a
b
A
c
B
ONE SOLUTION
Possible Outcomes
Case 2: If A is obtuse and a ≤ b
C
a
b
A
c
B
NO SOLUTION
EXAMPLE 1
Given: ABC where
a = 22 inches
a>b
b = 12 inches
mA > mB
SINGLE–SOLUTION CASE
(acute)
mA =
Find m B, m C, and c.
o
42
sin A = sin B
a
b
Sin B  0.36498
mB = 21.41o or 21o
Sine values of supplementary
angles are equal.
The supplement of B is B2.
 mB2=159o
mC = 180o – (42o + 21o)
o
mC = 117
sin A = sin C
a
c
c = 29.29 inches
SINGLE–SOLUTION CASE
EXAMPLE 2
Given: ABC where
c = 15 inches
b = 25 inches
c ? b sin C
c<b
15 < 25 sin 85o
NO SOLUTION CASE
(acute)
mC =
Find m B, m C, and c.
o
85
sin A = sin B
a
b
Sin B  1.66032
mB = ?
Sin B > 1 NOT POSSIBLE !
Recall: – 1  sin   1
NO SOLUTION CASE
/ctr
EXAMPLE 3
Given: ABC where
b = 15.2 inches
a = 20 inches
b<a
NO SOLUTION CASE
(obtuse)
mB =
Find m B, m C, and c.
o
110
sin A = sin B
a
b
Sin B  1.23644
mB = ?
Sin B > 1 NOT POSSIBLE !
Recall: – 1  sin   1
NO SOLUTION CASE
/ctr
EXAMPLE 4
Given: ABC where
a = 24 inches
b = 36 inches
a ? b sin A
a<b
24 > 36 sin 25o
TWO – SOLUTION CASE
o
mA = 25 (acute)
Find m B, m C, and c.
sin A = sin B
a
b
Sin B  0.63393
mB = 39.34o or 39o
The supplement of B is B2. 
mB2 = 141o
mC1 = 180o – (25o + 39o)
mC1 = 116o
mC2 = 180o – (25o+141o)
mC2 = 14o
sin A = sin C
a
c1
c1 = 51.04 inches
sin A = sin C
a
c2
c = 13.74 inches
/ctr
EXAMPLE 3
Final Answers:
mB1 = 39o mB2 = 141o
mC1 = 116o mC2 = 14o
c1 = 51.04 in.C2= 13.74 in.
TWO – SOLUTION CASE
/ctr
EXAMPLE 3
Final Answers:
mB1 = 39o mB2 = 141o
mC1 = 116o mC2 = 14o
c1 = 51.04 in.C2= 13.74 in.
TWO – SOLUTION CASE
/ctr
Answers:
1)Case 1:
mB=49o,mC=96o,c=15.78
Case 2:
mB=131o,mC=14o,c=3.84
2)No possible solution.
3)mB=38o,mC=76o,c=15.93
/ctr
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