Polar Coordinates

The foundation of the polar coordinate system is a horizontal ray that extends to the right.

The ray is called the polar axis.

The endpoint of the ray is called the pole.

The point P in the polar coordinate system is represented by an ordered pair of numbers 𝑟, 𝜃 .

r is a directed distance from the pole to P. (it can be positive, negative, or zero.)

𝜃 is an angle from the polar axis to the line segment from the pole to P.

The Sign of r and a Point’s Location in Polar Coordinates

The point 𝑃 = 𝑟, 𝜃 is located 𝑟 units from the pole.

-- 𝑟 > 0, the point lies along the terminal n

side of 𝜃.

-- 𝑟 < 0, the point lies along the ray opposite

the terminal side of 𝜃.

-- 𝑟 = 0, the point lies at the pole, regardless

of 𝜃.

Plot the point.

(2, 135°)

Plot the point.

(−3,3𝜋

2)

Plot the point.

−1,−𝜋

4

Multiple Representations of Points.

If n is any integer, the point (𝑟, 𝜃) can be represented as

𝑟, 𝜃 = (𝑟, 𝜃 + 2𝑛𝜋) or

𝑟, 𝜃 = (−𝑟, 𝜃 + 𝜋 + 2𝑛𝜋)

Find 3 representations

(2,𝜋

3)

a. r is positive and 2𝜋 < 𝜃 < 4𝜋

b. r is negative and 0 < 𝜃 < 2𝜋

c. r is positive and −2𝜋 < 𝜃 < 0

Find 3 representations

(5,𝜋

4)

a. r is positive and 2𝜋 < 𝜃 < 4𝜋

b. r is negative and 0 < 𝜃 < 2𝜋

c. r is positive and −2𝜋 < 𝜃 < 0

Graph of a Circle

𝑟 = 2

Graph of a Line

𝜃 =𝜋

6

Relations between Polar and Rectangular Coordinates

𝑥2 + 𝑦2 = 𝑟2

r 𝑠𝑖𝑛𝜃 =

y 𝑐𝑜𝑠𝜃 =

θ 𝑡𝑎𝑛𝜃 =

x

𝑥 =

𝑦 =

𝑡𝑎𝑛𝜃=

Find the rectangular coordinates

(2,3𝜋

2)

Find the rectangular coordinates:

−8,𝜋

3

What if it’s not on the Unit Circle?

(3, 52°)

Not on Unit Circle

(4, −168°)

Converting a Point from Rectangular to Polar Coordinates

1. Plot the point (𝑥, 𝑦).

2. Find r by computing the distance from

the origin to 𝑥, 𝑦 : 𝑟 = (𝑥2 + 𝑦2).

3. Find 𝜃 using 𝑡𝑎𝑛𝜃 =𝑦

𝑥 with the terminal

side of 𝜃 passing through (𝑥, 𝑦).

Find the Polar Coordinates

−1, 3

Find the Polar Coordinates

(1, − 3)

Find the polar coordinates

(0, -5)

Not on Unit Circle

(−3, 2)

Not on Unit Circle

(−4, 7)

One More

(4, −6.2)

Equation Conversion from Polar to Rectangular

Use one or more of these equations:

𝑟2 = 𝑥2 + 𝑦2 𝑟𝑐𝑜𝑠𝜃 = 𝑥 𝑟𝑠𝑖𝑛𝜃 = 𝑥 𝑡𝑎𝑛𝜃 =𝑥

𝑦

Convert to a rectangular equation: 𝑟 = 10

Convert to a rectangular equation

𝜃 =𝜋

3

Convert to a rectangular equation

𝑟𝑐𝑜𝑠𝜃 = 7

Convert to a rectangular equation

𝑟 = 6𝑠𝑒𝑐𝜃

Convert to a rectangular equation

𝑟 = 8𝑐𝑜𝑠𝜃𝜃 + 2𝑠𝑖𝑛𝜃

Convert to a rectangular equation

𝑟2𝑠𝑖𝑛2𝜃 = 4

Equation Conversion from Rectangular to Polar

To convert a rectangular equation in x and y to a polar equation that expresses r in terms of 𝜃:

--replace x with 𝑟𝑐𝑜𝑠𝜃

--replace y with 𝑟𝑠𝑖𝑛𝜃

Convert to a polar equation

𝑥 + 5𝑦 = 8

Convert to a polar equation

𝑦 = 3

Convert to a polar equation

𝑥2 + 𝑦2 = 16

Convert to a polar equation

𝑥2 + 𝑦 + 3 2 = 9

Convert to a polar equation

𝑥2 = 6𝑦