Transcript Document
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
3
tanθ =
1
4 2
tan1( 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