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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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)
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)
A) B) C)
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)
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.
(not famous – use calculator)
(if x > 0)
(if x < 0)
Ex 5) Find polar coordinates of
a) x = 3 to a polar equation
b) to a rectangular equation
x2 + y2
#1001 Pg 482 #1–53 odd, 34, 40, 54