§7.2 Law of Sines
Objectives
1. Solve SAA or ASA Triangles
2. Solve SSA Triangles
3. Solve Applied Problems
16 May 2017
1
Kidoguchi, Kenneth
§7.2 Law of Sines
Proof Law of Sines
A
b
c
h
C
B
a
C
a
h’
b
A
B
a
c
16 May 2017
2
Kidoguchi, Kenneth
§7.2 Law of Sines
ASA, SAA, SSA, SAS, and SSS Triangles
c
c
B
A
A
Case 1: ASA
c
Case 1: SAA
c
C
16 May 2017
a
c
A
b
Case 2: SSA
C
b
Case 3: SAS
4
b
Case 4: SSS
Kidoguchi, Kenneth
§7.2 Law of Sines
The Law of Sines
For a triangle with sides a, b, c, and opposite sides A, B, C, respectively:
sin( A) sin( B) sin(C)


a
b
c
c
B
c
c
A
A
C
C
b
Case 1: ASA
Case 1: SAA
Case 2: SSA
A + B + C = p = 180º
16 May 2017
5
Kidoguchi, Kenneth
§7.2 Law of Sines
1. Solve SAA Triangles
Solve the triangle: A = 40°, B = 60°, a = 4
B
c
a
A
C
b
16 May 2017
6
Kidoguchi, Kenneth
§7.2 Law of Sines
1. Solve SAA Triangles
Solve the triangle: A = 40°, B = 60°, a = 4
a
b
c


sin  A sin  B  sin  C 
B
C  180º  40º  60º  80º
c
a
sin  B 
sin  60º 
b
a
 4  5.39
sin  A
sin  40º 
A
sin  C 
sin 80º 
c
a
 4  6.13
sin  A
sin  40º 
16 May 2017
C
b
7
Kidoguchi, Kenneth
§7.2 Law of Sines
1. Solve ASA Triangles
Solve the triangle: B = 25°, C= 85°, a = 6
B
c
a
A
C
b
16 May 2017
8
Kidoguchi, Kenneth
§7.2 Law of Sines
1. Solve ASA Triangles
Solve the triangle: B = 25°, C= 85°, a = 6
a
b
c


sin  A sin  B  sin  C 
B
A  180º 25º 85º  70º
sin  B 
sin  25º 
a
 6  2.70
sin  A
sin  70º 
c
sin  C 
sin 85º 
c
a
 6  6.36
sin  A
sin  70º 
A
b
a
C
b
16 May 2017
9
Kidoguchi, Kenneth
§7.2 Law of Sines
2. Solve SSA Triangles (No Triangle)
16 May 2017
10
Kidoguchi, Kenneth
§7.2 Law of Sines
2. Solve SSA Triangles (One Right Triangle)
16 May 2017
11
Kidoguchi, Kenneth
§7.2 Law of Sines
2. Solve SSA Triangles (Two Triangles)
16 May 2017
12
Kidoguchi, Kenneth
§7.2 Law of Sines
2. Solve SSA Triangles
16 May 2017
13
Kidoguchi, Kenneth
§7.2 Law of Sines
2. Solve SSA Triangle (One Solution)
Solve the triangle: a = 3, b = 5, B = 35°
sin  A sin  B  sin  C 


a
b
c
B
c
a
sin  A  sin  B 
b

3

arcsin
sin
35º
A


 1 
 20.13º


A 
5

159.87º

 A2 
180º  A1

C  180º 35º  A  124.87º
a
C
A
b
sin  C 
c
b  7.15
sin  B 
16 May 2017
14
Kidoguchi, Kenneth
§7.2 Law of Sines
2. Solve SSA Triangle (Two Solutions)
Solve the triangle: a = 6, b = 8, A = 35°
sin  A sin  B  sin  C 


a
b
c
b
8
sin  B   sin  A  sin  35º 
a
6

4

 B1 arcsin  sin  35º    49.89º
B 
3

130.11º

 B2 
180º  B1

C2  180º 35º  B2  14.89º
C1  180º 35º  B1  95.11º
sin  C2 
c2 
a  2.69
sin  A
sin  C1 
c1 
a  10.42
sin  A
16 May 2017
15
Kidoguchi, Kenneth
§7.2 Law of Sines
2. Solve SSA Triangle (Two Solutions)
Solve the triangle: a = 6, b = 8, A = 35°
B1  49.89º
B2  130.11º
C1  95.11º
C2  14.89º
c1  10.42
c2  2.69
B1
c1
a
c2 B2
C1
A
b
16 May 2017
a
C2
b
16
Kidoguchi, Kenneth
§7.2 Law of Sines
2. Solve SSA Triangles (No Solutions)
Solve the triangle: a = 5, b = 8, A = 50°
a
b
c


sin  A sin  B  sin  C 
b
8
sin  B   sin  A  sin  50º   1.225  1
a
5
Since:  1  sin  B   1, B does not exist.
16 May 2017
17
Kidoguchi, Kenneth
§7.2 Law of Sines
3. Solve Applied Problems
A steel plate has the form of a
quarter circle with a radius of 24
inches. Two 3/8 inch holes are to 24
be drilled in the plate positioned
20
as shown. Present the analysis to
find exact values for:
a) the linear distance between the
centres of the two holes and
b) the coordinates of the centre of
each hole.
y
Linear Distance
(x2 , y2)
(x1 , y1)
a
a
a
x
20
16 May 2017
18
24
Kidoguchi, Kenneth
§7.2 Law of Sines
3. Solve Applied Problems
a) the linear distance between the centres
of the two holes and
b) the coordinates of the centre of each
hole.
y
24
20
Linear Distance
(x2 , y2)
a
(x1 , y1)
a
a
20 24
16 May 2017
19
x
Kidoguchi, Kenneth
§7.2 Law of Sines
4. Solve Applied Problems
Crankshafts on Cars: On a certain
automobile, the crankshaft is 3 inches
long and the connecting rod is 9 inches
long (see the figure). At the time when
∠OPQ is 15°, how far is the piston (P)
from the center (O) of the crankshaft?
Let: a  QOP &   OQP
sin  a  sin    sin 15


9
3
OP
16 May 2017

20
Kidoguchi, Kenneth
16 May 2017
21
Kidoguchi, Kenneth
§7.2 Law of Sines
4. Solve Applied Problems
Crankshafts on Cars: On a certain
automobile, the crankshaft is 3 inches
long and the connecting rod is 9 inches
long (see the figure). At the time when
∠OPQ is 15°, how far is the piston (P)
from the center (O) of the crankshaft?
Let: a  QOP &   OQP
sin  a  sin    sin 15


9
3
OP

a  arcsin(3sin(15 )), 0  a  90
 180 15  a
 180 15  arcsin(3sin(15 ))
OP  3
16 May 2017
sin  arcsin(3sin(15 )) 
sin 15

22
Kidoguchi, Kenneth
§7.2 Law of Sines
4. Solve Applied Problems
a  180º  arcsin(3sin(15 )), 0  a  90
 180  a 15
sin   
OP  3
sin 15 
16 May 2017
23
Kidoguchi, Kenneth
§7.2 Law of Sines
3. Solve Applied Problems
16 May 2017
24
Kidoguchi, Kenneth
§7.2 Law of Sines
3. Solve Applied Problems
16 May 2017
25
Kidoguchi, Kenneth