Chapter 9.5 Polar

Coordinates

Lesson Objective• After completing the lesson, you will be able to…

o Identify polar coordinates

o Name polar coordinates

o Find additional representations for polar coordinates

o Convert polar coordinates to rectangular coordinates and vice versa

o Convert equations from rectangular to polar form and vice versa

What are Polar Coordinates?

• Normally we plot points on a Rectangular Coordinate Systemo Coordinates appear as (X, Y)

o They represent a point by two distances from the origin

• In a Polar Coordinate system, a point O is called the pole (or the origin).

• A initial ray is formed from O, called the polar axis

• Each point P in the plane can be assigned polar coordinates (r, θ)o r is the directed distance from O to P

o θ is the directed angle, counterclockwise from the polar axis to segment OP

Identifying and Naming Polar Coordinates

• In order to name a polar coordinate as seen on a

polar graph one must:o To find r in the coordinate point, count the rings from the pole O to the

point P on the graph

o To find θ, see where the ray lies on the graph

• Ex: Name the polar coordinate of the point

• First count up

three units to find

r.

• Second, see

where the ray lies

according to the

graph in this case

60 degrees.

Plotting Polar Coordinates Example

• Ex 1: Plot the points (2, π/2) and (3, 11π/16)

(2, π/2)

(3, 11π/6)

Finding Additional

Representations of Coordinates • In rectangular coordinates (x , y) is a unique

representation.

• In polar coordinates this is not the case as they can

be altered as different representations that show

the same point.o When adding or subtracting by 2𝜋, leave r as is

o When adding or subtracting by 𝜋, make r, -r (take the opposite sign)

1) Finding Additional

Representations of Coordinates • For the Point (3, 5π/6) find three additional polar

representations using -2π<θ<2π.

(3, 5π/6)

(3, 5π/6 - 2π) = (3, -7π/6)

(-3, 5π/6 + π) = (-3, 11π/6)

(-3, 5π/6 - π) = (-3, -1π/6)

Coordinate Conversion• The polar axis coincides with the positive x-axis and

with the pole with the origin.

• As (x,y) lies on the circle of radius r, it follows that

𝑥2 + 𝑦2 = 𝑟2.

• The polar coordinates (r, θ ) are related to the

rectangular coordinates (x, y) in the following way:

Polar-to-Rectangular Rectangular-to-Polar

x = rcosθ tan θ =𝑦

𝑥

y = rsinθ 𝑟2 = 𝑥2 + 𝑦2

2) Coordinate Conversion Examples

• Convert the point (2, π) to rectangular coordinates.

x = r cos θ = 2 cos π = -2

y = r sin θ = 2 sin π = 0

The rectangular coordinates (x, y) = (-2, 0)

• Convert the point (-1, 1) to polar coordinates.

For the second quadrant point (x, y) = (-1, 1)

tan θ =𝑦

𝑥=

1

−1= −1

θ =3𝜋

4

Because θ is the same quadrant as (x, y):

r = 𝑥2 + 𝑦2 = (−1)2+(1)2= 2

Polar coordinates = ( 𝟐,𝟑𝝅

𝟒)

Equation Conversion• Both polar equations and rectangular equations

can be converted back and forth to one another.

• To convert a rectangular equation to polar form, x is

replaced by rcosθ and y by rsinθ.

• The same relationships as seen before are useful

when trying to complete these conversions.

Polar-to-Rectangular Rectangular-to-Polar

x = rcosθ tan θ =𝑦

𝑥

y = rsinθ 𝑟2 = 𝑥2 + 𝑦2

3) Equation Conversion Examples

• Convert 𝑦 = 𝑥2 to its polar form.

𝑦 = 𝑥2

r sinθ = (𝑟 cos 𝜃)2 - Replace the variables with their polar counterparts

𝑟𝑠𝑖𝑛𝜃

cos 𝜃2=

𝑟2 cos 𝜃2

cos 𝜃2- Divide both sides by cos 𝜃2

𝑟 sec 𝜃 tan 𝜃

𝑟=

𝑟2

𝑟- Divide both sides by r

𝑟 = sec 𝜃 tan 𝜃

3)Equation Conversion Examples Cont.

• Convert 𝑟 = sec 𝜃 to its rectangular form.

𝑟 cos 𝜃 = 1 - Replace 𝑟 sec 𝜃 𝑤𝑖𝑡ℎ 𝑟 cos 𝜃

X = 1 - Replace 𝑟 cos𝜃 with x referring to the previous table

Real World Application• One real world application of

polar coordinates and its

graphs is its use in radar.

Radar measures the things it is tracking by the angle and

distance from the antenna.