polar coordinates packet 1
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Polar Coordinates
Packet 1
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Polar Coordinates
Recording the position of an object using thedistance from a fixed point and an angle made from
that point uses a polar coordinate system.
When surveyors record the locations of objects using
distances and angles, they are using polarcoordinates.
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Polar Coordinate System
In a polar coordinatesystem, a fixed point O
is called the pole or
origin. The polar axis is
usually a horizontal raydirected toward the right
from the pole.
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Polar Coordinate System
The location of a point Pin the polar coordinate
system can be identified
by polar coordinates in
the form (r,
). If a ray is drawn from
the pole through point P,
the distance from the
pole to point P is r.
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Polar Coordinate System
The measure of theangle formed by andthe polar axis is . The
angle can be measured
in degrees or radians. This grid is sometimes
called the polar plane.
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Consider positive and negative values for r
Suppose r > 0. Thenis the measure of any
angle in standard
position that has as
its terminal side.
Suppose r < 0. Thenis the measure of any
angle that has the ray
opposite as its
terminal side.
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The angle
As you have seen, the r-coordinate can be any realvalue. The angle can also be negative. If > 0,
then is measured counterclockwise from the polar
axis. If < 0, then is measured clockwise from the
polar axis. Look at examples 1 and 2.
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Example 2
In this example, the point R(-2, -135) lies in thepolar plane 2 units from the pole on the terminal side
of a 45 angle in standard position.
This means that the point R could also be
represented by the coordinates (2, 45)
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Polar Coordinates
In general, the polar coordinates of a point are notunique. Every point can be represented by infinitely
many pairs of polar coordinates. This happens
because any angle in standard position is coterminal
with infinitely many other angles.
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Polar Coordinates
If a point has polar coordinates (r, ), then it also haspolar coordinates (r, + 2) in radians or (r, +360) in degrees.
In fact, you can add any integer multiple of 2to
and find another pair of polar coordinates for thesame point.
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Polar Coordinates
If you use the opposite r-value, the angle will changeby , giving (-r, + ) as another ordered pair for thesame point.
You can then find even more polar coordinates for
the same point by adding multiples of 2 to + .
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Polar Coordinates
The following graphs illustrate six of the differentways to name the polar coordinates of the same
point.
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In summary
Here is a summary of all the ways to represent apoint in polar coordinates:
If a point P has polar coordinates (r, ), then P can also
be represented by polar coordinates (r, + 2k) or (-r, +
(2k + 1)) , where k is any integer.
Note: In degrees, the representations are (r, +
360k) and (-r, + (2k + 1)180). For every angle
there are infinitely many representations.
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Polar Equations
An equation expressed in terms of polar coordinatesis called a polar equation. For example r = 2 sin is
a polar equation.
A polar graph is the set of all points whose
coordinates (r, ) satisfy a given polar equation.
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Graphing Polar Equations
You already know how to graph equations in theCartesian, or rectangular, coordinate system.
Graphs involving constants like x = 2 and y = -3 are
considered basic in the Cartesian coordinate system.
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Graphing Polar Equations
Similarly, the polar coordinate system has somebasic graphs. Graphs of the polar equations r = kand = k, where k is a constant, are considered
basic.
Look at example 4.
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Example
Graph each point.a. S(-4, 0)
b. R
c. Q(-2, -240)
2
3,2
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Example
Name four differentpairs of polar
coordinates that
represent point S
on the graph with
the restriction that -360 < < 360.
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Example: Graph each polar equation.
a. r = -3 b. 6
5
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HW: #17-39 odd