MTH 253 Calculus (Other Topics) Chapter 11 – Analytic Geometry in Calculus Section 11.1 – Polar Coordinates Copyright © 2006 by Ron Wallace, all rights reserved.

Rectangular (aka: Cartesian) Coordinates positive x-axisnegative x-axis positive y-axis negative y-axis x y (x, y) origin For any point there is a unique ordered pair (x, y) that specifies the location of that point.

Polar Coordinates polar axis (r,  ) r  pole Is (r,  ) unique for every point? NO! All of the following refer to the same point: (5, 120º) (5, 480º) (-5, 300º) (-5, -60º) etc... The angle  may be expressed in degrees or radians.

Polar Graph Paper Locating and Graphing Points 00 30  60  90  180  120  150  210  240  270  300  330  (5, 150  ) (6, 75  ) (3, 300  ) (3, -60  )(-3, 120  ) (-4, 30  ) (7, 0  ) (-7, 180  )

Converting Coordinates Polar  Rectangular x y (r,  )  (x, y)  r Recommendation: Find (r,  ) where r > 0 and 0 ≤  < 2  or 0  ≤  < 360 . Relationships between r, , x, & y R  P P  R

Examples: Converting Coordinates Polar  Rectangular

Examples: Converting Coordinates Polar  Rectangular Quadrant I

Examples: Converting Coordinates Polar  Rectangular Quadrant II OR

Examples: Converting Coordinates Polar  Rectangular Quadrant III OR

Examples: Converting Coordinates Polar  Rectangular Quadrant IV OR

Converting Equations Polar  Rectangular Use the same identities:

Converting Equations Polar  Rectangular x  Replace all occurrences of x with r cos . y  Replace all occurrences of y with r sin .  Simplify r Solve for r (if possible).

Converting Equations Polar  Rectangular  Express the equation in terms of sine and cosine only.  If possible, manipulate the equation so that all occurrences of cos  and sin  are multiplied by r.  Replace all occurrences of …  Simplify (solve for y if possible) r cos  with x r sin  with y r 2 with x 2 + y 2 Or, if all else fails, use:

Graphing Polar Equations  Reminder: How do you graph rectangular equations? Method 1:  Create a table of values.  Plot ordered pairs.  Connect the dots in order as x increases. Method 2:  Recognize and graph various common forms.  Examples: linear equations, quadratic equations, conics, … The same basic approach can be applied to polar equations.

Graphing Polar Equations Method 1: Plotting and Connecting Points 1.Create a table of values. 2.Plot ordered pairs. 3.Connect the dots in order as  increases. NOTE: Since most of these equations involve periodic functions (esp. sine and cosine), at some point the graph will start repeating itself (but not always).

Graphing Polar Equations Method 1: Plotting and Connecting Points wrt x-axis Replacing  with - doesn’t change the function Symmetry Tests (r,) (r,-)

Graphing Polar Equations Method 1: Plotting and Connecting Points wrt y-axis Replacing  with  -  doesn’t change the function Symmetry Tests (r,) (r,-)

Graphing Polar Equations Method 1: Plotting and Connecting Points wrt the origin Replacing r with –r doesn’t change the function. Replacing  with    doesn’t change the function. Symmetry Tests (r,) (-r,) (r,  )

Graphing Polar Equations Method 2: Recognizing Common Forms  Circles Centered at the origin: r = a  radius: a period = 360 Tangent to the x-axis at the origin: r = a sin   center: (a/2, 90) radius: a/2 period = 180  a > 0  above a < 0  below Tangent to the y-axis at the origin: r = a cos   center: (a/2, 90) radius: a/2 period = 180  a > 0  right a < 0  left r = 4 r = 4 sin  r = 4 cos 

Graphing Polar Equations Method 2: Recognizing Common Forms  Flowers (centered at the origin) r = a cos n or r = a sin n  radius: |a|  n is even  2n petals petal every 180/n period = 360  n is odd  n petals petal every 360/n period = 180  cos  1 st 0  sin  1 st 90/n r = 4 sin 2  r = 4 cos 3 

Graphing Polar Equations Method 2: Recognizing Common Forms  Spirals Spiral of Archimedes: r = k  |k| large  loose |k| small  tight r =  r = ¼  Other spirals … see page 726.

Graphing Polar Equations Method 2: Recognizing Common Forms  Heart (actually: cardioid if a = b … otherwise: limaçon) r = a ± b cos  or r = a ± b sin  r = cos  r = cos  r = sin  r = sin 

Graphing Polar Equations Method 2: Recognizing Common Forms  Lines Through the Origin: y = mx   = tan -1 m Horizontal: y = k  r sin  = k  r = k csc  Vertical: x = h  r cos  = h  r = h sec  Others: ax + by = c  y = mx + b 

Graphing Polar Equations Method 2: Recognizing Common Forms  Parabolas (w/ vertex on an axis) NOTE: With these forms, the vertex will never be at the origin.

Graphing Polar Equations Method 2: Recognizing Common Forms  Parabolas (w/ vertex at the origin)

Graphing Polar Equations Method 2: Recognizing Common Forms  Leminscate a = 16 Replacing the 2 w/ n will give 2n petals if n is odd and n petals if n is even. (these are not considered to be leminscates)