Download Section 9.1 The Law of Sines

Section 9.1 The Law of Sines
Note: A calculator is helpful on some exercises. Bring one to class for this lecture.
OBJECTIVE 1: Determining If the Law of Sines Can be Used to Solve an Oblique Triangle
Most triangles that we have worked with thus far in this text have been right triangles. We now turn our
attention to triangles that do not include a right angle. These triangles are called oblique triangles.
There are two types of oblique triangles. The first type is an oblique triangle with three acute angles.
The second type is an oblique triangle with one obtuse angle and two acute angles. The figure below
illustrates two such oblique triangles. Note that the angles in each triangle in are labeled with capital
letters and the sides opposite the angles are labeled with the respective lower case letter.
An
oblique
triangle
with
three
An
oblique
triangle
with
one
obtuse
acute
angles
angle
and
two
acute
angles
The goal of this section is to determine all angles and all sides of oblique triangles given certain
information. This process is called solving oblique triangles. In this section we will solve (or attempt to
solve) oblique triangles using the Law of Sines.
The
Law
of
Sines
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
a
b
c
sin A sin B sinC
or
.
=
=
=
=
sin A sin B sinC
a
b
c
In Class, Prove the Law of Sines for angles A and B and corresponding sides a and b.
€
€
Before we use the Law of Sines to solve oblique triangles, it is necessary to consider when the Law of
Sines can be used. To solve any oblique triangle, at least three pieces of information must be known. If
S represents a known side of a triangle and if A represents a known angle of a triangle, then there are six
possible situations where only three pieces of information can be known. The six possible cases are
shown in the table below.
Due to the fact that the Law of Sines uses proportions that involve both angles and sides, the following
pieces of information are needed in order to solve an oblique triangle using the Law of Sines:
1.
2.
3.
The measure of an angle must be known
The length of the side opposite the known angle must be known
At least one more side or one more angle must be known
The first three cases listed in Table 1 involve situations where this information is known. Therefore, the
Law of Sines can be used to solve the SAA, ASA, and SSA cases.
Organizing the information on a chart (like below) is helpful in determining if you can use Law of Sines.
Angles
Sides
a = _____ b = _____ c = _____ €
€ whether or not the Law of Sines can be used to solve each triangle. Do not
EXAMPLES. Decide
attempt to solve the €
triangle.
9.1.3
9.1.4
OBJECTIVE 2:
Using the Law of Sines to Solve the SAA Case or ASA Case
When the measure of any two angles of an oblique triangle are known and the length of any side is
known, always start by determining the measure of the unknown angle. Then use appropriate Law of
Sines proportions to solve for the lengths of the remaining unknown sides. Whenever possible, we will
avoid using rounded information to solve for the remaining parts of the triangle. When this cannot be
avoided, we will agree to use information rounded to one decimal place unless some other guideline is
stated. Use of the chart (above) to organize your information can avoid mistakes.
EXAMPLES. Solve each oblique triangle. Round the measures of all angles and the lengths of sides to
one decimal place.
9.1.8
9.1.10
OBJECTIVE 3:
Using the Law of Sines to Solve the SSA Case (the Ambiguous Case)
In the SAA and ASA cases (Objective 2), a unique triangle is always formed. In the SSA case (given
two sides and the angle opposite one of the sides) 3 possibilities exist:
1. No triangle fits the given information,
2. The triangle is a right triangle
3. There are one or two possible oblique triangles.
The table below summarizes the possibilities.
Again, organizing your information with the use of a chart is recommended.
EXAMPLES. Two sides and an angle are given. First determine whether the information results in no
triangle, one triangle, or two triangles. Solve each resulting triangle. Round all measures to one decimal
place.
9.1.13
9.1.14
9.1.15
OBJECTIVE 4:
Triangles
Using the Law of Sines to Solve Applied Problems Involving Oblique
The Law of Sines can be a useful tool to help solve many applications that arise involving triangles
which are not right triangles. Many areas such as surveying, engineering, and navigation require the use
of the Law of Sines.
Illustrates the navigation concept of bearing. In this text a bearing will be described as the
direction that one object is from another object in relation to north, south, east, and west. Two directions
and a degree measurement will be given to describe a bearing. For example, a bearing of N45°E (read as
“45 degrees east of north”) can be sketched by drawing the initial side of an angle along the positive yaxis which represents due north. The terminal side of the angle is then rotated away from the initial side
in an “easterly direction”
toward the positive x-axis. See the figures below.
9.1.20 To determine the width of a river, workers place markers on opposite sides of the river at points A
and B. A third marker is placed at point C, ________meters away from point A. If the angle between
point C and B is ________ and if the angle between point A and point C is __________, then determine
the width of the river to the nearest tenth of a meter.
9.1.23 A ship set sail from port at a bearing of ___________ and sailed ______ miles to point B. The
ship then turned and sailed an additional ________ miles to point C. Determine the distance from the
port to the ship if the bearing from the port to point C is ___________. Round to the nearest tenth of a
mile.