7
Applications of Trigonometry and
Vectors
7.4 Vectors, Operations, and the Dot
Product
7.5 Applications of Vectors
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7-1
7.4
Vectors, Operations, and the
Dot Product
Basic Terminology ▪ Algebraic Interpretation of Vectors ▪
Operations with Vectors ▪ Dot Product and the Angle Between
Vectors
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7-2
7.4 Example 1 Finding Magnitude and Direction Angle
Find the magnitude and direction angle for
Magnitude:
Direction angle:
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7-3
7.4 Example 2 Finding Horizontal and Vertical Components
Vector v has magnitude 14.5 and direction angle
220°. Find the horizontal and vertical components.
Horizontal component: –11.1
Vertical component: –9.3
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7-4
7.4
Example 3 Writing Vectors in the Form a, b
Write each vector in the form a, b.
u: magnitude 8, direction angle 135°
v: magnitude 4, direction angle 270°
w: magnitude 10, direction angle 340°
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7-5
7.4
Example 4 Finding the Magnitude of a
Resultant
Two forces of 32 and 48 newtons act on a point in the
plane. If the angle between the forces is 76°, find the
magnitude of the resultant vector.
because the
adjacent angles of a
parallelogram are supplementary.
Law of cosines
Find square
root.
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7-6
7.4
Example 6 Finding Dot Products
Find each dot product.
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7-7
7.4 Example 7 Finding the Angle Between Two Vectors
Find the angle θ between the two vectors u = 5, –12
and v = 4, 3.
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7-8
7.5
Applications of Vectors
The Equilibrant ▪ Incline Applications ▪ Navigation Applications
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7-9
7.5
Example 2 Finding a Required Force
Find the force required to keep a 2500-lb car parked
on a hill that makes a 12° angle with the horizontal.
The vertical force BA represents the force of gravity.
BA = BC + (–AC)
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7-10
7.5
Example 2 Finding a Required Force (cont.)
Vector BC represents the
force with which the
weight pushes against
the hill.
Vector BF represents the force that
would pull the car up the hill.
Since vectors BF and AC are equal,
magnitude of the required force.
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gives the
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7.5
Example 2 Finding a Required Force (cont.)
Vectors BF and AC are
parallel, so the measure
of angle EBD equals the
measure of angle A.
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 12°.
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7-12
7.5
Example 2 Finding a Required Force (cont.)
From right triangle ABC,
A force of approximately 520 lb will keep the car
parked on the hill.
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7-13
7.5
Example 3 Finding an Incline Angle
A force of 18.0 lb is required to hold a 74.0-lb crate on
a ramp. What angle does the ramp make with the
horizontal?
Vector BF represents the
force required to hold the
crate on the incline.
In right triangle ABC, the measure of angle B equals
θ, the magnitude of vector BA represents the weight
of the crate, and vector AC equals vector BF.
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7-14
7.5
Example 3 Finding an Incline Angle (cont.)
The ramp makes an angle of about 14.1° with the
horizontal.
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7-15
7.5
Example 4 Applying Vectors to a Navigation
Problem
A ship leaves port on a bearing of 25.0° and travels
61.4 km. The ship then turns due east and travels
84.6 km. 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.
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7-16
7.5
Example 4 Applying Vectors to a Navigation
Problem
(cont.)
Triangle PNA is a right
triangle, so the measure of
angle NAP = 90° − 25.0°
= 65.0°.
The measure of angle PAE = 180° − 65.0 = 115.0°.
Law of cosines
The ship is about 123.8 km from port.
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7-17
7.5
Example 4 Applying Vectors to a Navigation
Problem
(cont.)
To find the bearing of the
ship from port, first find the
measure of angle APE.
Law of sines
Now add 38.3° to 25.0° to find that the bearing is
63.3°.
