4.4 DeMoivre

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4.4 DeMoivres Thm

• In this section we will concentrate on finding
  the powers of complex numbers

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• Lets consider the complex number
• z r (cis ?)

Then z2 r (cis ?) r (cis ?) r2 (cis
2?)
Then z3 r (cis ?) r (cis ?) r (cis ?)
r3 (cis 3?)
Then z4 r (cis ?) r (cis ?) r (cis ?) r
(cis ?) r4 (cis 4?)
So what would z12
r12 (cis 12?)
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De Moivres Thm

• zn r(cos? isin ?)n rn(cos n? isinn?)

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Now you try

• Use DeMoives theorem to find the indicated
  power of the complex number. Write the result in
  standard form.

(2 2i)6
Step 1 Change the complex number into trig form.
Tan ? 2/2 ? ?/ 4
So, z 2 2(cis ?/4)6
Step 2 Use DeMoives thm.
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So, z 512(cis 6?/4) 512(cis 3?/2)
Last step changes this back to standard form.
So, z 512(cis 6?/4) 512(cis 3?/2)

• So, z 512(cos 3?/2 isin 3?/2)

• 512( 0 i(-1))

-512i
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Now you try

• Use DeMoives theorem to find the indicated
  power of the complex number. Write the result in
  standard form.

(3 - 2i)8
Step 1 Change the complex number into trig form.
Tan ? -2/3 ? 33.69 However, this is
in quad IV, so ? 360-33.69 326.31
Step 2 Use DeMoives thm.
So, z (-0.0084) i(.99996)
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(r, ?)
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We are going to look at a new coordinate system
called the polar coordinate system.
You are familiar with plotting with a rectangular
coordinate system.
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The center of the graph is called the pole.
Angles are measured from the positive x axis.
Points are represented by a radius and an angle
radius
angle
(r, ?)
To plot the point
First find the angle
Then move out along the terminal side 5
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A negative angle would be measured clockwise like
usual.
To plot a point with a negative radius, find the
terminal side of the angle but then measure from
the pole in the negative direction of the
terminal side.
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Let's plot the following points
Notice unlike in the rectangular coordinate
system, there are many ways to list the same
point.
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Let's take a point in the rectangular coordinate
system and convert it to the polar coordinate
system.
(3, 4)
Based on the trig you know can you see how to
find r and ??
r
4
?
3
r 5
We'll find ? in radians
(5, 0.93)
polar coordinates are
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Let's generalize this to find formulas for
converting from rectangular to polar coordinates.
(x, y)
r
y
?
x
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Now let's go the other way, from polar to
rectangular coordinates.
Based on the trig you know can you see how to
find x and y?
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y
x
rectangular coordinates are
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Let's generalize the conversion from polar to
rectangular coordinates.
r
y
x
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Polar coordinates can also be given with the
angle in degrees.
(8, 210)
(6, -120)
(-5, 300)
(-3, 540)
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Convert the rectangular coordinate system
equation to a polar coordinate system equation.
?
Here each r unit is 1/2 and we went out 3 and did
all angles.
r must be ? 3 but there is no restriction on ? so
consider all values.
Before we do the conversion let's look at the
graph.
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Convert the rectangular coordinate system
equation to a polar coordinate system equation.
What are the polar conversions we found for x and
y?
substitute in for x and y
We wouldn't recognize what this equation looked
like in polar coordinates but looking at the
rectangular equation we'd know it was a parabola.