Polar Coordinates and Polar Graphs

Graph Q with polar coordinates:

Q = (3, 450)

Q = (-2, -300)

Q = (4, p / 2)

Name 3 other polar coordinates that will be point Q in the last example.
 

Rectangular to Polar Coordinates:

The rectangular coordinates (x,y) of a point P that has polar coordinates (r,q) can be found using the following conversion formulas:
x = r cos q                 x2 + y2 = r2  may also be used.
y = r sin q

Example 1:
Change (2, -400) to rectangular coordinates:

x = 2 cos (-400)  =  1.532                           (x,y)= (1.532, -1.286)
y = 2 sin (-400) = -1.286

Example 2:

Change (-2, 3p/4) to rectangular coordinates:

x = -2 cos (3p/4)  =  -2(-\/2 / 2) = \/2                           (x,y)= (\/2 , \/2 )
y = 2 sin (3p/4) = 2(\/2 / 2) = \/2

Example 3:

Express each equation in rectangular form:

(1)  r = 5 cos q
      Multiply both sides by r.
      Then  r2 = 5 r cos q
      So   x2 + y2 = 5x

(2)  r =   ___   6_______
                  2 – 3 sin  q
      Multiply both sides by the denominator.
      2 r - 3 r sin q = 6
      2\/( x2 + y2) - 3y = 6
      2\/( x2 + y2) = 3y + 6
      4( x2 + y2) = 9y2 + 36y + 36
      4 x2 + 4 y2 = 9y2 + 36y + 36
      4 x2 - 5 y2 -  36y = 36

Polar to Rectangular Coordinates:

A pair of polar coordinates (r,q) of a point named by the rectangular coordinates (x,y) can be found using the following formulas:
x2 + y2 = r2        q = Arctan y/x

Example 1:
Change (-3,3) to polar coordinates:
 9 + 9 = r2             arctan (3/ -3) = arctan (-1) = 135o  since the point is in the second quadrant
     18 = r2
   3\/2 = r                     (3\/2 , 135o)

Example 2:
Find the polar equation for each of the following::
(1)  x2 + y2 = 4
              r2 = 4

(2)  y = 5
      r sin q = 5
      r = 5 / sin q

(3)  2x – y = 4
      2r cos q - r sin q = 4
      r = 4 / (2 cos q - sin q)
 

Graph and find the rectangular equation:

r = 3 makes a circle of radius 3.  (x2 + y2 = 9)

q = p/4  makes a straight line along the angle  (y = x)

r = sec q  makes a vertical line  (x = 1)
 

Symmetries:

x-axis:   If  (r, q) is on the graph, so is (r, -q) or (-r, p - q).

y-axis:   If  (r, q) is on the graph, so is (-r, -q) or (r, p - q).

origin:    If  (r, q) is on the graph, so is (r, p + q) or (-r, q).
 

Types of Graphs:

r = a + b cos q  or  r = a + b cos q               limacon

                                If  0 < a/b  < 1, the limacon will have an inner loop of radius |a| while the
                                   maximum radius of the limacon is |a| + |b|.
                                If 1 < a/b < 2, the limacon will have a dimple of |a| - |b| and the maximum
                                   radius is |a| + |b|.  (also called a cardioid)
                                If a/b = 1, the limacon will have a dimple of 1 and the maximum
                                   radius is |a| + |b|.  (also called a cardioid)
                                If a/b > 2, the limacon will be convex of |a| - |b| and a maximum radius
                                   of |a| + |b|.

r = a cos q  or  r = a sin q                                  circle

                                 The cos will be oriented along the x-axis.
                                 The sin will be oriented along the y-axis.
                                 a is the diameter.

r = a cos nq   or   r = a sin nq                               roses

                     The cos will have one petal on the positive x-axis.
                                 The sin will have a petal on the y-axis.
                                a is the radius of each petal.
                                 n is the number of petals if n is odd.
                                2n is the number of petals if n is even.

q = a                                                                      line

r = a                                                                       circle

                                  radius is a and centered at origin

r = q                                                                      Archimedian spiral

r2 = b2 sin 2q or r2 = b2 cos 2q                          lemniscate

                       b is the radius of one petal
                                   sin is oriented through first and third quadrants
                                  cos lies on the x-axis.

r = a/q                                                                    hyperbolic spiral

r = sec q                                                                vertical line

r = csc q                                                                  horizontal line