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

O (the pole)

ray (polar axis)

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 = aCircles of radius a,

- centered at point (0,a): r = 2a sin - centered at point (a,0): r = 2a cos

r = 2 sin r = 2 cos

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

dfA 2

21 )(

ExampleFind the area of the region bounded by the equation

r = 3 + 2 cos , 0 2.

Solution:

112sinsin12112

1

2cos22cos1292

1get we

2cos222

2cos14cos4 Because

cos4cos129 2

1

cos23

20

2

0

2

2

0

2

2

0

2

212

21

dA

d

ddrA

r = 3 + 2 cos , , 0 2