1. POLAR COORDINATES (Ch.10.2-10.3)
Given the pole O and the polar axis, the point P with polar coordinates (r, ) is located :
-  degree angle from the x-axis ( is measured counter clockwise)
- at distance r from the origin.
r: radial coordinate ( if r<0, then P lies opposite direction)
: angular coordinate
ray (polar axis)
O (the pole)

2. POLAR COORDINATES
Any point has more than one representation in polar coordinates;
(r, ) = (- r, + )
Example:
the following polar coordinates represent the same point
(2, /3), (-2, 4/3), (2, 7/3), (-2, -2/3).
Convert polar coordinates into rectangular coordinates, use the relations:
x = r cos  , y = r sin 
Then r2 = x2 + y2, tan  = y/x, if x  0

3. POLAR COORDINATE EQUATIONS
Polar equation of a circle with radius a: r = a
Circles of radius a,
- centered at point (0,a): r = 2a sin 
- centered at point (a,0): r = 2a cos 
r = 2 cos 
r = 2 sin 

4. Transform the equation r = 2 sin  into rectangular coordinates:
Multiply both sides by r:
r2 = 2r sin 
x2 + y2 = 2y
x2 + y2 - 2y = 0
Complete the square in y : x2 + (y -1)2 = 1

5. Find a point of intersection of the equations
r = 1 + sin  and r2 = 4 sin .
Solution:
(1 + sin  )2 = 4 sin 
1 + 2 sin  + sin2  - 4 sin  = 0
sin2  - 2 sin  + 1 = 0
(sin  - 1)2 = 0  sin  = 1
So  is the angle of the form: 1/2 + 2n, where n is an integer.
Point: (2,  /2)

6. Area Computations in Polar Coordinates
Definition:
The area A of the region R bounded by the lines  =  and  =  and the curve r = f( ) is

7. Example Solution: Find the area of the region bounded by the equation
r = cos  , 0    2.
Solution:

8. r = cos , , 0    2