Polar Coordinates

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Polar Coordinates

• We Live on a Sphere

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Polar Coordinates

• Up till now, we have graphed on the Cartesian
  plane using rectangular coordinates
• In the rectangular coordinate system a point is
  plotted as (x, y).

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Polar Coordinate

• In a polar coordinate system, we select a pint,
  called to pole, and then a ray with vertex at the
  pole, called the polar axis.
• We still use an ordered pair to graph.
• The new ordered pair is (r, ?).
• If r gt 0, then r is the distance of the point
  from the pole (like the origin)

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Polar Coordinates

• ? is the angle (in degrees or radians) formed by
  the polar axis and a ray from the pole.
• We call the ordered pair (r, ?) the polar
  coordinates of the point.

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Polar Coordinates

• Since angles have several different ways to name
  them, there are an infinite number of polar
  coordinates for each point. (Unlike rectangular
  coordinates which have only one name for point on
  the Cartesian plane.)

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Polar Coordinates

• Find four names for the point
• We are given a positive radius and a positive
  angle. We want to find a positive angle and a
  negative r, a pos r and neg angle, and a neg
  angle and neg r.

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Steps for finding other polar coordinates

• 1. Subtract 360o (or 2p) to get a negative angle
• 2. Add 180o (or p) to change the r to negative
  (half-way around the circle to be on the other
  side of the polar graph)
• 3. Add or subtract 360o (or p) to find the other
  angle

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Example

• Graph
• Examples

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Conversion from Polar Coordinates to Rectangular
Coordinates

• If P is a point with polar coordinates (r, q),
  the rectangular coordinates
• (x, y) of P are given by
• x r cos ? y r sin ?
• Remember

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Examples

• Find the rectangular coordinates of the points
  with the following polar coordinates

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Examples
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Polar to Rectangular Coordinates

• You can check your answers using your calculator.
• First do 2nd APPS
• Choose
• Put in polar coordinates
• Hit enter

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Steps for Converting from Rectangular to Polar
Coordinates

• 1. Always plot the point (x, y) first
• 2. To find r, r2 x2 y2 (Look familiar?)
• 3. To find q, remember that we only know x and y.
  Therefore, the trig value that we can use
  involves only x and y tangent.

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Converting from Rectangular Coordinates to Polar
Coordinates

• Find polar coordinates of a point whose
  rectangular coordinates are
• a. (0, 3)
• b. (2, -2)
• c. (-3, 3)

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Transforming an Equation from Polar to
Rectangular Form

• Transform the equation r cos q.
• We do not have a formula for just
• cos q, but we do have one for r cos q.
• Multiply both sides by r to get r cos q.
• That gives us r2 r cos q

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Transforming an Equation from Polar to
Rectangular Form

• r2 x2 y2 and r cos q x
• So, x2 y2 x
• This is the equation of a circle
• Find the answer by completing the square

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Transforming an Equation from Polar to
Rectangular Form

• (x2 x ) y2 0 (complete the square)
• (x2 x ¼) y2 ¼
• (x - ½)2 y2 ¼
• This is a circle whose center is (½, 0) and whose
  radius is ½

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Transforming an Equation from Rectangular to
Polar Form
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Transforming from Rectangular to Polar Form

• y2 2x
• (r sin q)2 2 r cos q
• r2 sin2 q 2r cos q

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Examples

• More Examples
• Tutorials