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7-18
7.5
Example 5 Applying Vectors to a Navigation
Problem
A plane with an airspeed of 355 mph is headed on a
bearing of 62°. A west wind is blowing (from west to
east) at 28.5 mph. Find the groundspeed and the
actual bearing of the plane.
The groundspeed is represented by |x|.
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7-19
7.5
Example 5 Applying Vectors to a Navigation
Problem
(cont.)
Law of cosines
The plane’s groundspeed is about 380 mph.
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7-20
7.5
Example 5 Applying Vectors to a Navigation
Problem
(cont.)
Use the law of sines to find
α, and then determine the
bearing, 62° + α.
The bearing is about 62° + 2° = 64°.
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7-21
8
Complex Numbers, Polar Equations, and
Parametric Equations
8.5 (part I) Polar Coordinates
8.1 Complex Numbers
8.2 Trigonometric (Polar) Form of Complex
Numbers
8.3 The Product and Quotient Theorems
8.4 De Moivre’s Theorem; Powers and Roots of
Complex Numbers
8.5 (part II) Polar Equations and Graphs
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8-22
8.5 Polar Coordinates (part I)
Polar Coordinate System ▪ Converting Polar and Rectangular
Coordinates
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8-23
8.5 Example 1 Plotting Points With Polar Coordinates
Plot each point by hand in the polar coordinate
system. Then, determine the rectangular coordinates
of each point.
The rectangular coordinates
of P(4, 135°) are
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8-24
8.5
Example 1 Plotting Points With Polar
Coordinates (cont.)
The rectangular coordinates of
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8-25
8.5
Example 1 Plotting Points With Polar
Coordinates (cont.)
The rectangular coordinates of
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are (0, –2).
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8.5
Example 2(a) Giving Alternative Forms for
Coordinates of a Point
Give three other pairs of polar coordinates for the
point P(5, –110°).
Three pairs of polar coordinates for the point
P(5, −110º) are (5, 250º), (−5, 70º), and (−5, −290º).
Other answers are possible.
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8-27
8.5
Example 2(b) Giving Alternative Forms for
Coordinates of a Point
Give two pairs of polar coordinates for the point with
the rectangular coordinates
The point
Since
300°.
lies in quadrant II.
, one possible value for θ is
Two pairs of polar coordinates are (12, 300°) and
(−12, 120°).
Other answers are possible.
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8-28
8.1
Complex Numbers
Basic Concepts of Complex Numbers ▪ Complex Solutions of
Equations ▪ Operations on Complex Numbers
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8-29
8.1
Example 1 Writing √–a as i√a
Write as the product of real number and i, using the
definition of
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8-30
8.1
Example 2 Solving Quadratic Equations for Complex
Solutions
Solve each equation.
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8-31
8.1
Example 3 Solving a Quadratic Equation for Complex
Solutions
Write the equation in standard form,
then solve using the quadratic formula with a = 2,
b = –2, and c = 5.
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8-32
8.1
Example 4 Finding Products and Quotients Involving
Negative Radicands
Multiply or divide as indicated. Simplify each answer.
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8-33
8.1
Example 5 Simplifying a Quotient Involving a Negative
Radicand
Write
in standard form.
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8-34
8.1 Example
6 Adding and Subtracting Complex Numbers
Find each sum or difference.
(a) (4 – 5i) + (–5 + 8i) = [4 + (–5)] + (–5i + 8i)
= –1 + 3i
(b) (–6 + 3i) + (12 – 9i) = 6 – 6i
(c) (–10 + 7i) – (5 – 3i) = (–10 – 5) + [7i + (3i)]
= –15 + 10i
(d) (15 – 8i) – (–10 + 4i) + (–25 + 12i)
= [15 – (–10) + (–25)] + [–8i – 4i + 12i]
= 0 + 0i
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8-35
8.1
Example 7 Multiplying Complex Numbers
Find each product.
