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=
!
!
a = _____ !
B=
!
!
b = _____ !
C=
!
!
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 A = 48°, B = 53° , and b = 6
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 a = 8.1, b = 7.3, and A = 40°
9.1.14 a = 21.5, b = 29.3, and A = 70.1°
9.1.15 a = 3.7, b = 7.4, and A = 30°
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.
Figure 8 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” 45otoward the positive x-axis. See the figures below.
!
!
!
!
!
N 45! oE
!
$
N 45oW
S 45oE
S 45oW
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, 40 meters away from point A. If the angle between point C
and B is 42° and if the angle between point A and point C is 112° ,!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 N! 20° W and sailed 35 miles to point B. The ship then
turned and sailed an additional 40 miles to point C. Determine the distance from the port to the ship if
the bearing from the port to point C is N! 51° W. Round to the nearest tenth of a mile.
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$$
$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$(The$Ambiguous$Case)$
The other cases in which three pieces of information can be known are the SAS, SSS, and AAA cases.
We will ignore the AAA case because the AAA case never defines a unique triangle. The Law of Sines
cannot be used to begin to solve the ASA or SSS cases because in either case, no angle opposite a known
side is given. However, we can use the Law of Cosines to obtain this needed information.
#
The$Law$of$Cosines$
If#A,#B,#and#C"are#the#measures#of#the#angles#of#any#triangle#and#if#a,#b,#and#c#are#the##
lengths#of#the#sides#opposite#the#corresponding#angles,#then#
a 2 = b 2 + c 2 − 2bc cos A ,#
b 2 = a 2 + c 2 − 2ac cos B ,#and#
c 2 = a 2 + b 2 − 2ab cos C .#
#
In Class derive the first equation in the Law of Cosines.
#
#
The$Alternate$Form$of$the$Law$of$Cosines$
If#A,#B,#and#C"are#the#measures#of#the#angles#of#any#triangle#and#if#a,#b,#and#c#are#the##
lengths#of#the#sides#opposite#the#corresponding#angles,#then#
cos A =
b2 + c 2 − a 2
,#
2bc
a 2 + c 2 − b2
,#and#
cos B =
2ac
a 2 + b2 − c 2
.#
cos C =
2ab
The three equations seen above are useful when determining the measure of a missing angle provided
that the length of each of the three sides is known. Note that the angle on the left-hand side of the
equation lies opposite the side of the triangle that is squared and subtracted on the right-hand side of the
equation
Again, in solving oblique triangles it is helpful to first organize the given information.
########
#Angles$$
Sides$
A=
#
#
a=
B=
#
#
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# 180o to#
determine#the#measure#of#the#remaining#angle.#
$
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 b = 2, c = 9 and A = 130°
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# 180o #to#
determine#the#measure#of#the#remaining#angle.#
9.2.13
9.2.16 a = 18, b = 14 and c = 13
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 300 mph with a bearing of N# 29° E. The other plane flies at an average speed
of 350 mph with a bearing of S# 47° W. How far are the planes from each other 3 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 54° in a circle with a radius of
18 cm. Round to the nearest tenth of a centimeter.