Transcript Notes 7.7 - TeacherWeb

Chapter 7
Applications of
Trigonometric
Functions
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All rights reserved
© 2010
2011 Pearson Education, Inc. All rights reserved
1
SECTION 7.7
Polar Coordinates
OBJECTIVES
1
2
3
4
Plot points using polar coordinates.
Convert points between polar and rectangular
forms.
Convert equations between rectangular and
polar forms.
Graph polar equations.
POLAR COORDINATES
In a polar coordinate system, we draw a
horizontal ray, called the polar axis, in the
plane; its endpoint is called the pole.
A point P in the plane is described by an
ordered pair of numbers (r,), the polar
coordinates of P.
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POLAR COORDINATES
The figure below shows the point P(r,) in
the polar coordinate system, where r is the
“directed distance” from the pole O to the
point P and θ is a directed angle from the
polar axis to the line segment OP.
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POLAR COORDINATES
The polar coordinates of a point are not unique.
The polar coordinates (3, 60º), (3, 420º), and
(3, −300º) all represent the same point.
In general, if a point P has polar coordinates (r, ),
then for any integer n,
(r,  + n · 360º) or (r,  + 2nπ)
are also polar coordinates of P.
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EXAMPLE 1
Finding Different Polar Coordinates
a. Plot the point P with polar coordinates
(3, 225º).
Find another pair of polar coordinates of P for
which the following is true.
b. r < 0 and 0º <  < 360º
c. r < 0 and –360º <  < 0º
d. r > 0 and –360º <  < 0º
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EXAMPLE 1
Finding Different Polar Coordinates
Solution
a.
b. r < 0 and 0º <  < 360º
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EXAMPLE 1
Finding Different Polar Coordinates
Solution continued
c. r < 0 and –360º <  < 0º
d. r > 0 and –360º <  < 0º
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POLAR AND RECTANGULAR COORDINATES
Let the positive x-axis of the rectangular
coordinate system serve as the polar axis and
the origin as the pole for the polar coordinate
system.
Each point P has both
polar coordinates (r, )
and rectangular
coordinates (x, y).
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RELATIONSHIPS BETWEEN
POLAR AND RECTANGULAR COORDINATES
x y r
2
2
2
y
sin  
r
x
cos 
r
y
tan  
x
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CONVERTING FROM
POLAR TO RECTANGULAR COORDINATES
To convert the polar coordinates (r, ) of a
point to rectangular coordinates (x, y), use the
equations
x  r cos
and
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y  r sin  .
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EXAMPLE 2
Converting from Polar to Rectangular
Coordinates
Convert the polar coordinates of each point to
rectangular coordinates.


b.   4, 
a.  2, 30º 

3
Solution
a. x  r cos
y  r sin 
x  2 cos  30º 
y  2 sin  30º 
x  2cos  30º 
y  2 sin 30º
 3
 1
y  2    1
x  2
  3
 2
2


The rectangular coordinates of (2, –30º) are 3, –1 .

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
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EXAMPLE 2
Converting from Polar to Rectangular
Coordinates


Solution continued b.   4, 

3
b. x  r cos
 
x   4cos  
3
 1
x  4    2
 2
y  r sin 
 
y   4sin  
3
 3
y  4 
  2 3
 2 


The rectangular coordinates of   4, 

3
are 2, 2 3 .


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CONVERTING FROM
RECTANGULAR TO POLAR COORDINATES
To convert the rectangular coordinates (x, y) of a
point to polar coordinates, follows these steps:
1. Find the quadrant in which the given point
(x, y) lies.
2. Use r  x  y to find r.
y
3. Find  by using tan   and choose  so
x
that it lies in the same quadrant as the point
(x, y).
2
2
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EXAMPLE 3
Convert from Rectangular to Polar
Coordinates
Find polar coordinates (r, ) of the point P
with r > 0 and 0≤ θ < 2π, whose rectangular
coordinates are (x, y) =  2,2 3 .

