Law of Cosines
Lesson 4.2
Who's Law Is It, Anyway?

Murphy's Law:


Anything that can possibly go wrong, will
go wrong (at the worst possible moment).
Cole's Law ??

Finely chopped cabbage
2
Solving an SAS Triangle

The Law of Sines was good for




ASA
AAS
SSA
- two angles and the included side
- two angles and any side
- two sides and an opposite angle
(being aware of possible ambiguity)
Why would the Law of Sines not work for
an SAS triangle?
No side opposite
from any angle to
get the ratio
15
26°
12.5
3
Deriving the Law of Cosines
C
h  b  sin A
k  b  cos A

Write an equation
using Pythagorean
theorem for shaded
triangle.
b
h
k
c-k
A
a   b  sin A    c  b  cos A 
2
a
2
B
c
2
a 2  b 2 sin 2 A  c 2  2  c  b  cos A  b 2 cos 2 A
a 2  b 2  sin 2 A  cos 2 A   c 2  2  c  b  cos A
a 2  b 2  c 2  2  c  b  cos A
4
Law of Cosines

Similarly
a  b  c  2  c  b  cos A
2
2
2
b  a  c  2  a  c  cos B
2
2
2
c  b  a  2  a  b  cos C
2

2
2
Note the pattern
5
Applying the Cosine Law

Now use it to solve the triangle we
started with
C
15

Label sides
and angles

Side c first
A
26°
c
12.5
B
c  b  a  2  a  b  cos C
2
2
2
c  152  12.52  2 15 12.5  cos 26
6
Applying the Cosine Law
C
15
A

26°
c = 6.65
12.5
B
Now calculate the angles
2
2
2
b

a

c
 2  a  c  cos B
 use
and solve for B
b2  a 2  c 2
cos B 
2  a  c
2
2
2


b

a

c
1
B  cos 

 2  a  c 
7
Applying the Cosine Law
C
15
A

26°
c = 6.65
12.5
B
The remaining angle
determined by subtraction

180 – 93.75 – 26 = 60.25
Experiment with
Cosine Law
Spreadsheet
8
Wing Span
C


The leading edge of
each wing of the
B-2 Stealth Bomber
measures 105.6 feet
A
in length. The angle between the
wing's leading edges is 109.05°.
What is the wing span (the distance
from A to C)?
Hint … use the law of cosines!
9
Assignment A



Lesson 4.2A
Page 308
Exercises 1 – 27 odd,
and 41 - 51 odd
10
Using the Cosine Law to Find Area


Recall that
We can use
the value for h
to determine
the area
1
Area  c  b  sin A
2
h  b  sin A
C
b
h
a
A
B
c
11
Using the Cosine Law to Find Area

We can find the area knowing two
sides and the included angle
C
1
Area  a  b  sin C
2
1
 c  a  sin B
2
b
A

a
c
B
Note the pattern
12
Try It Out

Determine the area of these triangles
42.8°
17.9
12
127°
24
76.3°
13
Cost of a Lot

An industrial piece of real estate is
priced at $4.15 per square foot. Find,
to the nearest $1000, the cost of a
triangular lot measuring 324 feet by
516 feet by 412 feet.
516
14
Assignment B



Lesson 4.2B
Page 309
Exercises 29 – 39 odd
and 53 – 61 odd
15