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).
P = (r, θ)
r
θ
polar axis
pole
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).
Plotting Points in Polar Coordinates.
To plot the point P(r, θ ), go a distance of
r at 00 then move θ0 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.
0
Plot each point (r, θ)
a) A(3, 450)
C
b) B(-5, 1350)
A
B
c) C(-3, -π/6)
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)
3
a) x = rcos(-30⁰)  2(
) 3
2
y  2sin( 30 )  2( 1/ 2)   1
The rectangular coordinates of (2,  30 ) are( 3 ,-1)
b) x= -4cos(π/3) = -4(1/2) = -2
3
)  2 3
y= -4 sin(π/3) = 4(
2
The rectangular coordinates of (-4,

3
) are(-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 =
x 2  y 2 to find r .
y
3) Find  by using tan 
and choose  so that it lies in the
x
same quadrant as the point (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.
r  (1)2  ( 3)2  4  2
3
tanθ =
1
  tan1(  3 )
2

3
The required polar coordinates are (2, 2π/3)
Give polar coordinates for the point shown.
a ) r  0, 0    360
b) r  0,  360    0
c) r  0, 0    360
d) r  0, 360    720
Now give each with  in radians.
EQUATION CONVERSION FROM RECTANGULAR TO
POLAR COORDINATES.
A polar equation is an equation whose variables are r and θ.
5
and r  3csc  . To convert a
Examples are r 
cos   sin 
rectangular coordinate equation in x and y to a polar equation
in r and θ, replace x with rcosθ and y with rsinθ.
Example: Convert each rectangular equation to a polar
equation that expresses r in terms of θ’
5
ans. r 
cos   sin 
a) x + y = 5
b) ( x  1)  y  1
2
2
ans. r= 2cosθ
EQUATION CONVERSION FROM POLAR TO
RECTANGULAR COORDINATES.
Use one or more of the following equations:
r x  y
2
2
2
r cos   x
r sin   y
y
tan  
x
Examples:
Convert each polar equation to a rectangular equation in x and y:
a) r = 5
b)  =
Ans. a) x 2  y 2  25

c) r = 3csc
4
b) y  x
c) y  3
d) r = -6cos
d )  x  3  y 2  9
2