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Polar Coordinates

Dr .Hayk Melikyan/ Departmen of Mathematics and CS/ melikyan@nccu.edu

1. Plotting Points Using Polar Coordinates

2. Determining Different Representations of a Point

3. Converting a Point from Polar Coordinates to Rectangular Coordinates

4. Converting a Point from Rectangular Coordinates to Polar Coordinates

5. Converting an Equation from Rectangular Form to Polar Form

6. Converting an Equation from Polar Form to Rectangular Form

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Plotting Points Using Polar Coordinates

Plot the following points in a polar coordinate system.

a.

b.

c.

4,4

A

3,120B

32,

4C

A

B

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Determining Different Representations of the Same Point

The point is shown. Determine a different

representation of point P that have the specified conditions.

54,

6P

a. 0, 2 0r

b. 0,0 2r

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Determining Different Representations of the Same PointThe point is shown.

r > 0; negative angle; coterminal with given angle

54,

6P

a. 0, 2 0r

52

65

4, 2 ( )6

5 5 12 74, 2 4, 4,

6 6 6

1

6

k

P

P P P

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Determining Different Representations of the Same Point

The point is shown.

r < 0; positive angle; coterminal with given angle

54,

6P

b. 0,0 2r 5

26

54, 2 ( )

6

5 5 6 114, 2 4, 4,

6 6 6 6

0

k

P

P P P

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Relationships Used when Converting a Point from Polar Coordinates to Rectangular Coordinates

cos and sinx r y r

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Converting a Point from Polar Coordinates to Rectangular Coordinates

Determine the rectangular coordinates for the given polar coordinates.

6,6

cos

6cos6

36 3 3

2

x r

x

x

sin

6sin6

16 3

2

y r

y

y

6, 3 3,36

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Converting a Point from Polar Coordinates to Rectangular Coordinates

Determine the rectangular coordinates for the given polar coordinates.

4,4

cos

4cos4

24 2 2

2

x r

x

x

sin

4sin4

24 2 2

2

y r

y

y

4, 2 2,2 24

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Converting Rectangular Coordinates to Polar Coordinates for Points Lying Along an Axis

Determine the polar coordinates for the given rectangular coordinates.

The point lies along the negative x-axis. So The distance from the origin to the point is 4.5 units, so r = 4.5.

4.5,0

4.5,

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Converting a Point from Rectangular Coordinates to Polar Coordinates

Determine the polar coordinates for the given rectangular coordinates.

Find the value of r.

2, 2

4.5,

2 2

2 22 ( 2)

8 2 2

r x y

r

r

The point (2, 2) lies in

Quadrant IV.

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Converting a Point from Rectangular Coordinates to Polar Coordinates-cont

Find the value of .

tan

2tan 1

2

R

R

y

x

Use inverse tangent.1tan (1)R

Determine the point.12 tan (1) 5.50

Polar Coordinates

2 2,5.50

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Converting an Equation from Rectangular Form to Polar Form

Convert the equation given in rectangular form into polar form.

Replace x with r cos .

9x

9

cos 9

9

cos

x

r

r

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Converting an Equation from Rectangular Form to Polar Form

Convert the equation given in rectangular form into polar form.

Replace x with r cos and y with r sin

3 12x y

3 12

3 cos sin 12

(3cos sin ) 12

12

3cos sin

x y

r r

r

r

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Converting an Equation from Polar Form to Rectangular Form

Convert the equation given in rectangular form into polar form.

Replace r cos with x and r sin with y

3 cos 7 sin 15r r

3 cos 7 sin 15

3 7 15

3 15

7 7

r r

x y

or y x

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Converting an Equation from Polar Form to Rectangular Form

Convert the equation given in rectangular form into polar form.

First multiply both sides of the equation by r.

8cosr

2

2 2

8cos

8 cos

8

r

r r

x y x

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Graphs in Polar coordinates Sketching Equations of the Form

Sketching Equations of the Form

Sketching Equations of the Form

Sketching Equations of the Form

Sketching Equations of the Form

, cos ,r

sin , and cos sinr ar br c

, sin ,r a r a and cosr a

sinr a b and cosr a b

sinr a n and cosr a n2 2 sin 2r a 2 2and cos2r a

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Sketching the Graph of a Polar Equation

Sketch the graph of the polar equation

The graph is a line passingthrough the pole that makes an angle of 2/3 with the polar axis.

2

3

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Sketching the Graph of a Polar Equation

Sketch the graph of the polar equation cos 2r cos 2

2

cos

r

r

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Sketching the Graph of a Polar Equation

Sketch the graph of the polar equation

The graph is a horizontal line equivalent to the equation y = 3.

sin 3r

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Sketching the Graph of a Polar Equation

Sketch the graph of the polar equation

Convert the equation to rectangular form.3 cos 2 sin 6r r

3 cos 2 sin 6

3 2 6

33

2

r r

x y

y x

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Sketching the Graph of a Polar Equation

Sketch the graph of the polar equation

The graph is a circle centered at the pole with radius of length |3| = 3.

3r

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Sketching the Graph of a Polar Equation

Sketch the graph of the polar equation Convert the equation to rectangular form.

4sinr

2

2 2

4sin

4 sin

4

r

r r

x y y

Complete the square.

2 2

2 2

2 2

2 2

4

4 0

4 4 4

( 2) 4

x y y

x y y

x y y

x y

The equation is a circle with center located 2 units directly above the pole with a radius of 2 units.

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Sketching the Graph of a Polar Equation

Sketch the graph of the polar equation Make a table of values. 3 3sinr

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Sketching the Graph of a Polar Equation-cont

Sketch the graph of the polar equation 3 3sinr

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Sketching the Graph of a Polar Equation-cont

Sketch the graph of the polar equation 3 3sinr

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Sketching the Graph of a Polar EquationSketch the graph of the polar equation 1 2cosr

The graph is symmetric; complete

the graph by drawing the reflection.

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Sketching the Graph of a Polar Equation

Sketch the graph of the polar equation 3sin 2r

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Sketching the Graph of a Polar Equation

Sketch the graph of the polar equationThe equation is a lemniscate. The length of each loop is 3 units.

2 9cos2r