Polar Coordinates
• a different system of plotting points and
coordinates than rectangular (x , y)
• it is based on the ordered pair (r , θ), where r
is the distance from the origin and θ is the
angle in standard position
• unlike for trig. problems r can be positive or
negative (θ can also be either)
• each point can be named with different polar
coordinates (an infinite number of them)
Example: Plot the point (3 , 150º)
3
Some other ways of naming that same point: (3 , -210º),
 5 
3 ,
 and
6 

 - 7 
3 ,
 and others
6 

What about negative values of r?
answer: to graph (-3 , 150º), go 3 units out in the
opposite direction from 150º
3
Finding all polar coordinates of (r , θ)
Positive r: add multiples or 360º or 2π
(r ,   2n) or (r ,  360n)
Negative r: add 180º or π, then you can
add multiples of 360º or 2π
(r ,  (2n  1) ) or (r ,  (2n  1)180)
Coordinate Conversion
• Use the following to convert (x , y)  (r , θ)
r x y
2
2
y

  tan  x  & adjust if necessary


-1
• Use the following to convert (r , θ)  (x , y)
x  r cos 
y  r sin 
Example #1
3 

Convert to (x , y):  4 ,

4 

3
x  4 cos
4
3
y  4 sin
4

2
  2 2
 4 

2


 2
  2 2
 4

2


(2 2 , 2 2 )
Example #2 Convert into (r , θ): (-3 , -7)
r  (3)  (7)
2
2

58
  tan (7 / 3)  66.8
1
add 180 because its in quadrant III  246.8
( 58 , 246.8)
Practice Problems
1.) convert into (x , y): (6 , 240)
2.) convert into (r , θ) : (4 , -2)
Practice Problems
1.) convert into (x , y): (6 , 240)
1
x  6 cos 240  6( )   3
2
3
y  6 sin 240  6( )   3 3
2
(3 ,  3 3)
Practice Problems
2.) convert into (r , θ) : (4 , -2)
r  4  (2) 
2
2
20  2 5
2
  tan    26.6 or 333.4
 4 
1
(2 5 ,  26.6)
or (2 5 , 333.4)
Equation Conversion
• equations in polar form have r in terms of θ
example : r = 4cosθ
• these equations can be graphed using
the calculator or by hand (section 6-5)
• To convert equations between
rectangular form and polar form use:
x  r cos 
y  r sin 
r x y
2
2
2
Example #3 Convert into a rect. equation
r  4 cos
r  4r cos 
(multiply both sides by r)
x  y  4x
(substitut e w/ converters )
2
2
2
x2  4x 
y2  0
(subtract 4x)
x 2  4 x  4  y 2  4 (complete the square)
(x  2 )  y  4
2
2
(factor)
This is the equation of a circle w/ center at (2 , 0) and radius 2
Example #4 Convert into polar equation
( x  3)  ( y  2)  13
2
2
this is the equation of a circle with center at (3 , 2)
and radius of 13
I’ll do this problem on the board.
Practice Problem #3 Convert into a
rectangular equation: r  5 csc
r  5 csc 
r5
sin 
r sin   5
y5
a horizontal line
Distance Between Two Polar Coordinates
• use Law of Cosines
• the two r values are the sides and θ can be
found by taking the difference between the
two angles
• See textbook example #7 for details
r1
θ
r2