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Polar Coordinates

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Title: Polar Coordinates


1
Polar Coordinates
  • We Live on a Sphere

2
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).

3
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)

4
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.

5
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.)

6
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.

7
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

8
Example
  • Graph
  • Examples

9
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

10
Examples
  • Find the rectangular coordinates of the points
    with the following polar coordinates

11
Examples
12
Polar to Rectangular Coordinates
  • You can check your answers using your calculator.
  • First do 2nd APPS
  • Choose
  • Put in polar coordinates
  • Hit enter

13
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.

14
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)

15
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

16
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

17
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

18
Transforming an Equation from Rectangular to
Polar Form
19
Transforming from Rectangular to Polar Form
  • y2 2x
  • (r sin q)2 2 r cos q
  • r2 sin2 q 2r cos q

20
Examples
  • More Examples
  • Tutorials
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