Post on 27-Dec-2015
transcript
10.5: Polar Coordinates
Greg Kelly, Hanford High School, Richland, Washington
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
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.
x-axis
r A polar coordinate pair
2
4
4,2
is a polar function in which the length r is a function of the angle
Some curves are easier to describe with polar coordinates:
)(fr
…but how to plot such a function…
sin()
/ 6 sin(/6) r =
/ 4 sin(/4) r =
/ 3 sin(/3) r =
/ 2 sin(/2) r = 1
2/ 3 sin(2/3) r =
3/ 4 sin(3/4) r =
5/ 6 sin(5/6) r =
sin() r = 0
r
2
12
1
2
3
2
1
2
1
2
3
x
y
is a polar function in which the length r is a function of the angle
Some curves are easier to describe with polar coordinates:
sinr
x
y
x
y
x
y
x
y
x
y
x
y
x
y
x
y
x
y
sin()
7/ 6 sin(7/6) r =
5/ 4 sin(5/4) r =
4/ 3 sin(4/3) r =
3/ 2 sin(3/2) r = 1
5/ 3 sin(5/3) r =
7/ 4 sin(7/4) r =
/ 6 sin(11/6) r =
2 sin(2) r = 0
r
2
12
1
2
3
2
1
2
1
2
3
is a polar function in which the length r is a function of the angle
Some curves are easier to describe with polar coordinates:
)(fr
x
y…but how can this be?
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.
x-axisr
2
4
5
4
5,2
4
5
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.
x-axis
2
4
5,2
But what about this coordinate pair?
sin()
7/ 6 sin(7/6) r =
5/ 4 sin(5/4) r =
4/ 3 sin(4/3) r =
3/ 2 sin(3/2) r = 1
5/ 3 sin(5/3) r =
7/ 4 sin(7/4) r =
/ 6 sin(11/6) r =
2 sin(2) r = 0
r
2
12
1
2
3
2
1
2
1
2
3
is a polar function in which the length r is a function of the angle
Some curves are easier to describe with polar coordinates:
)(fr
x
y…but how can this be?
1 2 02
r
r a
o
(Circle centered at the origin)
Some curves are easier to describe with polar coordinates:
(Try graphing it on your calculator)
(Line through the origin whose slope is )0tan
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
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, +) .
tan
cosr
Tests for Symmetry:
If a graph has two symmetries, then it has all three:
2cos 2r
x
y
),( yx
Converting to Cartesian Coordinates
Remember the unit circle?
22 yxr
x
y
cosr
sinr r
Converting to Cartesian Coordinates
Convert 2xy to polar coordinates
2xy
2cossin rr
2cossin rr
2)cos(sin r
cossin
2
r
Graph this on your calculators for comparison