6.4 Polar Coordinates
I.
Polar Coordinates

The polar coordinate system is basically a way of plotting points and graphing when you don’t have the
unit circle – you have circles of larger radius.

The Cartesian coordinates of (x, y) are sometimes called “rectangular coordinates” due to the
rectangular nature of movement when plotting a point.

Instead of (x, y), our plotted points have the coordinates (𝑟, 𝜃) where r = radius & 𝜃 = the angle
formed.
initial side of the angles lies on the “polar axis”
pole
Example 1: Graphically express the point (x, y) as (r, 𝜃).
𝜋
Example 2: The point P(3, ) is the point on a circle of radius 3 that is on the terminal side of the angle of
6
radians.
P

𝜋
6
P(3,
−11𝜋
6
6
6
) would also be the same coordinate.
Note that we cannot have an infinite number of coordinates for (x, y) as there is only one way to get to
that point from the origin.
II. Graphing Polar Coordinates:




6
Trigonometric functions are circular functions that repeat. So, we can have infinite number of
𝜋
13𝜋
coordinates for P (as there are infinite # of angles that are coterminal with ). For example, P(3,
)
or

.
𝜋
Polar coordinate graph paper is a series of concentric circles representing different radii.
It is easier to graph the angle first, then do the radius.
if r > 0 the point is on the terminal side…move forward
if r < 0 the point is on the ray opposite the terminal side…move backwards (this is like adding
±180° 𝑜𝑟 𝜋)
Plot the points and represent it 2 other ways.
𝜋
1. Plot (−4, − 6 )
2. Plot (−1,
4𝜋
3
)
III. Converting between Polar & Rectangular Coordinates:
𝜋
A) Convert (3, 2 ) to rectangular.
B) Convert (7.5, 110°) to rectangular.
C) Convert (3, 0) to polar.
D) Convert (1, -1) to polar.
E) Convert (−12, −5) to polar.
IV.


Converting between Polar & Rectangular Equations:
Rectangular equations have x’s and y’s and should be solved for y, where possible
Polar equations have r’s and 𝜃’s and should be solved for r, where possible
A) Given 𝑟 = tan 𝜃 sec 𝜃, rewrite as a rectangular equation.
B) Given 𝑟 = cos 𝜃, rewrite as a rectangular equation.
C) Given 2𝑥 2 + 2𝑦 2 = 3, rewrite as a polar equation.
D) Given 2𝑥𝑦 = 1, rewrite as a polar equation.
E) Given (𝑥 − 3)2 + 𝑦 2 = 9, rewrite as a polar equation.
6.5 Polar Equations & Graphs
Convert each polar equation to a rectangular equation and sketch the graph.
1) 𝑟 = 3
2) 𝜃 =
𝜋
3
*Keep in mind no matter what 𝜃 is, r=3.
3) 𝑟𝑠𝑖𝑛𝜃 = 2 and 𝑟𝑐𝑜𝑠𝜃 = −3
4) Factor & solve for r in order to graph, 𝑟 2 − 3𝑟 + 2 = 0.
Ex) Graph 𝑟 = −3 cos 𝜃
θ
What picture did it make? __________________________
r
Ex) Graph 𝑟 = 3 + 3 sin 𝜃
θ
r
θ
r
What picture did it make? __________________________
Ex) Graph 𝑟 = 5 cos 2𝜃
What picture did it make? __________________________
𝜃
𝜃
𝜃
2
2
2
Ex) Graph 𝑟 = 6 cos … since , we need to go around circle twice so 𝜃 = 720° →
θ
r
What picture did it make? __________________________

In calculator, you can graph these, MODEPOLRADIAN
(You can adjust 𝜽. For example, you may want to start with 𝜽max as 2π.)
Ex) Graph 𝑟 = 4 sin 3𝜃 + 2
𝜃
Ex) Graph 𝑟 = 𝑒 5 (called logarithmic spiral)
Ex) Graph 𝑟 = 1 + 2 cos3 (4𝜃) − sin(2𝜃)
= 360°.
θ
r
6.6 Complex Plane & DeMoivre’s Theorem
Complex #:
z:
Complex Plane:
r:
Rectangular Form
of a complex #:
Polar Form (aka: trig. form)
of a complex #:
x=
a=
y=
b=
z=
z=
r=
r=
tanθ =
tanθ =
Expressing a complex # in polar form (trigonometric form) or rectangular form:
Ex) Plot 1 + 𝑖 and write it in polar form.
Ex) Write 2(cos 120° + 𝑖 sin 120°) in rectangular form & plot it.

Using identities, we can find the products and quotients of complex numbers to be:
Ex) Given 𝑧 = 2(cos 40° + 𝑖 sin 40°) and 𝑤 = 4(cos 20° + 𝑖 sin 20°)
a) Find zw
b) Find
𝑧
𝑤
IV. DeMoivre’s Theorem is used to easily find a complex number raised to a power.
Ex) Write [4(cos 40° + 𝑖 sin 40°)]3
in rectangular or standard
form (𝑥 + 𝑦𝑖)…
Ex) Write [2(cos 60° + 𝑖 sin 60°)]2
in rectangular or standard
form (𝑥 + 𝑦𝑖)…