1. 10-1: The Law of Cosines
Learning Goals:
©2008 Roy L. Gover (www.mrgover.com)
•Derive The Law of Cosines
•Solve triangles using The
Law of Cosines

2. Try This
A
B
C
7
49.2°
b
c
Solve triangle
ABC. Round
angle and
side measures
to the nearest
tenth.
B=40.8°, b=6.0,c=9.2

3. Important Idea
The Law of Cosines
extends right triangle
trigonometry to the
measurement of sides and
angles of triangles that are
not right triangles.

4. Definition
Law of Cosines in 3 parts:
2 2 2
2 cos
a b c bc A
  
2 2 2
2 cos
b a c ac B
  
2 2 2
2 cos
c a b ab C
  
Do you
see a
pattern?
A
C
B a
b
c

5. Analysis A
B C
c b
a
x
a-x
h
2 2 2
b x h
 
2 2 2
( )
c a x h
  

6. A
B C
c b
a
x
a-x
h
2 2 2
( )
c a x h
  
2 2 2 2
2
c a ax x h
   
2 2 2
b x h
 
2 2 2
2
c a ax b
  

7. A
B C
c b
a
x
a-x
h
2 2 2
( )
c a x h
  
2 2 2 2
2
c a ax x h
   
2 2 2
2
c a ax b
  
cos
x
C
b

cos
x b C


8. A
B C
c b
a
x
a-x
h
2 2 2
( )
c a x h
  
2 2 2 2
2
c a ax x h
   
2 2 2
2
c a ax b
  
2 2 2
2 cos
c a ab C b
  
2 2 2
2 cos
c a b ab C
  
aka the
Law of
Cosines

9. Example
You are building a house.
The roof has the dimensions
in the diagram. Find the
width, c, of the lower beam.
100°
60’ 90’
c

10. Example
100°
60’ 90’
c
2 2 2
2 cos
c a b ab C
  
2 2 2
60 90 (60)(90)cos100
c    

11. Important Idea
•Don’t forget to set the
mode on your calculator.
•Throughout this
section, no rounding is
done in the
computation. Round only
the final answer.

12. Example
Solve triangle ABC (find the
missing sides & angles) if
A=39.4°,b=12 & c=14

13. Try This
Solve triangle ABC (find the
missing sides & angles) if
A=52°10’,b=6 & c=8.Round
angle and side measures to
the nearest tenth.
a=6.4,B=46.8°’,C=80.1’

14. Example
Solve triangle PQR if p=14,
q=15 & r=16.
Note: we are given 3
sides and asked to find 3
angles.

15. Try This
Solve triangle ABC if a=
21,b=16.7 & c =10.3. Round
angle measures to the
nearest tenth.
A=99.4°
B=51.7°
C=28.9° A
B
C
21
16.7
10.3

16. Example
•Natalie is
flying from
Dallas to
Little Rock, a distance of 319
miles. She starts her flight
15° off course and flies for
75 miles. How far is she
from Little Rock?

17. d
15°

18. Lesson Close
The Law of Cosines is used
to solve triangles when you
know either:
•Two sides and an included
angle (SAS)
•Three sides (SSS)
or