Polar Coordinates Packet 1

8/12/2019 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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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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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

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Polar Coordinates Packet 1