Solution



1. The point P 2, 2 3 lies in quadrant II with
x  2 and y  2 3.
2. r  x  y
2
r
 2 
2
2

 2 3

2
r  4  12  16  4
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EXAMPLE 3
Convert from Rectangular to Polar
Coordinates
Solution continued
y
tan  
3.
x
2 3
tan  
 3
2
 5
 2
or   2  
So     
3
3
3
3
2
Choose
because it lies in quadrant II.
3
 2 
The polar coordinates of 2,2 3 are  4,
.
 3 

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
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CONVERTING EQUATIONS BETWEEN
RECTANGULAR AND POLAR FORMS
An equation that has the rectangular
coordinates x and y as variables is called a
rectangular (or Cartesian) equation.
An equation where the polar coordinates r and
 are the variables is called a polar equation.
Some examples of polar equations are
r  sin  ,
r  1  cos  , and r   .
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CONVERTING EQUATIONS BETWEEN
RECTANGULAR AND POLAR FORMS
To convert a rectangular equation to a polar
equation,
replace x with rcos  and y with rsin ,
and then simplify where possible.
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Converting an Equation from Rectangular
to Polar Form
EXAMPLE 5
Convert the equation x2 + y2 – 3x + 4 = 0 to
polar form.
x 2  y 2  3x  4  0
Solution
 r cos    r sin  
2
 3 r cos   4  0
2
r cos   r sin   3r cos   4  0
2
2
2
2
r  cos   sin    3r cos  4  0
2
2
2
r 2  3r cos   4  0
2
r
The equation  3r cos   4  0 is the polar
form of the given rectangular equation.
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CONVERTING AN EQUATION FROM
POLAR TO RECTANGULAR FORM
Converting an equation from polar to
rectangular form frequently requires some
ingenuity in order to use the substitutions
r 2  x2  y 2 ,
r sin   y,
r cos   x,
and
y
tan   .
x
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EXAMPLE 6
Converting an Equation from Polar to
Rectangular Form
Convert each polar equation to a rectangular
equation and identify its graph.
b.   45º
a. r  3
c. r  csc 
d. r  2 cos
Solution
a.
r3
r 2  32
x 2  y2  9
Circle:
center (0, 0)
radius = 3 units
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EXAMPLE 6
Converting an Equation from Polar to
Rectangular Form
Solution continued
b.   45º
tan   tan 45º
y
1
x
yx
Line through the origin
with a slope of 1
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EXAMPLE 6
Converting an Equation from Polar to
Rectangular Form
Solution continued
c. r  csc 
1
r
sin 
r sin   1
y 1
Horizontal line with
y-intercept = 1
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EXAMPLE 6
Converting an Equation from Polar to
Rectangular Form
Solution continued
d. r  2 cos
2
r  2r cos 
x 2  y 2  2x
2
2
x  2x  y  0
2
2
 x  2 x  12  y  1
2
x

1

y
1
 
Circle:
Center (1, 0)
radius = 1
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THE GRAPH OF A POLAR EQUATION
To graph polar equations, we plot points in polar
coordinates. The graph of a polar equation
r  f  
is the set of all points P(r, ) that have at least one
polar coordinate representation that satisfies the
equation.
Make a table of several ordered pair solutions (r, )
of the equation, plot the points, and join them with
a smooth curve.
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TESTS FOR SYMMETRY IN
POLAR COORDINATES
Symmetry with respect to the polar axis
Replace (r, )
with (r, –) or
(–r, π – ).
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TESTS FOR SYMMETRY IN
POLAR COORDINATES
Symmetry with respect to the line  

2
Replace (r, )
with (r, π – ) or
(–r, –).
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TESTS FOR SYMMETRY IN
POLAR COORDINATES
Symmetry with respect to the pole
Replace (r, )
with (r, π + ) or
(–r, ).
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EXAMPLE 8
Sketching the Graph of a Polar Equation
Sketch the graph of the polar equation
r  2 1  cos .
Solution
Because cos  is even, the graph is symmetric
about the polar axis. Thus, compute values for
0 ≤  ≤ π.
θ
0

6
2+ 3
r = 2(1 + cos θ) 4
≈ 3.73
  2
3 2 3
3
2
1
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5
6
π
2 – 3 ≈ 0.27 0
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EXAMPLE 8
Sketching the Graph of a Polar Equation
Solution continued
r  2 1  cos 
This type of
curve is called a
cardioid
because it
resembles a
heart.
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LIMAÇONS
The graph of an equation of the form
r  a  b cos  , r  a  b sin  , a  0, b  0
is a limaçon. If b > a, the limaçon has a loop. If
b = a, the limaçon is a cardioid.
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LIMAÇONS
r  a  b cos  , r  a  b sin  , a  0, b  0
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ROSE CURVES
The graph of an equation of the form
r  a cos n , r  a sin n , a  0
is a rose curve. If n is odd, the rose has n petals.
If n is even, the rose has 2n petals.
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ROSE CURVES
r  a cos n , r  a sin n , a  0
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CIRCLES
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LEMNISCATES
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SPIRALS
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