Section 9.2 The Law of Cosines
Note: A calculator is helpful on some exercises. Bring one to class for this lecture.
OBJECTIVE 1: Determining If the Law of Sines or the Law of Cosines Should be Used to Begin to
Solve an Oblique Triangle
Use the Law of Sines for the following cases:
If
S
represents
a
given
side
of
a
triangle
and
if
A
represents
a
given
angle
of
a
triangle,
then
the
Law
of
Sines
can
be
used
to
solve
the
three
triangle
cases
shown
below.
The
SAA
Case
The
ASA
Case
The
SSA
Case
(The
Ambiguous
Case)
The other cases in which three pieces of information can be known are the ASA, SSS, and AAA cases.
We will ignore the AAA case because the AAA case never defines a unique triangle. The Law of Sines
cannot be used to begin to solve the ASA or SSS cases because in either case, no angle opposite a known
side is given. However, we can use the Law of Cosines to obtain this needed information.
The
Law
of
Cosines
If
A,
B,
and
C
are
the
measures
of
the
angles
of
any
triangle
and
if
a,
b,
and
c
are
the
lengths
of
the
sides
opposite
the
corresponding
angles,
then
,
,
and
.
In Class derive the first equation in the Law of Cosines.
The
Alternate
Form
of
the
Law
of
Cosines
If
A,
B,
and
C
are
the
measures
of
the
angles
of
any
triangle
and
if
a,
b,
and
c
are
the
lengths
of
the
sides
opposite
the
corresponding
angles,
then
,
,
and
.
The three equations seen above are useful when determining the measure of a missing angle provided
that the length of each of the three sides is known. Note that the angle on the left-hand side of the
equation lies opposite the side of the triangle that is squared and subtracted on the right-hand side of the
equation
Again, in solving oblique triangles it is helpful to first organize the given information.
Angles
Sides
b = ____
C = ____
c = ____
€
€ whether the Law of Sines or the Law of Cosines should be used to begin to solve
€ EXAMPLES. Decide
the given triangle. Do not solve the triangle.
9.2.1
9.2.2
9.2.3
OBJECTIVE 2:
Using the Law of Cosines to Solve the SAS Case
Solving
a
SAS
Oblique
Triangle
Step
1
Use
the
Law
of
Cosines
to
determine
the
length
of
the
missing
side.
Step
2
Determine
the
measure
of
the
smaller
of
the
remaining
two
angles
using
the
Law
of
Sines
or
using
the
alternate
form
of
the
Law
of
Cosines.
(This
angle
will
always
be
acute.)
Step
3
Use
the
fact
that
the
sum
of
the
measures
of
the
three
angles
of
a
triangle
is
determine
the
measure
of
the
remaining
angle.
to
EXAMPLES for Objectives 2 and 3. Solve each oblique triangle. Round the measures of all angles and
the lengths of all sides to one decimal place.
9.2.8
9.2.11
OBJECTIVE 3:
Using the Law of Cosines to Solve the SSS Case
Solving
a
SSS
Oblique
Triangle
Step
1
Use
the
alternate
form
of
the
Law
of
Cosines
to
determine
the
measure
of
the
largest
angle.
This
is
the
angle
opposite
the
longest
side.
Step
2
Determine
the
measure
of
one
of
the
remaining
two
angles
using
the
Law
of
Sines
or
the
alternate
form
of
the
Law
of
Cosines.
(This
angle
will
always
be
acute.)
Step
3
Use
the
fact
that
the
sum
of
the
measures
of
the
three
angles
of
a
triangle
is
determine
the
measure
of
the
remaining
angle.
to
9.2.13
9.2.16
OBJECTIVE 4:
Triangles
Using the Law of Cosines to Solve Applied Problems Involving Oblique
As with the Law of Sines, the Law of Cosines can be a useful tool to help solve many applications that
arise involving triangles which are not right triangles. You may want to review the concept of bearing
that was introduced in Section 9.1.
9.2.17 Two planes take off from the same airport at the same time using different runways. One plane
flies at an average speed of _____________ mph with a bearing of _________. The other plane flies at
an average speed of __________ with a bearing of ___________. How far are the planes from each other
_______ hours after takeoff? Round to the nearest then of a mile.
9.2.21 Determine the length of the chord intercepted by a central angle of _______ in a circle with a
radius of _____________. Round to the nearest tenth of a centimeter.