10.1 Polar Coordinates

The Cartesian system of rectangular coordinates is not the only graphing system. This chapter explores the polar coordinate system.

O A

P

θ

(r, θ)r

fixed point (pole or origin)

(polar axis)fixed ray OA

A polar coordinate is the ordered pair (r, θ)

r = distance from pole to point

θ = angle from polar axis (deg or rad)

(pos or neg)

onterminal

side

on opposite of terminal side

(pos or neg)

counterclockwise clockwise

We will graph in what is called the rθ-plane

Ex 1) Graph each point on the rθ-plane. (Just sketch)

a) 2, 2, 2,3 3 3

P Q R

b) c)

O

P 2,3

3

O

Q 2,3

3

O

R 2,3

3

Note: Since θ and θ + 2πn, n will produce equal angles, a point can be represented in infinitely many polar coordinate pairs. r can also be positive or negative, adding to the options Note: If r > 0 and 0 ≤ θ < 2π, then (r, θ) represents exactly 1 point.

Ex 2) Plot2

1,3

1 2

3

Which of these does NOT represent the same point?(Identify and fix it)

A) B) C)5 4

1, 1, 1,3 3 3

An equation with polar coordinates is a polar equation. We will graph with constants today, r = c and θ = k, and explore more complicated ones tomorrow.Ex 3) Graph each polar equation. a) r = 3

(length always 3angle is anything)

1 2 3

b) 3

(angle always r can be anything positive or negative)

3

3

OR

Your turn. Graph on whiteboard.

c) d) r = –44

4

1 2 3 4

*same as r = 4

If we superimposed the rectangular coordinate system on the rθ-plane we can discover their relationships.

(r, θ)

θ

r

In cartesian: (x, y)

x

y

2 2 2

cos sin

tan

x y

r r

r x y

y

x

and

= r cosθ

= r sinθ

also

2 2x y

You will use these relationships to change equations from one system to another system.

Ex 4) Find the rectangular coordinates. Round to nearest hundredth. (if necessary)

a)2

3, 10,6 9

3 3 33cos 3

6 2 21 3 2

3sin 3 10cos 7.666 2 2 9

3 3 3 2, 10sin 6.43

2 2 9(7.66, 6.43)

P Q

x

y x

y

b)

(not famous – use calculator) **RAD mode

To convert from rectangular to polar:

2 2 1 1tan tany y

r x yx x

(if x > 0) (if x < 0)

Ex 5) Find polar coordinates of 2

2

1

3, 1

3 (1) 3 1 4 2

1 6 50 tan

6 6 63

52,

6

r

x

Ex 6) Convert:

a) x = 3 to a polar equation

cos 3

3

cos3sec

r

r

r

2

2 2

4sin

4 sin

4 sin

4

r

r r r

r r

x y y

b) to a rectangular equation

x2 + y2 y

Homework

#1001 Pg 482 #1–53 odd, 34, 40, 54