1. 9.6 Polar Coordinates

2. Objectives
Plot points and find multiple representations of points in the polar coordinate system
Convert points from rectangular to polar form and vice-versa
Convert equations from rectangular to polar form and vice- versa

3. Rectangular Coordinate Plane
point (x, y) in the rectangular plane represent the directed distance from the coordinate axes to the point
Each point (x, y) has a unique representation

4. Polar Coordinate System
To form the polar coordinate system in the plane:
fix a point O (called the pole or origin)
construct from O an initial ray (polar axis)
each point P in the plan can be assigned polar coordinates (r, θ)
P = (r, θ)
r = directed distance from O to P
θ= directed angle, counterclockwise from the polar axis to segment OP
r=directed distance
θ=directed angle
Polar Axis

5. Plotting Points in the Polar Coordinate System
(2, π/3)
(1, 2π/3)
2. (3, -π/6)
3. (3, 11π/6)

6. Polar Coordinates – Multiple Representations
Unlike rectangular coordinates, polar coordinates are NOT unique
the point (r, θ) and (r, θ + 2π) represent the same point
the point (r, θ) and (-r, θ + π) are the same
In general, (r, θ) =
(r, θ ± 2nπ)
(-r, θ ± (2n+1)π)
where n is any integer
The pole is (o, θ), where θ is any angle

7. Multiple Representations of Points
Plot the point (3, -3π/4)
Find 3 additional polar
coordinates representations
of this point using -2π<θ<2π

8. Coordinate Conversion
(x, y) lies on a circle of radius r, it follows that r2 = x2 + y2
coordinate conversion
the polar coordinates (r, θ) are related to the rectangular coordinates (x, y) as follows
x = r cosθ
y = r sinθ
tanθ = y/x
r2 = x2 + y2

9. Convert: Polar  Rectangular
Convert each point to rectangular coordinates
(2, π)
(√3, π/6)

10. Convert: Rectangular  Polar
Convert each point to polar coordinates.
(-1, 1)
(0,2)