ee101 l11 (a)polar coordinate and complex numbers

7/25/2019 EE101 L11 (a)Polar Coordinate and Complex Numbers

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EE101Calculus and AnalyticalGeometry 2

Polar Coordinates andComplex No.

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Polar Coordinates


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Polar Coordinates

Coordinates is a pair of numbers which

represents a point in the plane

Usually, we use Cartesian coordinates, which are

directed distances from two perpendicular axes.

Coordinate system is introduced by Newton,

called polar coordinate system. (It is more

convenient for many purposes)


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POLAR COORDINATES

In this section, we will learn :

How to represent points in polar coordinates


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Pole

Pole (or origin)A point in the plane which

labeled with O.

o


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Polar Axis

Then, we draw a ray (half-line) start at Ocalled

the polar axis.

oPolar axis x

This axis is usually drawn horizontally to the rightcorresponding to the positive x-axis in Cartesian

coordinates


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If Pis any other point in the plane, let

rbe the distance from Oto P.

be the angle (usually measured in radians)between the polar axis and the line OP.


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Polar Coordinates

Pis represented by the ordered pair (r,).

r,are called polar coordinates of P


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Polar Coordinates

An angle () is :

PositiveIf measured in the counter-clockwisedirection from the polar axis.

NegativeIf measured in the clockwise direction

from the polar axis.


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Polar Coordinates

The points (-r,) and (r,) lie on the same line

through O and at the same distance |r| from O,

but on opposite sides of O


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Polar Coordinates

If r > 0, the point (r,) lies in the same quadrat

as .

If r < 0, it lies in the quadrant on the opposite

side of the pole.

Notice that (-r,)represents the same

point as (r, +).


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Polar Coordinates

Example 1

Plot the points whose polar coordinates aregiven:

a. (1, 5/4)

b. (2, 3)

c. (2, -2/3)

d. (-3, 3/4)


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Solution

Example 1 a

The point (1, 5/4) is plotted here.


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Example 1 b

The point (2, 3) is plotted.


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Example 1 c

The point (2, -2/3) is plotted.


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Example 1 d

The point (-3, 3/4) is plotted.

It is located three units from

the pole in the fourthquadrant.

This is because the angle ()

3/4 is in the secondquadrant and r= -3 is

negative.


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Relating Polar toCartesianCoordinates


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Cartesian VS. Polar Coordinates

In the Cartesian coordinate system, every point

has only one representation.

However, in the polar coordinate system, each

point has many representations.


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Cartesian VS. Polar Coordinates

For instance, the point (1, 5/4) could be

written as:

(1, -3/4), (1, 13/4) or (-1, /4)


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Cartesian & Polar Coordinates

The relation between polar and Cartesian

coordinates can be seem here.

The pole corresponds to the

origin.

The polar axis coincides with

the positive x-axis.

i l i l


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Cartesian to Polar Conversion Formulas

If the point Phas Cartesian coordinates (x, y)and polar coordinates (r, ).

From the figure, we have:

cos sinx y

r r


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Therefore,

Equation formed are:

cos

sin

x r

y r

These two equations allow us

to find the Cartesian

coordinates of a point when

the polar coordinates are

known.


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Polar to Cartesian Conversion Formulas

To findrand whenxandyare known, we usethe equations

2 2 2tan

yr x y

x

These can be

simply read from

the figure


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Copyright 2005 PearsonEducation, Inc. Publishing as

Pearson Addison-Wesley


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ComplexNumbers


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38

The Imaginary Number

By definition

Consider powers if i

21 1i i

2

3 2

4 2 2

5 4

1

1 1 1

1

...

i

i i i i

i i i

i i i i i

It's any

number

you can

imagine


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39

Using

Now we can handle quantities thatoccasionally show up in mathematicalsolutions

1a a i a


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Complex Numbers

Combine real numbers withimaginary numbers

a + bi

Realpart

Imaginarypart

3 4i

3

6 2 i

4.5 2 6i

Example:


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42

Operations on omplex

Numbers

Complex numbers can be combined with

addition

subtraction

multiplication

division

Consider

3 8 2i i 9 12 7 15i i

2 4 4 3i i


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Operations on omplex

Numbers

Division technique

Multiply numerator and denominator by theconjugate of the denominator

3

5 2

i

i

2

2

3 5 2

5 2 5 2

15 6

25 4

6 15 6 15

29 29 29

i i

i i

i i

i

ii


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52

Warning

Consider

It is tempting to combine them

The multiplicative property of radicals only works for

positive values under the radical sign Instead use imaginary numbers

16 49

16 49 16 49 4 7 28

216 49 4 7 4 7 28i i i


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Relating Polar form with

Complex Number(Cartesian form)


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r = |z|

E 5 + 2i l b i l f


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Express 5 + 2i complex number in polar form

sinicosrz Express in

38.0sini38.0cos39.5z ANS:

E th P l F t


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6

7sini

6

7cos2z

oo210sini210cos2

i2

1

2

3

2z

i3

Express the Polar Form toComplex Number


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Express in polar form the following complex

numbers

i31z (a)

(b) i22z


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SUMMARY

To convert from Polar Coordinates (r, ) to CartesianCoordinates (x, y):

x = r cos ; y = r sin

To convert from Cartesian Coordinates (x, y) to PolarCoordinates (r, ):

22

yxr or 22 ba

a

btan

x

ytan 11


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Thank You

Prepared by:

Dr. Ng Ching Yin

[email protected]

ee101 l11 (a)polar coordinate and complex numbers

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