LAW OF SINES:

1
LAW OF SINES

• THE AMBIGUOUS CASE

2
Review

• 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 650
Law of Cosines
3. a 78.3, b 23.5 and c 36.8
Law of Cosines
3
Law of Sines The Ambiguous Case

• Given
• lengths of two sides and the angle opposite one
  of them (S-S-A)

4

• AMBIGUOUS
• Open to various interpretations
• Having double meaning
• Difficult to classify, distinguish, or
  comprehend

5
Always set your triangle up this way
Other given side here
6
Possible Outcomes
Case 1 If ? is acute and the side opposite the
given angle lt the other given side.
C
a. If a lt h
C
a
b
a
h b sin A
b
B
A
h
c
A
B
c
NO SOLUTION
7
Possible Outcomes
Case 1 If ? is acute and the side opposite the
given angle lt the other given side.
b. If a h
C
h b sin A
b
a
h
A
B
c
1 SOLUTION
8
Possible Outcomes
Case 1 If ? is acute and the side opposite the
given angle lt the other given side.
b. If a gt h
C
b
h
a
a
?
?
180 - ?
A
B
B
c
2 SOLUTIONS
9
Possible Outcomes
Case 2 If ? is obtuse and the side opposite the
given angle gt the other given side.
C
a
b
B
A
c
ONE SOLUTION
10
Possible Outcomes
Case 2 If ? is obtuse and the side opposite the
given angle ? the other given side.
C
a
b
A
B
c
NO SOLUTION
11
SUMMARY
Side opposite lt other side
Side opposite gt other side
FIND HEIGHT h other side ? sin ?
1 Solution
No Solution
Side opposite h
1 Solution
Side opposite gt h
2 Solutions
Side opposite gt other side
Side opposite lt other side
No Solution
1 Solution
12
EXAMPLE 1
Given ?ABC where a 22 inches b 12 inches
m?A 42o
SINGLESOLUTION CASE
(acute)
Find m ?B, m ?C, and c.
13
Sin B ? 0.36498 m?B 21.41o or 21o
Sine values of supplementary angles are
equal. The supplement of ?B is ?B2. ? m?B2159o
14
m?C 180o (42o 21o) m?C 117o
sin A sin C a c
c 29.29 inches
SINGLESOLUTION CASE
15
EXAMPLE 2
Given ?ABC where c 15 inches b 25 inches
m?C 85o
15 lt 25 sin 85o
c ? b sin C
NO SOLUTION CASE
(acute)
Find m ?B, m ?C, and c.
16
Sin B ? 1.66032 m?B ? Sin B gt 1 NOT POSSIBLE
! Recall 1 ? sin ? ? 1
NO SOLUTION CASE
17
EXAMPLE 3
Given ?ABC where b 15.2 inches a 20 inches
m?B 110o
NO SOLUTION CASE
(obtuse)
Find m ?B, m ?C, and c.
18
Sin B ? 1.23644 m?B ? Sin B gt 1 NOT POSSIBLE
! Recall 1 ? sin ? ? 1
NO SOLUTION CASE
19
EXAMPLE 4
Given ?ABC where a 24 inches b 36 inches
m?A 25o
a ? b sin A
24 gt 36 sin 25o
TWO SOLUTION CASE
(acute)
Find m ?B, m ?C, and c.
20
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 (25o141o) m?C2 14o
21
sin A sin C a c1
c1 51.04 inches
sin A sin C a c2
c 13.74 inches
22
EXAMPLE 3
Final Answers m?B1 39o m?C1 116o c1 51.04
in.
m?B2 141o m?C2 14o C2 13.74 in.
TWO SOLUTION CASE
23
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  
24
Answers  1)Case 1 m?B49o,m?C96o,c15.78
Case 2   m?B131o,m?C14o,c3.84
2)No possible solution. 3)m?B38o,m?C76o,c15.
93