This document provides an overview of using the Law of Cosines to solve oblique triangles in three cases: when given three sides (SSS), two sides and the angle between them (SAS), and two angles and a non-included side (AAS). It includes the standard and alternative forms of the Law of Cosines, an example of finding the three angles of an SSS triangle, an example of solving an SAS triangle, and an application problem about ships traveling at different bearings and speeds to find how far apart they will be at a given time. Homework assignments are provided at the end to practice these concepts.

1. 6.2: Law of
Cosines
Mr. Fjelstrom
1/16/09

2. Quick Review of the 4 cases for solving oblique triangles:
1. Two angles and any side (AAS or ASA)
A
A
c
c
B
C
2. Two sides and an angle opposite one of them (SSA)
c
C a
3. Three sides (SSS)
c
b
c
a
4. Two sides and their included angle (SAS) B
a
2

3. The SSS and SAS cases can be solved using the …
Law of Cosines
Standard Form Alternative Form
In general:
opp2 = adj 2 + adj 2 − 2(adj)(adj) cos(Angle)
3

4. Example 1:
Find the three angles of the triangle.
SSS C
117.3°
8
6
26.4°
36.3°
A B
12
Find the angle
opposite the longest
side first.
Law of Sines:
4

5. Example 2:
C
Solve the triangle.
SAS 67.8° 6.2
9.9
Law of Cosines: 75°
37.2°
A B
9.5
Law of Sines:
5

6. Heron’s Area Formula
Given any triangle with sides of lengths a, b, and c, the
area of the triangle is given by:
Example 3:
10
8
Find the area of the triangle.
5
6

7. Application:
Two ships leave a port at 9 A.M. One travels at a bearing of N
53° W at 12 mph, and the other travels at a bearing of S 67° W at
16 mph. How far apart will the ships be at noon?
N
N
At noon, the ships have traveled for 3 hours. 53°
36 mi
53°
Angle C = 180° – 53° – 67° = 60° 43 mi c 60° P
67°
48 mi
67°
The ships will be approximately 43 miles apart.
7

8. Homework:
#3 Law of Cosines WS
#4 p. 443 #1, 3, 7, 8, 23, 24,
32, 34-36, 41