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)

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.

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

Try graphing this.(Pol mode)

2sin 2.15

0 16

r

Remember from trig, in polar coordinates,

x = r cosΘ

y = r sinΘ

To find the slope of a polar curve:

dy

dy ddxdxd

sin

cos

dr

ddr

d

sin cos

cos sin

r r

r r

We use the product rule here.

A lot like parametric slope.

Example: 1 cosr sinr

sin sin 1 cos cosSlope

sin cos 1 cos sin

2 2sin cos cos

sin cos sin sin cos

2 2sin cos cos

2sin cos sin

cos 2 cos

sin 2 sin

The length of an arc (in a circle) is given by r. when is given in radians.

Area Inside a Polar Graph:

For a very small , the curve could be approximated by a straight line and the area could be found using the triangle formula: 1

2A bh

r dr

21 1

2 2dA rd r r d

We can use this to find the area inside a polar graph.

21

2dA r d

21

2dA r d

21

2A r d

Example: Find the area enclosed by: 2 1 cosr

2 2

0

1

2r d

2 2

0

14 1 cos

2d

2 2

02 1 2cos cos d

2

0

1 cos 22 4cos 2

2d

This graph is called a limaƈon.

2

0

1 cos 22 4cos 2

2d

2

03 4cos cos 2 d

2

0

13 4sin sin 2

2

6 0

6

Notes:

To find the area between curves, subtract:

2 21

2A R r d

Just like finding the areas between Cartesian curves, establish limits of integration where the curves cross.

When finding area, negative values of r cancel out:

2sin 2r

22

0

14 2sin 2

2A d

Area of one leaf times 4:

2A

Area of four leaves:

2 2

0

12sin 2

2A d

2A

To find the length of a curve:

Remember: 2 2ds dx dy

Again, for polar graphs: cos sinx r y r

If we find derivatives and plug them into the formula, we (eventually) get:

22 dr

ds r dd

So: 22Length

drr d

d

22Length

drr d

d

There is also a surface area equation similar to the others we are already familiar with:

22S 2

dry r d

d

When rotated about the x-axis:

22S 2 sin

drr r d

d