(a) (5 + 3i)(2 – 7i) = 5(2) + (5)(–7i) + (3i)(2) + (3i)(–7i)
= 10 – 35i + 6i – 21i2
= 10 – 29i – 21(–1)
= 31 – 29i
(b) (4 – 5i)2 = 42 – 2(4)(5i) + (5i)2
= 16 – 40i + 25i2
= 16 – 40i + 25(–1)
= –9 – 40i
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8-36
8.1
Example 7 Multiplying Complex Numbers (cont.)
(c) (3 – i)(–3 + i)
= –9 + 3i + 3i – i2
= –9 + 6i – (–1)
= –9 + 6i + 1
= –8 + 6i
(d) (9 – 8i)(9 + 8i) = 92 – (8i)2
= 81 – 64i2
= 81 – 64(–1)
= 81 + 64
= 145 or 145 + 0i
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8-37
8.1
Example 8 Simplifying Powers of i
Simplify each power of i.
(a)
(b)
Write the given power as a product involving
or
.
(a)
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(b)
8-38
8.1
Example 8 Simplifying Powers of i (cont.)
(c)
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8-39
8.1
Example 9(a) Dividing Complex Numbers
Write the quotient in standard form a + bi.
Multiply the numerator and
denominator by the complex
conjugate of the denominator.
Multiply.
i2 = –1
Combine terms.
Lowest terms; standard form
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8-40
8.1
Example 9(b) Dividing Complex Numbers
Write the quotient in standard form a + bi.
Multiply the numerator and
denominator by the complex
conjugate of the denominator.
Multiply.
–i2 = 1
Lowest terms; standard form
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8-41
8.2
Trigonometric (Polar) Form of Complex
Numbers
The Complex Plane and Vector Representation ▪ Trigonometric
(Polar) Form ▪ Converting Between Rectangular and
Trigonometric (Polar) Forms ▪ An Application of Complex
Numbers to Fractals
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8-42
8.2
Example 1 Expressing the Sum of Complex Numbers
Graphically
Find the sum of 2 + 3i and –4 + 2i. Graph both
complex numbers and their resultant.
(2 + 3i) + (–4 + 2i) = –2 + 5i
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8-43
8.2
Example 2 Converting From Trigonometric Form to
Rectangular Form
Express 10(cos 135° + i sin 135°) in rectangular form.
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8-44
8.2
Example 3(a) Converting From Rectangular Form to
Trigonometric Form
Write 8 – 8i in trigonometric form.
The reference angle for θ is 45°.
The graph shows that θ is in
quadrant IV, so θ = 315°.
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8-45
8.2
Example 3(b) Converting From Rectangular Form to
Trigonometric Form
Write –15 in trigonometric form.
–15 = –15 + 0i
–15 + 0i is on the negative
x-axis, so θ = 180°.
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8-46
8.2
Example 4 Converting Between Trigonometric and
Rectangular Forms Using Calculator
Approximations
Write each complex number in its alternative form,
using calculator approximations as necessary.
(a) 7(cos 205° + i sin 205°) ≈ –6.3442 – 2.9583i
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8-47
8.2
Example 4 Converting Between Trigonometric and
Rectangular Forms Using Calculator
Approximations (cont.)
(b) –7 + 2i
x = −7 and y = 2
The reference angle for θ is approximately 15.95°.
The graph shows that θ is in quadrant II, so
θ = 180° – 15.95° = 164.05°.
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8-48
8.3
The Product and Quotient Theorems
Products of Complex Numbers in Trigonometric Form ▪
Quotients of Complex Numbers in Trigonometric Form
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8-49
8.3
Product Theorem
 r1 (cos 1  i sin 1 ) r2 (cos  2  i sin  2 )
Can be written as
r1r2 [cos(1   2 )  i sin(1   2 )]
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8-50
8.3
Example 1 Using the Product Theorem
Find the product of 4(cos 120° + i sin 120°) and
5(cos 30° + i sin 30°). Write the result in rectangular
form.
