Polar Coordinates

1
Polar Coordinates

• 6.4

2
Why polar coordinates??

• Radar systems use polar coordinates. A rotating
  antennae sends out pulsed radio beams. The beams
  return after they are reflected. The time it
  takes for the signal to return is known and so
  the distance from the radar system to the object
  can be calculated. Because the antennae is
  rotating, the angle is also know. This data is a
  polar coordinate.

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Polar Coordinates

• A polar coordinate is (r,?). This is the point
  where the angle ? intersects a circle of radius r.

4
Polar Coordinates--r

• r is directed, and therefore will follow the
  angle if positive and travel opposite to the
  angle if negative.

5
Polar Coordinates-- ?

• As with the angles in trigonometry, positive ?
  travels counter-clockwise and negative ? travels
  clockwise.

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Polar Coordinates Examples

• (3,p/4)to plot, draw a circle of radius three
  centered at the origin. Mark the point where the
  angle drawn in standard position intersects the
  circle.

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Polar Coordinates Examples

• (-2,-p/6)to plot, draw a circle of radius two
  centered at the origin. Mark the point where the
  angle drawn in standard position intersects the
  circle. Connect this point through the origin to
  the other side of the circle. Mark this point as
  the result.

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Polar Coordinates Examples

• (-2,-p/6)Try plotting (2, 5p/6). Notice that
  you end up in the same place, so the two angles
  are equivalent. Therefore, each polar coordinate
  has many equivalent coordinates.

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Polar Coordinate Equivalents

• (-2,-p/6) is equivalent to (2, 5p/6).
• (r, ? 2np) in this case (-2, 11p/6),
  (-2,23p/6), etc.
• (-r, ? (2n1)p) in this case (2, 5p/6), (2,
  17p/6), etc.

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Polar Coordinate Equivalents

• (r, ?) is equivalent to
• (r, ? 2np) and
• (-r, ? (2n1)p)

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Converting From Polar to Cartesian

• Let the point P have polar coordinates (r, ?) and
  rectangular coordinates (x,y). Then
• x r cos ? r2 x2 y2
• y r sin ? tan ? y/x

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Converting From Polar to Cartesian

• Examples2,4

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Equivalent Polars Example

• 24,28

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Polar Graphing on the Calculator

• Select the mode button and choose Pol.
• The put the equation in y, which is r1.
• Try zoom 6 and then press the window key to
  change ?min, ?max, and ?Step.
• ?Step just changes the number of calculations the
  calculator makes for plots.

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Equation Converting

• Examples--36,38,44,46

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Graphs of Polar Equations

• 6.5

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Horizontal Line--r sin ? a

• Is a horizontal line a units above the pole or
  origin if agt0 and a units below the pole if
  alt0.
• Assume that a is not zero.

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Vertical Line--r cos ? a

• Is a vertical line a units to the right of the
  pole or origin if a is positive and a units to
  the left of the pole if a is negative.

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Line--? ?o

• The line which makes an angle ?o of with the
  x-axis. The angle is fixed at a given value, but
  the radius can be anything, so this creates a
  line that creates the fixed angle with the x-axis.

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Circle--ra

• Circle of radius a centered at the origin. The
  radius is fixed at a, but the angle can be
  anything.

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Circle--r /-2a sin ?

• Circle with radius a and passes through the pole.
• Sine will orient along the y-axis
• Centered at (0,/-a) in rectangular coordinates

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Circle--r /-2a cos ?

• Circle with radius a and passes through the pole.
• Cosine will orient on the x-axis.
• In rectangular coordinates, the center will be at
  (/-a,0)

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Archimedes Spiral--Ra?

• The angle varies as the radius varies. In fact,
  as the angle gets larger, so does the radius.

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Rose Curves

• r a cos n? or r a sin n?

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Cardiods

• ra(1/- cos ?)
• Or
• ra(1 /- sin ?)

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Limacons

• R a /- b cos ?
• R a /- b sin ?
• If agt0, and bgt0, and agtb, then there is no inner
  loop.
• If agt0, and bgt0, and altb, then there is an inner
  loop.

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Lemniscates

• r2 a2sin(2?) or
• r2 a2cos(2?)

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Symmetry

• With Respect to the Polar Axis
• In the polar equation, replace ? with ?. If an
  equivalent equation results, the graph is
  symmetric with respect to the polar axis.
• With Respect to the Line ? p/2
• In a polar equation, replace ? with (p ?). If
  an equivalent equation results, the graph is
  symmetric with respect to the polar axis.
• With Respect to the Pole
• In a polar equation, replace r with r. If an
  equivalent equation results, then graph is
  symmetric with respect to the pole.

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Examples

• 14-20 evens

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Maximum R-Value

• The r value of the polar coordinate that is the
  maximum distance from the pole.
• Use the calculator to graph the function and use
  trace to help determine this value.

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Examples

• 22,24