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).
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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.
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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