Download Chapter 8 Section 4

8
Applications
of
Trigonometry
Copyright © 2009 Pearson Addison-Wesley
8.4-1
8 Applications of Trigonometry
8.1 The Law of Sines
8.2 The Law of Cosines
8.3 Vectors, Operations, and the Dot Product
8.4 Applications of Vectors
8.5 Trigonometric (Polar) Form of Complex
Numbers; Products and Quotients
8.4 De Moivre’s Theorem; Powers and Roots of
Complex Numbers
8.5 Polar Equations and Graphs
8.6 Parametric Equations, Graphs, and
Applications
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8.4-2
8.4 Applications of Vectors
The Equilibrant ▪ Incline Applications ▪ Navigation Applications
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The Equilibrant
Sometimes it is necessary to find a vector that will
counterbalance a resultant.
 This opposite vector is called the equilibrant.
 The equilibrant of vector u is the vector –u.
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Example 1
FINDING THE MAGNITUDE AND
DIRECTION OF AN EQUILIBRANT
Find the magnitude of the equilibrant of forces of 48
newtons and 60 newtons acting on a point A, if the
angle between the forces is 50°. Then find the angle
between the equilibrant and the 48-newton force.
The equilibrant is –v.
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Example 1
FINDING THE MAGNITUDE AND
DIRECTION OF AN EQUILIBRANT (cont.)
The magnitude of –v is the same as the magnitude of v.
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Example 1
FINDING THE MAGNITUDE AND
DIRECTION OF AN EQUILIBRANT (cont.)
The required angle, , can be found by subtracting
the measure of angle CAB from 180°. Use the law of
sines to find the measure of angle CAB.
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Example 1
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FINDING THE MAGNITUDE AND
DIRECTION OF AN EQUILIBRANT (cont.)
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Example 2
FINDING A REQUIRED FORCE
Find the force required to keep a 50-lb wagon from
sliding down a ramp inclined at 20° to the horizontal.
(Assume there is no friction.)
The vertical force BA
represents the force of gravity.
BA = BC + (–AC)
Vector BC represents the
force with which the weight
pushes against the ramp.
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Example 2
FINDING A REQUIRED FORCE (cont.)
Vector BF represents the force
that would pull the weight up
the ramp.
Since vectors BF and AC are
equal, |AC| gives the
magnitude of the required
force.
Vectors BF and AC are parallel, so the measure of
angle EBD equals the measure of angle A.
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Example 2
FINDING A REQUIRED FORCE (cont.)
Since angle BDE and angle C
are right angles, triangles CBA
and DEB have two
corresponding angles that are
equal and, thus, are similar
triangles.
Therefore, the measure of angle ABC equals the
measure of angle E, which is 20°.
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Example 2
FINDING A REQUIRED FORCE (cont.)
From right triangle ABC,
A force of approximately 17 lb will keep the wagon
from sliding down the ramp.
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Example 3
FINDING AN INCLINE ANGLE
A force of 16.0 lb is required
to hold a 40.0 lb lawn mower
on an incline. What angle
does the incline make with
the horizontal?
Vector BE represents the
force required to hold the
mower on the incline.
In right triangle ABC, the measure of angle B equals
θ, the magnitude of vector BA represents the weight
of the mower, and vector AC equals vector BE.
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Example 3
FINDING AN INCLINE ANGLE (cont.)
The hill makes an angle of about 23.6° with the
horizontal.
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Example 4
APPLYING VECTORS TO A NAVIGATION
PROBLEM
A ship leaves port on a bearing
of 28.0° and travels 8.20 mi. The
ship then turns due east and
travels 4.30 mi. How far is the
ship from port? What is its
bearing from port?
Vectors PA and AE represent the ship’s path. We are
seeking the magnitude and bearing of PE.
Triangle PNA is a right triangle, so the measure of
angle NAP = 90° − 28.0° = 62.0°.
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Example 4
APPLYING VECTORS TO A NAVIGATION
PROBLEM (continued)
Use the law of cosines to find
|PE|.
The ship is about 10.9 miles from port.
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Example 4
APPLYING VECTORS TO A NAVIGATION
PROBLEM (continued)
To find the bearing of the
ship from port, first find the
measure of angle APE.
Use the law of sines.
Now add 28.0° to 20.4° to find that the bearing is
48.4°.
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Airspeed and Groundspeed
The airspeed of a plane is its
speed relative to the air.
The groundspeed of a plane
is its speed relative to the
ground.
The groundspeed of a plane
is represented by the vector
sum of the airspeed and
windspeed vectors.
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Example 5
APPLYING VECTORS TO A NAVIGATION
PROBLEM
A plane with an airspeed of
192 mph is headed on a
bearing of 121°. A north wind
is blowing (from north to
south) at 15.9 mph. Find the
groundspeed and the actual
bearing of the plane.
The groundspeed is represented by |x|.
We must find angle  to determine the bearing, which
will be 121° + .
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Example 5
APPLYING VECTORS TO A NAVIGATION
PROBLEM (continued)
Find |x| using the law of
cosines.
The plane’s groundspeed is about 201 mph.
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Example 5
APPLYING VECTORS TO A NAVIGATION
PROBLEM (continued)
Find  using the law of
sines.
The plane’s groundspeed is about 201 mph on a
bearing of 121° + 4° = 125°.
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