Section 6.3

Polar Coordinates

The foundation of the polar coordinate system is a horizontal ray that extends to the right. This ray is called the polar axis. The endpoint of the polar axis is called the pole. A point P in the polar coordinate system is designated by an ordered pair of numbers (r, θ).

r

pole

polar axis

θ

P = (r, θ)r is the directed distance form the pole to point P ( positive, negative, or zero).

θ is angle from the pole to P (in degrees or radians).

0 0

0

To plot the point P( r,θ) , go a distance of

r at 0 then move θ along a circle of

radius r.

If r > 0, plot a point at that location. If r < 0, the

point is plotted on a circle of the same radius,

but 180 in the opposite direction.

Plotting Points in Polar Coordinates.

Plot each point (r, θ)

a) A(3, 450)

A b) B(-5, 1350)

Bc) C(-3, -π/6)

C

CONVERTING BETWEEN POLAR AND RECTANGULAR FORMS

CONVERTING FROM POLAR TO RECTANGULAR COORDINATES.To convert the polar coordinates (r, θ) of a point to rectangular coordinates (x, y), use the equations

x = rcosθ and y = rsinθ

Convert the polar coordinates of each point to its rectangular coordinates.a) (2, -30⁰ ) b) (-4, π/3)

a) x = rcos(-30⁰) 3

2( ) 32

2sin( 30 ) 2( 1/ 2) 1y

The rectangular coordinates of (2, 30 ) are ( 3,-1)

b) x= -4cos(π/3) = -4(1/2) = -2

y= -4 sin(π/3) = 3

4( ) 2 32

The rectangular coordinates of (-4, ) are 3

( -2, -2 3)

CONVERTING FROM RECTANGULAR TO POLAR COORDINATES:

To convert the rectangular coordinates (x, y) of a point to polar coordinates:

1)Find the quadrant in which the given point (x, y) lies.

2) Use r = 2 2 to find .x y r

3) Find by using tan and choose so that it lies in the

same quadrant as the point ( , ).

y

x

x y

Find the polar coordinates (r, θ) of the point P with r > 0 and 0 ≤ θ ≤ 2π, whose rectangular coordinates are (x, y) = ( 1, 3)

The point is in quadrant 2.

2 2( 1) ( 3) 4 2r

tanθ =

1tan ( 3)

2

3

The required polar coordinates are (2, 2π/3)

3

1

Give polar coordinates for the point shown.

) 0, 0 360

) 0, 360 0

) r 0, 0 360

d) r 0, 360 720

Now give each with in radians.

a r

b r

c

EQUATION CONVERSION FROM RECTANGULAR TO POLAR COORDINATES.

A polar equation is an equation whose variables are r and θ.

Examples are To convert a

rectangular coordinate equation in x and y to a polar equation in r and θ, replace x with rcosθ and y with rsinθ.

5and 3csc .

cos sinr r

Example: Convert each rectangular equation to a polar equation that expresses r in terms of θ’

a)x + y = 5

b)

5ans.

cos sinr

2 2( 1) 1x y ans. r= 2cosθ

EQUATION CONVERSION FROM POLAR TO RECTANGULAR COORDINATES.

2 2 2

Use one or more of the following equations:

cos sin tany

r x y r x r yx

Examples:

Convert each polar equation to a rectangular equation in and :

a) = 5 b) = c) = 3csc d) = -6cos4

x y

r r r

22 2 2. a) 25 b) c) 3 ) 3 9Ans x y y x y d x y