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)

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

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 

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

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)

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

Example Solution: Find the area of the region bounded by the equation r = 3 + 2 cos  , 0    2. Solution:

r = 3 + 2 cos , , 0    2