Transcript Document

Digital Lesson
Polar Coordinates and
Graphs of Polar
Equations
The polar coordinate system is formed by fixing a point, O,
which is the pole (or origin).
The polar axis is the ray constructed from O.
Each point P in the plane can be assigned polar coordinates (r, ).
P = (r, )
O
 = directed angle
Pole (Origin)
Polar
axis
r is the directed distance from O to P.
 is the directed angle (counterclockwise) from the polar axis
to OP.
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2
Plotting Points
The point (r, )  2,  lies two units from the pole on the
3

terminal side of the angle    .
2
3
 
3
 
 2, 3 

1
2
3
0
3,  34 
   3
4
3 units from
the pole
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3
2
3
 
There are many ways to represent the point 2, 3 .
 2, 3    2,  53    2,  23    2, 43 
additional ways
to represent the

2
 2,  53 
 2, 3 

1
2
3
0
 
point 2, 
3
(r, )   r,  2n 
(r, )   r,  (2n  1) 
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3
2
4
The relationship between rectangular and polar
coordinates is as follows.
y
The point (x, y) lies on a
circle of radius r, therefore,
r2 = x2 + y2 .
(x, y)
(r, )
r
y
Pole

(Origin)
x
x
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Definitions of
trigonometric functions
y
sin  
r
cos   x
r
y
tan  
x
5
Coordinate Conversion
cos   x
x  r cos
r
y
y  r sin 
sin  
r
y
2
2
2
(Pythagorean Identity)
tan  
r

x

y
x
Example:


Convert the point 4,   into rectangular coordinates.
3
x  r cos   4 cos    4 1  2
3
2


y  r sin   4 sin    4   3   2 3
3
 2 
 x, y   2, 2 3


  
 
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6
Example:
Convert the point (1,1) into polar coordinates.
 x, y   1,1
y 1
tan     1
x 1
 
4
r  x 2  y 2  12  12  2
One set of polar coordinates is (r, ) 


Another set is (r, )   2, 5 .
4
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

2,  .
4
7
Example:
Convert the polar equation r  4sin  into a rectangular
equation.
r  4sin 
Polar form
r 2  4r sin 
Multiply each side by r.
x2  y 2  4y
Substitute rectangular
coordinates.
x2  y 2  4y  0
x   y  2  4
2
2
Equation of a circle with
center (0, 2) and radius of 2
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8
Example:
Graph the polar equation r = 2cos .

0
r
2
6
3
3

2
2
3
5
6

7
6
3
1
0
–1


2
11
6
2

2

1
2
3
0
 3
–2
 3
0
3
2
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3
2
The graph is a
circle of radius 2
whose center is at
point (x, y) = (0, 1).
9
If substitution leads to equivalent equations, the graph
of a polar equation is symmetric with respect to one
of the following.
1. The line   2
Replace (r,  ) by (r,  –  ) or (–r, – ).
2. The polar axis
Replace (r,  ) by (r, – ) or (–r,  – ).
3. The pole
Replace (r,  ) by (r,  +  ) or (–r,  ).
The graph is
Example:
In the graph r = 2cos , replace (r,  ) by (r, – ). symmetric with
respect to the
r = 2cos(–) = 2cos 
cos(–) = cos 
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polar axis.
10
Example:
Find the zeros and the maximum value of r for the
graph of r = 2cos .

2
The maximum value of r is 2.
It occurs when  = 0 and 2.
r  0 when
   and 3 .
2
2

1
2
3
0
These are the zeros of r.
3
2
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11
Each polar graph below is called a Limaçon.
r  1  2cos
r  1  2sin 
3
3
–5
5
–3
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–5
5
–3
12
Each polar graph below is called a Lemniscate.
r  3 cos 2
r  2 sin 2
2
2
2
3
3
–5
5
–3
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2
–5
5
–3
13
Each polar graph below is called a Rose curve.
r  2cos3
r  3sin 4
3
3
a
–5
5
–5
5
a
–3
–3
The graph will have n petals if n is odd, and 2n
petals if n is even.
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14