10.3 polar coordinates. one way to give someone directions is to tell them to go three blocks east...
TRANSCRIPT
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10.3Polar Coordinates
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One way to give someone directions is to tell them to go three blocks East and five blocks South.
Another way to give directions is to point and say “Go a half mile in that direction.”
Polar graphing is like the second method of giving directions. Each point is determined by a distance and an angle.
Initial ray
r A polar coordinate pair
determines the location of a point.
,r
r – the directed distance from the origin to a point
Ө – the directed angle from the initial ray (x-axis) to ray OP.
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1 2 02
r
r a
o
(Circle centered at the origin)
(Line through the origin)
Some curves are easier to describe with polar coordinates:
(Ex.: r = 2 is a circle of radius 2 centered around the origin)
(Ex. Ө = π/3 is a line 60 degrees above the x-axis extending in both directions)
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30o
2
More than one coordinate pair can refer to the same point.
2,30o
2,210o
2, 150o
210o
150o
All of the polar coordinates of this point are:
2,30 360
2, 150 360 0, 1, 2 ...
o o
o o
n
n n
Each point can be coordinatized by an infinite number of polar ordered pairs.
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Tests for Symmetry:
x-axis: If (r, q) is on the graph,
r
2cosr
r
so is (r, -q).
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Tests for Symmetry:
y-axis: If (r, q) is on the graph,
r
2sinr
r
so is (r, p-q)
or (-r, -q).
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Tests for Symmetry:
origin: If (r, q) is on the graph,
r
r
so is (-r, q) or (r, q+p) .
tan
cosr
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Tests for Symmetry:
If a graph has two symmetries, then it has all three:
2cos 2r
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Try graphing this.(Pol mode)
2sin 2.15
0 16
r
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SPECIAL GRAPHS
Circles:r = a cosθr = a sinθ
Lemniscates:r2 = a2sin(2θ)r2 = a2cos(2θ)
Limaçons:
r = a ± b(cosθ)
r = a ± b(sinθ)
a > 0, b > 0Types of Limaçons:
If , limaçon has an inner loop1b
a
If , limaçon called a cardiod (heart shaped)1b
a
If , limaçon with a dimple.21 b
a
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SPECIAL GRAPHS
Types of Limaçons:
If , limaçon has an inner loop1b
a
If , limaçon called a cardiod (heart shaped)1b
a
If , limaçon with a dimple.21 b
a
If , convex limaçon.2b
a
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SPECIAL GRAPHS
Rose curves:
r = a cos(nθ)
r = a sin(nθ)
If n is odd, the rose will have n petals.
If n is even, the rose will have 2n petals.
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CONVERTING TO RECTANGULAR COORDINATES:
1.) x = r cosΘ y = r sinΘ
2.)x
ytan 222 yxr
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Example:
Convert the point represented by the polar coordinates (2, π) to rectangular coordinates.
x = r cos(θ)
x = 2cos(π)x = –2
y = r sin(θ)
y = 2 sin(π)y = 0
So, (–2, 0)
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Example:
Convert the point represented by the rectangular coordinates (–1, 1) to polar coordinates.
x
ytan
1tan
4
3
222 yxr
22 )1()1( r
2r
4
3,2
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Converting Polar Equations• You can convert polar equations to parametric
equations using the rectangular conversions.
Example:
3cos2r
cosrx cos)3cos2(x
sinry sin)3cos2(y
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Homework
• Section 10.4– #1, 3, 11, 13, 23, 25, 27, 29, 31, 34, 35, 37, 41