

Polar Coordinate System

Polar
and Cartesian coordinates relations 
Conversion from polar
to rectangular coordinates 
Conversion from
rectangular to polar coordinates 
Polar coordinates of a point 
Equation
of a line in polar form 
Lines
parallel to the axes, horizontal and vertical lines 
Lines
running through the origin or pole (radial lines) 
Polar
equation
of a line 
Equation
of a circle in polar form 
General
equation of a circle in polar coordinates 
Polar
equation of a circle with a center on the polar axis running
through the pole 
Polar
equation of a circle with a center at the pole 





Polar coordinate system

The
polar coordinate system is a twodimensional coordinate system
in which each point P
on a plane is determined by the length of its position vector r
and the angle q
between it and the positive direction of the xaxis,
where 0 <
r
< + oo
and 0
<
q
< 2p. 


Polar
and Cartesian coordinates relations, 

Note,
since the inverse tangent function (arctan or tan^{}^{1})
returns values in the range p/2
< q <
p/2, then 
for points lying in the 2nd or 3rd quadrant 


and for points lying in the 4th quadrant 



Example:
Convert Cartesian coordinates
(1,
Ö3)
to polar coordinates. 
Solution: 

and since the point lies in the 3rd quadrant,
then 










Equation
of a line in polar form 
Lines
parallel to the axes, horizontal and vertical lines 
Lines
parallel to the yaxis 
A
vertical line, x
=
c is
represented by the equation 
r
cosq
=
c
or 





Lines
parallel to the xaxis 
A
horizontal line, y
=
c is
represented by the equation 
r
sinq
=
c
or 







Lines
running through the origin or pole (radial lines) 
The
equation of a line through the origin or pole that makes an angle
a with the positive
xaxis 

is
represented by the equation 
q
=
a 
in polar coordinates. 


Polar
equation
of a line 
As 



the
polar equation of a line 

where
p
is the distance of the line from the pole O
and j
is the angle that the segment p
makes with the polar axis. 




Example:
Write polar equation of the
line passing through points (4,
0)
and (0, 4). 
Solution:
Using polar equation of a line 

Proof,
use of Cartesian to polar
conversion formulas. 
The
intercept form of the line 

or 


x
+ y =
4,
y
=
x
+ 4 



r
cosq
+ r sinq
=
4,
r (sinq

cosq
)
=
4, 





Equation
of a circle in polar form 
General
equation of a circle in polar coordinates 
The
general equation of a circle with a center
at 
(r_{0},
j)
and radius R. 
Using
the law of cosine, 

r^{2}
+ r_{0}^{2}

2rr_{0}
cos(q

j)
=
R^{2} 






Polar
equation of a circle with a center on the polar axis running
through the pole 
Polar
equation of a circle with radius R
and a center on the polar axis running through the pole O
(origin). 
Since 

then, 






Polar
equation of a circle with a center at the pole 
Since,
r^{2}
=
x^{2}
+ y^{2} and
x^{2}
+ y^{2}
=
R^{2} then 
r
=
R 


is
polar
equation of a circle with radius R
and a center at the pole (origin). 

Example:
Convert the polar
equation of a circle r
=
4
cosq
into Cartesian coordinates. 
Solution:
As, r
=
4
cosq 
then r^{2}
=
4r
cosq, 
and
by using polar to Cartesian conversion formulas, r^{2}
=
x^{2}
+ y^{2} and
x
=
r cosq 
obtained
is
x^{2}
+ y^{2}
=
4x 
x^{2}
+ 4x
+ y^{2}
= 0

or
(x
+ 2)^{2} + y^{2}
= 4

the
equation of a circle with radius R
= 2 and
the

center
(2,
0).












College
algebra contents A




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