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Polar Coordinates

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2,-p/6) to plot, draw a circle of radius two centered at the origin. ... Rose Curves. r = a cos n? or r = a sin n?. Cardiods. r=a(1 /- cos ?) Or. r=a(1 /- sin ?) ... – PowerPoint PPT presentation

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Title: 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.

3
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.

6
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.

7
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.

8
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.

9
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.

10
Polar Coordinate Equivalents
  • (r, ?) is equivalent to
  • (r, ? 2np) and
  • (-r, ? (2n1)p)

11
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

12
Converting From Polar to Cartesian
  • Examples2,4

13
Equivalent Polars Example
  • 24,28

14
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.

15
Equation Converting
  • Examples--36,38,44,46

16
Graphs of Polar Equations
  • 6.5

17
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.

18
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.

19
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.

20
Circle--ra
  • Circle of radius a centered at the origin. The
    radius is fixed at a, but the angle can be
    anything.

21
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

22
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)

23
Archimedes Spiral--Ra?
  • The angle varies as the radius varies. In fact,
    as the angle gets larger, so does the radius.

24
Rose Curves
  • r a cos n? or r a sin n?

25
Cardiods
  • ra(1/- cos ?)
  • Or
  • ra(1 /- sin ?)

26
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.

27
Lemniscates
  • r2 a2sin(2?) or
  • r2 a2cos(2?)

28
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.

29
Examples
  • 14-20 evens

30
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.

31
Examples
  • 22,24
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