10.3 Polar Coordinates
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Transcript of 10.3 Polar Coordinates
10.3Polar Coordinates
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.
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)
30o2
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.
Tests for Symmetry:
x-axis: If (r, ) is on the graph,
r
2cosr
r
so is (r, -).
Tests for Symmetry:
y-axis: If (r, ) is on the graph,
r
2sinr
r
so is (r, -)
or (-r, -).
Tests for Symmetry:
origin: If (r, ) is on the graph,
r
r
so is (-r, ) or (r, +) .
tancos
r
Tests for Symmetry:
If a graph has two symmetries, then it has all three:
2cos 2r
Try graphing this.(Pol mode)
2sin 2.150 16
r
SPECIAL GRAPHSCircles: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 loop1ba
If , limaçon called a cardiod (heart shaped)1ba
If , limaçon with a dimple.21 ba
SPECIAL GRAPHSTypes of Limaçons:If , limaçon has an inner loop1ba
If , limaçon called a cardiod (heart shaped)1ba
If , limaçon with a dimple.21 ba
If , convex limaçon.2ba
SPECIAL GRAPHSRose 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.
CONVERTING TO RECTANGULAR COORDINATES:
1.) x = r cosΘ y = r sinΘ
2.)xy
tan 222 yxr
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)
Example:Convert the point represented by the rectangular
coordinates (–1, 1) to polar coordinates.
xy
tan
1tan
43
222 yxr
22 )1()1( r
2r
43,2
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