OBJECTIVES:
Solve triangles by using the Law of Sines
Find the area of a triangle...
Determine whether a triangle has 0, 1, or 2 solutions
Dude, that's not right!
C
b
a
A
c
B
LAW OF SINES
1
LAW OF SINES
The ratio of the length of any side of a triangle to the sine of the
angle opposite that side is a constant for a given triangle.
C
b
a
A
c
Solve
B
C
32 cm
b
a
12 cm
A
42o
c
B
2
Area of Triangles
Let ΔABC be any triangle with a, b, and c representing
the measures of the sides opposite the angles with
measurements A, B, and C. Then the area K can be determined
using one of the formulas below:
K=
1
2 bc
sinA
K=
1
ab
2
sinC
K=
1
2
C
4.7
b
a
ac sin B
How can this work???
o
A 47 20'
c 12.4
B
F
13.9
34.4o
E
D
14.8o
K=
1
2
K=
1
ab
2
K=
1
2
K=
1 2 sinA sinC
b
2
sin B
K=
1 2 sinB sinC
a
2
sin A
K=
1 2 sinA sinB
c
2
sin C
bc sinA
sinC
ac sin B
Modified
Formulas
3
BONUS QUESTION
IF
and
IF
CAN
YOU
PROVE
IT??
then
=
if more than one solution, you must solve ALL possible solutions
The Ambiguous Case for the Law of Sines
CASE 1: A < 90o
b
a<b
a = b sinA
a > b sinA
b sinA
a < b sinA
a
A
4
The Ambiguous Case for the Law of Sines
b
a>b
b sinA
CASE 1: A < 90o
a
A
The Ambiguous Case for the Law of Sines
CASE 2: A > 90o
a
a<b
a>b
b
A
5
Homework
pg. 316 #11­23 Every 3rd
pg. 324 #11­23 Every 3rd, 24, 27
6