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 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.
The Sign of r and a Point’s
Location in Polar Coordinates
(2, 135°)
Plot the point.
3𝜋
(−3, )
2
Plot the point.
−𝜋
−1,
4
Plot the point.
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)
Multiple Representations of
Points.
𝜋
(2, )
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
𝜋
(5, )
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
Find 3 representations
𝑟=2
Graph of a Circle
𝜋
𝜃=
6
Graph of a Line
r
y
θ
𝑥2 + 𝑦2 = 𝑟2
𝑠𝑖𝑛𝜃 =
𝑐𝑜𝑠𝜃 =
𝑡𝑎𝑛𝜃 =
x
𝑥=
𝑦=
𝑡𝑎𝑛𝜃=
Relations between Polar and
Rectangular Coordinates
3𝜋
(2, )
2
Find the rectangular coordinates
𝜋
−8,
3
Find the rectangular coordinates:
(3, 52°)
What if it’s not on the Unit Circle?
(4, −168°)
Not on Unit Circle
Plot the point (𝑥, 𝑦).
2. Find r by computing the distance from
the origin to 𝑥, 𝑦 : 𝑟 = (𝑥 2 + 𝑦 2 ).
𝑦
3. Find 𝜃 using 𝑡𝑎𝑛𝜃 = with the terminal
𝑥
side of 𝜃 passing through (𝑥, 𝑦).
1.
Converting a Point from
Rectangular to Polar Coordinates
−1, 3
Find the Polar Coordinates
(1, − 3)
Find the Polar Coordinates
(0, -5)
Find the polar coordinates
(−3, 2)
Not on Unit Circle
(−4, 7)
Not on Unit Circle
(4, −6.2)
One More
Use one or more of these equations:
𝑟2 = 𝑥2 + 𝑦2
𝑟𝑐𝑜𝑠𝜃 = 𝑥
𝑟𝑠𝑖𝑛𝜃 = 𝑥
𝑡𝑎𝑛𝜃 =
𝑥
𝑦
Convert to a rectangular equation: 𝑟 = 10
Equation Conversion from Polar to
Rectangular
𝜋
𝜃=
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
𝑟 2 𝑠𝑖𝑛2𝜃 = 4
Convert to a rectangular equation
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
Equation Conversion from
Rectangular to Polar
𝑥 + 5𝑦 = 8
Convert to a polar equation
𝑦=3
Convert to a polar equation
𝑥 2 + 𝑦 2 = 16
Convert to a polar equation
𝑥2 + 𝑦 + 3
2
=9
Convert to a polar equation
𝑥 2 = 6𝑦
Convert to a polar equation