Transcript Notes Polar Coordintates

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,q ) = 2, p lies two units from the pole on the
3
p
terminal side of the angle q = p .
2
3
q=p
3
The point (r,q ) = 3,- 3p lies three
4
p
2,
units from the pole on the terminal
3
p
side of the angle q = p .
4
0
1 2 3
( )
(
)
(
( )
)
3,- 3p
4
q = - 3p
4
3 units from
the pole
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3p
2
3
( 3)
(2,- 53p )
There are many ways to represent the point 2, p .
p
( )(
)(
)(
2, p = 2,- 5p = -2,- 2p = -2, 4p
3
3
3
3
additional ways
)
2
 2, 3 
p
1
to represent the
( )
2
3
0
point 2, p
3
(
)
(r,q ) = (-r,q ± (2n +1)p )
(r,q ) = r,q ± 2np
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3p
2
4
Find the other representations for the
point (3,
p
2
3
)
4
5
(3, )
4
7
(3, )
4
p
1
2
3
0
3
( 3, )
4
3p
2
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
(3, )
4
Stop
5
2
Graph (2, 3 )
• Warm Up.
representations.
and find the other 3
p
2
p
1
2
3
0
3p
2
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6
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 q =
r
cosq = x
r
tan q = y
x
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Coordinate Conversion
cosq = x
x
=
r
cos
q
r
sin q = y
r
tan q = y
x
Example:
y = rsin q
r2 = x2 + y2
(Pythagorean Identity)
( )
Convert the point 4,- p into rectangular coordinates (x, y).
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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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
Stop
9
Warm Up
• Convert the following point from polar to
rectangular (1, 2 )
3
• Convert the following point from
rectangular to polar: (-4, 1)
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10
Convert rectangular to
polar equations and polar
to rectangular equations.
Graph polar equations
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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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12
Example:
5
Convert the polar equation   into a rectangular
3
equation.
x
tan  
y
5
tan
 3
3
 3
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y
x
 3x  y
13
Example:
Convert the rectangular equation x2 + y2 – 6x = 0
into a polar equation.
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14
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
p
2
p
1
2
3
0
 3
–2
 3
0
3
2
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3p
2
The graph is a
circle of radius 1
whose center is at
point (x, y) = (1, 0).
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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.
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Example:
Find the zeros and the maximum value of r for the
graph of r = 2cos .
p
2
The maximum value of r is 2.
It occurs when  = 0 and 2.
r  0 when
   and 3 .
2
2
p
1
2
3
0
These are the zeros of r.
3p
2
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17
Each polar graph below is called a Limaçon.
r = a ± b cosq
r = a ± b sinq
r  1  2cos
(a > 0,b > 0)
3
3
–5
5
–3
r  1  2sin 
–5
5
–3
Note the symmetry of each graph.
What does the symmetry have in common with the trig function?
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18
Each polar graph below is called a Rose curve.
r = 2cos3q
r = 3sin 4q
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. And, again, note the symmetry.
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