Transcript Document

9.2 - 9.3
The Law of Sines and The Law of Cosines
In these sections, we will study the following
topics:

Solving oblique triangles (4 cases)

Solving a triangle using Law of Sines

Solving a triangle using Law of Cosines
1
2
In this chapter, we will work with oblique triangles
 triangles that do NOT contain a right angle.
An oblique triangle has either:

three acute angles
or

two acute angles and one obtuse angle
3
Every triangle has 3 sides and 3 angles.
To solve a triangle means to find the lengths of its
sides and the measures of its angles.
To do this, we need to know at least three of these
parts, and at least one of them must be a side.
4
Here are the four possible combinations of parts:
1.
Two angles and one side (ASA or SAA)
2.
Two sides and the angle opposite one of them
(SSA)
3.
Two sides and the included angle (SAS)
4.
Three sides (SSS)
5
Case 1:
Two angles and one side (ASA or SAA)
6
Case 2:
Two sides and the angle opposite one of
them (SSA)
7
Case 3:
Two sides and the included angle (SAS)
8
Case 4:
Three sides (SSS)
9
C
b
a
B
A
c
The Law of Sines
a
sin A

b
sin B

c
sin C
Three equations for the price of one!
10
Solving Case 1: ASA or SAA
11
Solving Case 1: ASA or SAA
12
Example using Law of Sines
A ship takes a sighting on two buoys. At a certain instant,
the bearing of buoy A is N 44.23° W, and that of buoy B is N
62.17° E. The distance between the buoys is 3.60 km, and
the bearing of B from A is N 87.87° E. Find the distance of
the ship from each buoy.
13
Continued from above
14
Solving Case 2: SSA
In this case, we are given two sides and an
angle opposite.
This is called the AMBIGUOUS
CASE.
That is because it may yield no solution, one
solution, or two solutions, depending on the
given information.
15
SSA --- The Ambiguous Case
16
If a  h  b sin A, then side a is not
sufficiently long enough to form a triangle.
No Triangle
17
If a  h  b sin A, then
side a is just long enough to form a right triangle.
One Right Triangle
18
If h  b sin A  a and a  b, two
distinct triangles can be formed from the given
information.
Two Triangles
19
One Triangle
If a  b, only one triangle can be
formed.
20
21
Continued from above
22
23
Continued from above
24
25
Making fairly accurate sketches can help you to determine the
number of solutions.
26
Example: Solve ABC where A = 27.6, a =112, and c = 165.
27
Continued from above
28
To deal with Case 3 (SAS) and Case 4 (SSS), we
do not have enough information to use the Law
of Sines.
So, it is time to call in the Law of Cosines.
29
C
b
a
B
A
The Law of Cosines
c
a  b  c  2 b c co s A
2
2
2
b  a  c  2 a c co s B
2
2
2
c  a  b  2 a b co s C
2
2
2
30
Using Law of cosines to Find the Measure
of an Angle
*To find the angle using Law of Cosines, you will need to solve
the Law of Cosines formula for CosA, CosB, or CosC.
For example, if you want to find the measure of angle C, you
would solve the following equation for CosC:
c  a  b  2 ab cos C
2
2
2
2 ab cos C  a  b  c
2
a b c
2
cos C 
2
2
2 ab
2
2
To solve for C, you would
take the cos-1 of both
sides.
31
Guidelines for Solving Case 3: SAS
When given two sides and the included angle, follow
these steps:
1.
Use the Law of Cosines to find the third side.
2.
Use the Law of Cosines to find one of the
remaining angles.
{You could use the Law of Sines here, but you must be careful due to
the ambiguous situation. To keep out of trouble, find the SMALLER of
the two remaining angles (It is the one opposite the shorter side.)}
3.
Find the third angle by subtracting the two
known angles from 180.
32
Solving Case 3: SAS
Example: Solve ABC where a = 184, b = 125, and C = 27.2.
33
Continued from above
34
Solving Case 3: SAS
Example: Solve ABC where b = 16.4, c = 10.6, and A = 128.5.
35
Continued from above
36
Guidelines for Solving Case 4: SSS
When given three sides, follow these steps:
1.
Use the Law of Cosines to find the LARGEST
ANGLE *(opposite the largest side).
2.
Use the Law of Sines to find either of the two
remaining angles.
3.
Find the third angle by subtracting the two
known angles from 180.
37
38
Solving Case 4: SSS
Example: Solve ABC where a = 128, b = 146, and c = 222.
39
Continued from above
40
When to use what……
(Let bold red represent the given info)
SAS
AAS
ASA
Be careful!!
May have 0, 1, or
2 solutions.
SSS
SSA
Use Law of Sines
Use Law of Cosines
41
42
43
Continued from above
44
End of Sections 9.2 – 9.3
45