Polar Coordinates
Given:r: Directed distance from the Polar axis (pole) to point PƟ: Directed angle from the Polar axis to ray OP
),( rP
OInitial ray
Each point P in the plane can be assigned polar coordinates (r, Ɵ), as follows.
r = directed distance from O to PƟ = directed angle, counterclockwise from polar axis to segment OP
Polar Coordinates
In general, the point (r, Ɵ) can be written as
(r, Ɵ) = (r, Ɵ + 2nπ)or
(r, Ɵ) = (–r, Ɵ + (2n + 1)π)
where n is any integer. Moreover, the pole is represented by (0, Ɵ), where Ɵ is any angle.
Polar Coordinates
To establish the relationship between polar and rectangular coordinates, let the polar axis coincide with the positive x-axis and the pole with the origin
Because (x, y) lies on a circle ofradius r, it follows that r2 = x2 + y2.
Coordinate Conversion
rryx
y
x
so
so sin
so cos
222
5) Convert the polar equation to rectangular form
6) Convert the polar equation to rectangular form
4cos r
sincos2
4
r
7) Convert the rectangular equation to polar form
8) Convert the rectangular equation to polar form
3 yx
2xy
The graph of r = a is a circle of radius a centered at zero
Ɵ = α is a Line through O making angle α with the initial ray
Polar Graphs
• Symmetric about the x-axis: if the point (r, Ɵ) lies on the graph, the point (-r, Ɵ) or (r, π-Ɵ) lies on the graph
• Symmetric about the y-axis: if the point (r, Ɵ) lies on the graph, the point (-r, -Ɵ) or (r, π-Ɵ) lies on the graph
• Symmetric about the origin: if the point (r, Ɵ) lies on the graph, the point (-r, Ɵ) or (r, π+Ɵ) lies on the graph
Symmetry
9) Graph without a graphing calculator with the values of Ɵ from 0 to 2π.This curve is called a cardioid.To plot points use
Polar Graphs
sin45 r
sin and cos ryrx
The following are simpler in polar form than in rectangular form. The polar equation of a circle having a radius of a and centered at the origin is simply
Special Polar Graphs
cos sin barbarar
Using the parametric form of dy/dx we have
Slope and Tangent Lines
sincos
cossin
rddr
rddr
ddxd
dy
dx
dy
sin)(sin
cos)(cos
fry
frx
• Horizontal
• Vertical
Horizontal and Vertical Tangent Lines
0 where0 d
dx
d
dy
0 where0 d
dy
d
dx
Cusp at (0, 0)
If
then
Then the lineIs tangent to the pole to the graph of
Tangent Lines at the Pole0)(' where0)( ff
fr
6
5 ,
2 ,
6
pole at the tangents
tan
0cos
0sin
sincos
cossin
ddrddr
rddr
rddr
dx
dy