10.1 Polar Coordinates. The Cartesian system of rectangular coordinates is not the only graphing system. This chapter explores the polar coordinate system. P. ( r , θ ). r. θ. O. (polar axis). fixed ray OA. A. We will graph in what is called the r θ -plane.
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10.1 Polar Coordinates
The Cartesian system of rectangular coordinates is not the only graphing system. This chapter explores the polarcoordinatesystem.
fixed ray OA
We will graph in what is called the rθ-plane
fixed point (pole or origin)
A polar coordinate is the ordered pair (r, θ)
r = distance from pole to point
θ = angle from polar axis (deg or rad)
(pos or neg)
on opposite of terminal side
(pos or neg)
Ex 1) Graph each point on the rθ-plane. (Just sketch)
Note: Since θ and θ + 2πn, n will produce equal angles, a point can be represented in infinitely many polar coordinate pairs.
r can also be positive or negative, adding to the options
Note: If r > 0 and 0 ≤ θ < 2π, then (r, θ) represents exactly 1 point.
Ex 2) Plot
Which of these does NOT represent the same point?
(Identify and fix it)
An equation with polar coordinates is a polarequation. We will graph with constants today, r = c and θ = k, and explore more complicated ones tomorrow.
Ex 3) Graph each polar equation.
a) r = 3
(length always 3
angle is anything)
1 2 3
r can be anything
positive or negative)
Your turn. Graph on whiteboard.
c)d) r = –4
*same as r = 4
1 2 3 4
If we superimposed the rectangular coordinate system on the rθ-plane we can discover their relationships.
= r sinθ
= r cosθ
You will use these relationships to change equations from one system to another system.
Ex 4) Find the rectangular coordinates. Round to nearest hundredth.
(not famous – use calculator)
To convert from rectangular to polar:
(if x > 0)
(if x < 0)
Ex 5) Find polar coordinates of
Ex 6) Convert:
a) x = 3 to a polar equation
b)to a rectangular equation
x2 + y2
#1001 Pg 482 #1–53 odd, 34, 40, 54