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, q) is on the graph,

r

2cosr

r

so is (r, -q).

Tests for Symmetry:

y-axis: If (r, q) is on the graph,

r

2sinr

r

so is (r, p-q)

or (-r, -q).

Tests for Symmetry:

origin: If (r, q) is on the graph,

r

r

so is (-r, q) or (r, q+p) .

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

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

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

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.

CONVERTING TO RECTANGULAR COORDINATES:

1.) x = r cosΘ y = r sinΘ

2.)x

ytan 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.

x

ytan

1tan

4

3

222 yxr

22 )1()1( r

2r

4

3,2

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

Homework

• Section 10.4– #1, 3, 11, 13, 23, 25, 27, 29, 31, 34, 35, 37, 41