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Section 9-4
The Law of Cosines
Recall…

In section 9-2 we mentioned that by the SAS
condition for congruence, a triangle is
uniquely determined if the lengths of two
sides and the measure of the included angle
are known. By the side-side-side (SSS)
condition for congruence, a triangle is also
uniquely determined if the lengths of three
sides are known. The Law of Cosines can be
used to solve a triangle in either of these two
cases.
The Law of Cosines


In ∆ABC,
To help you remember the law of
cosines, notice that is has the basic
pattern:
c 2  a 2  b 2  2ab cos C
2
2
 side

 side
  other side



 
 opposite    adjacent    adjacent
 angle

 to angle   to angle



 
2

 one
 other 




  2 adjacent  adjacent  cosangle 

 side
 side





Special cases of the Law of
Cosines
** Notice that when  C = 90° the law
2
2
2
a

b

c
of cosines reduces to
.
Therefore the law of cosines includes
the Pythagorean Theorem as a special
case and is more flexible and useful
than the Pythagorean Theorem.
2
2
2

c

a

b
** When C is acute,
by the
amount 2ab cos C
** When  C is obtuse, cos C is negative
2
2
2
c

a

b
and so
.
If we solve the law of cosines
for cos C, we obtain:
a2  b2  c2
cos C 
2ab
b2  c2  a2
cos A 
2bc
a2  c2  b2
cos B 
2ac
In this form, the law of cosines can be
used to find the measures of the angles
of a triangle when the lengths of the
three sides are known. The basic
pattern for this form of the law of
cosines is:
2
2
2

adjacent   adjacent   opposite 
cosangle  
2  adjacent   adjacent 

The Law of Cosines

Using the Law of Cosines, we can easily
identify acute and obtuse angles. In
example 2, since cos α is negative and
cos β is positive, we know that α is an
obtuse angle and β is an acute angle.
The law of sines does not distinguish
between acute and obtuse angles
because both types of angles have
positive sine values.
When to use what:
Caution…

When you use the law of sines,
remember that every acute angle and
its supplement have the same sine
value.
Example:
A triangle has sides of lengths 6, 12,
and 15.



Find the measure of the smallest angle.
Find the length of the median to the
longest side.
Example:
A parallelogram has diagonals of lengths
20 cm and 12 cm. If the diagonals
intersect to form a 60° angle, find the
perimeter of the parallelogram.