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8-51
8.3
Quotient Theorem
 r1 (cos 1  i sin 1 ) 


 r2 (cos  2  i sin  2 ) 
Can be written as
r1
[cos(1   2 )  i sin(1   2 )]
r2
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8-52
8.3
Example 2 Using the Quotient Theorem
Find the quotient
rectangular form.
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Write the result in
8-53
8.3
Example 3 Using the Product and Quotient Theorems
With a Calculator
Use a calculator to find the following. Write the results
in rectangular form.
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8-54
8.3
Example 3 Using the Product and Quotient Theorems
With a Calculator (cont.)
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8-55
8.4
De Moivre’s Theorem; Powers and
Roots of Complex Numbers
Powers of Complex Numbers (De Moivre’s Theorem) ▪ Roots of
Complex Numbers
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8-56
8.4
Example 1 Finding a Power of a Complex Number
Find
form.
First write
and express the result in rectangular
in trigonometric form.
and
Because x and y are both positive, θ is in quadrant I,
so θ = 45°.
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8-57
8.4
Example 1 Finding a Power of a Complex Number
(cont.)
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8-58
8.4
Example 2 Finding Complex Roots
Find the three cube roots of 8 (cos 180o + i sin 180o).
Write the roots in rectangular form.
Note that [2 (cos 60o + i sin 60o)]3 = 8 (cos 180o + i sin 180o).
So one cube root is 2 (cos 60o + i sin 60o)
The other 2 are
360
 120 apart
3
2 (cos 180o + i sin 180o) and 2 (cos 300o + i sin 300o)
If cubed all 3 will yield 8 in rectangular form
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8-59
8.4
Example 3 Finding Complex Roots
Find all fourth roots of
Write the roots in rectangular form.
First write
in trigonometric form.
Because x and y are both negative, θ is in quadrant
III, so θ = 240°.
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8-60
8.4
Example 3 Finding Complex Roots
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(cont.)
8-61
8.5 Polar Equations and Graphs (part II)
Graphs of Polar Equations ▪ Converting from Polar to
Rectangular Equations ▪ Classifying Polar Equations
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8-62
8.5
Example 3 Examining Polar and Rectangular Equations
of Lines and Circles
For each rectangular equation, give the equivalent
polar equation and sketch its graph.
(a) y = 2x – 4
In standard form, the equation is 2x – y = 4, so a = 2,
b = –1, and c = 4.
The general form for the polar equation of a line is
y = 2x – 4 is equivalent to
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8-63
8.5
Example 3 Examining Polar and Rectangular Equations
of Lines and Circles (cont.)
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8-64
8.5
Example 3 Examining Polar and Rectangular Equations
of Lines and Circles (cont.)
This is the graph of a circle with center at the origin
and radius 5.
Note that in polar coordinates it is possible for r < 0.
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8-65
8.5
Example 3 Examining Polar and Rectangular Equations
of Lines and Circles (cont.)
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8-66
8.5
Example 8 Converting a Polar Equation to a Rectangular
Equation
#53
r  2sin( )
#58
2
r
4 cos   sin 
#60
r  5csc( )
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8-67
8.5
Example 4 Graphing a Polar Equation (Cardioid)
Find some ordered pairs to determine a pattern of
values of r.
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8-68
8.5
Example 4 Graphing a Polar Equation (Cardioid) (cont.)
Connect the points in order from (1, 0°) to (.5, 30°) to
(.1, 60°) and so on.
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8-69
8.5
Example 5 Graphing a Polar Equation (Rose)
Find some ordered pairs to determine a pattern of
values of r.
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8-70
8.5
Example 5 Graphing a Polar Equation (Rose) (cont.)
Connect the points in order from (4, 0°) to (3.6, 10°)
to (2.0, 20°) and so on.
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8-71
8.5
Example 7 Graphing a Polar Equation (Spiral of
Archimedes)
Graph r = –θ (θ measured in radians).
Go to wzgrapher. Use domain [0, 20π]
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8-72