Polar Coordinates Polar and Rectangular Coordinates

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Polar CoordinatesPolar and Rectangular
Coordinates

• 9.3

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Polar vs. Rectangular Coordinates

• For some real-world phenomena, it is useful to be
  able to convert between polar coordinates and
  rectangular coordinates.

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Polar vs. Rectangular Coordinates

• Polar Coordinates P(r, ?)
• Rectangular Coordinates P(x, y)

• Suppose a rectangular coordinate system is
  superimposed on a polar coordinate system so that
  the origins coincide and the x-axis aligns with
  the polar axis, as shown at the right. Let P be
  any point in the plane.

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Polar to Rectangular Coordinates

• Trigonometric functions can be used to convert
  polar coordinates to rectangular coordinates.
• The rectangular coordinates (x, y) of a point
  named by the polar coordinates (r, ?) can be
  found by using the following formulas
• x r cos ?
• y r sin ?

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Rectangular to Polar Coordinates

• If a point is named by the rectangular
  coordinates (x, y), you can find the
  corresponding polar coordinates by using the
  Pythagorean Theorem and the Arctangent function
  (Arctangent is also known as the inverse tangent
  function).

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Rectangular to Polar Coordinates

• Since the Arctangent function only determines
  angles in quadrants 1 and 4 (because tangent has
  an inverse in quadrants 1 and 4) you must add p
  radians to the value of ? for points with
  coordinates (x, y) that are in quadrants 2 or 3.

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Rectangular to Polar Coordinates

• When x gt 0, ?

• When x lt 0, ?

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Rectangular to Polar Coordinates

• When x is zero, . Why?
• The polar coordinates (r, ?) of a point named by
  the rectangular coordinates (x, y) can be found
  by the following formulas
• r
• When x gt 0
• When x lt 0

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Converting Equations

• The conversion equations can also be used to
  convert equations from one coordinate system to
  the other.

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Example 1

• Find the rectangular coordinates of each point.
• a. b. C(3, 270)

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Example 2

• Find the polar coordinates of
• a) E(2, -4) b) F(-8, -12)

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Example 3

• Write the polar equation r -3 in rectangular
  form.
• Write the polar equation r 5 cos? in
  rectangular form.

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Example 4

• Write the rectangular equation x² (y 1)² 1
  in polar form.