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Transcript of Polar Coordinates and Graphs of Polar Equations. Copyright by Houghton Mifflin Company, Inc. All...
Polar Coordinates and Graphs of Polar
Equations
Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 2
The polar coordinate system is formed by fixing a point, O, which is the pole (or origin).
= directed angle Polar axis
r = directed distance
OPole (Origin)
The polar axis is the ray constructed from O.
Each point P in the plane can be assigned polar coordinates (r, ).
P = (r, )
r is the directed distance from O to P. is the directed angle (counterclockwise) from the polar axis to OP.
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The point lies two units from the pole on the
terminal side of the angle
1 2 3 0
3 units from the pole
Plotting Points
The point lies three
units from the pole on the terminal
side of the angle
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There are many ways to represent the point
1 2 3 0
2, 3
additional ways
to represent the
point
Find the other representations for the point
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)4
3,3(
1 2 3 0
)4
3,3(
)4
5,3(
)4
7,3(
)4
,3(
Stop
• Warm Up. Graph and find the other 3 representations.
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)3
2,2(
1 2 3 0
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(r, )(x, y)
Pole x
y
(Origin)
yr
x
The relationship between rectangular and polar coordinates is as follows.
The point (x, y) lies on a circle of radius r, therefore,
r2 = x2 + y2.
Definitions of trigonometric functions
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Coordinate Conversion
(Pythagorean Identity)
Example:Convert the point into rectangular coordinates (x, y).
1cos co 3 24 s 4 2x r
3sin sin 4 23 24 3y r
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Example:Convert the point (1,1) into polar coordinates.
, 1,1x y
1tan 11yx
4
2 2 2 21 1 2r x y
set of polar coordinates is ( , ) 2, .4One r
5Another set is ( , ) 2, .4r Stop
Warm Up
•Convert the following point from polar to rectangular
•Convert the following point from rectangular to polar: (-4, 1)
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)3
2,1(
Convert rectangular to polar equations and polar to rectangular equations.
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Graph polar equations
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Example:Convert the polar equation into a rectangular equation.
4sinr
4sinr 2 4 sinr r Multiply each side by r.2 2 4x y y Substitute rectangular
coordinates.
22 2 4x y Equation of a circle with center (0, 2) and radius of 2
Polar form
2 2 4 0x y y
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Example:Convert the polar equation into a rectangular equation. 3
5
yx
tan
33
5tan
xy
3 yx 3
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Example:Convert the rectangular equation x2 + y2 – 6x = 0 into a polar equation.
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Example:Graph the polar equation r = 2cos .
1 2 3 0
2
0
–2
–101
20r
6
3
2
23
56
76
32
116
2
3
3
3
3 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
2. The polar axis 3. The pole
Replace (r, ) by (r, – ) or (–r, – ).
Replace (r, ) by (r, – ) or (–r, – ).
Replace (r, ) by (r, + ) or (–r, ).
Example:In the graph r = 2cos , replace (r, ) by (r, – ).
r = 2cos(–) = 2cos
The graph is symmetric with respect to the polar axis. cos(–) = cos
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Example:Find the zeros and the maximum value of r for the graph of r = 2cos .
1 2 3 0
The maximum value of r is 2.
It occurs when = 0 and 2. 0 when 3 and .2 2
r
These are the zeros of r.
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Each polar graph below is called a Limaçon.
1 2cosr 1 2sinr
–3
–5 5
3
–5 5
3
–3Note the symmetry of each graph.
What does the symmetry have in common with the trig function?
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Each polar graph below is called a Rose curve.
The graph will have n petals if n is odd, and 2n petals if n is even. And, again, note the symmetry.
–5 5
3
–3
–5 5
3
–3
a
a