Math 1B
§ 10.3 Polar Coordinates
Overview: Up to this point we’ve dealt exclusively with the Cartesian (or rectangular) coordinate
system. In this section we will start looking at the polar coordinate system.
We choose a point in the plane that is called the pole (or origin)
and is labeled O. Then we draw a ray (half-line) starting at O called
the polar axis. If P is any other point in the plane, let r be the distance
from O to P and let 𝜃 be the angle (usually measured in radians)
between the polar axis and the line OP. Then the point P is
represented by the ordered pair 𝑟, 𝜃 and r, 𝜃 are called polar
coordinates of P.
The connection between Cartesian coordinates and polar coordinates is shown below. The pole
corresponds to the origin and the polar axis coincides with the positive x-axis.
!
!
We also allow r to be negative. Below is a picture of the two points 2, ! and −2, ! .
We see that if r is positive the point will be in the same quadrant as 𝜃. If r is negative the point will end
!
up in the quadrant exactly opposite 𝜃. Notice that the coordinates −2, ! describe the same point as
the coordinates 2,
!!
!
.
In the Cartesian coordinate system every point has only one representation, but in the polar coordinate
!
system every point has many representations. For example, the point 5, ! can be written as
__________
__________
__________
Let’s think about converting between the two coordinate systems (see above figure).
If the point P has Cartesian coordinates (𝑥, 𝑦) and polar coordinates (𝑟, 𝜃) then we have
!
cos 𝜃 = !
sin 𝜃 =
!
!
which gives us …
Polar to Cartesian Conversion Formulas:
𝑥 = 𝑟 cos 𝜃
𝑦 = 𝑟 sin 𝜃
To find the Cartesian coordinates of a point when the polar coordinates are known we use
Cartesian to Polar Conversion Formulas:
𝑟! = 𝑥! + 𝑦!
!
tan 𝜃 = !
Example: Plot the point whose polar coordinates are given. Then find the Cartesian coordinates of the
point.
!!
−4, !
Example: Find polar coordinates (𝑟, 𝜃) of the point (−1, −1).
We can also use the above formulas to convert equations from one coordinate system to the other.
Example: Convert 4𝑦 ! = 𝑥 into polar coordinates.
Example: Convert 𝑟 = −8 cos 𝜃 into Cartesian coordinates.
Polar Curves
Example: Graph 𝑟 = 7, 𝑟 = 4 cos 𝜃 and 𝑟 = −7 sin 𝜃 on the same axis system.
Graphs from the previous example:
Example: Graph 𝑟 = 2 − 2 sin 𝜃.
𝜃
r
0
𝜋
6
𝜋
2
5𝜋
6
𝜋
3𝜋
2
2𝜋
2
1
0
1
2
4
2
Example: Graph 𝑟 = 1 + 2 cos 𝜃.
𝜃
r
0
𝜋
3
𝜋
2
2𝜋
3
𝜋
4𝜋
3
3𝜋
2
5𝜋
3
2𝜋
3
2
1
0
-1
0
1
2
3
Tangents to Polar Curves
To find a tangent line to a polar curve 𝑟 = 𝑓 𝜃 , we regard 𝜃 as a parameter and write its parametric
equations as
𝑥 = 𝑟 cos 𝜃 = 𝑓(𝜃) cos 𝜃
𝑦 = 𝑟 sin 𝜃 = 𝑓(𝜃) sin 𝜃
Then, using the method for finding slopes of parametric curves and the Product Rule, we have
Derivative with Polar Coordinates:
𝑑𝑦 𝑑𝑟
𝑑𝑦 𝑑𝜃 𝑑𝜃 sin 𝜃 + 𝑟 cos 𝜃
=
=
𝑑𝑥 𝑑𝑥 𝑑𝑟 cos 𝜃 − 𝑟 sin 𝜃
𝑑𝜃 𝑑𝜃
!"
!"
Note: The curve has a horizontal tangent when !" = 0 (provided that !" ≠ 0)
!"
!"
The curve has a vertical tangent when !" = 0 (provided that !" ≠ 0)
!
Example: For the cardioid 𝑟 = 2 − 2 sin 𝜃, find the slope of the tangent line when 𝜃 = ! .
Example: Find the points on the cardioid 𝑟 = 1 − sin 𝜃 where the tangent line is vertical or horizontal.