section 6.5 notes
DESCRIPTION
Section 6.5 Notes. 1 st Day. Today we will learn how to change from rectangular (standard) complex form to trigonometric (polar) complex form and vice versus. A complex number z = a + bi can be represented as a point ( a , b ) in a coordinate plane called the complex plane . - PowerPoint PPT PresentationTRANSCRIPT
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Section 6.5 Notes
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1st Day
Today we will learn how to change from rectangular (standard) complex form to trigonometric (polar) complex form and vice versus.
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A complex numberz = a + bi
can be represented as a point (a, b) in a coordinate plane called the complex plane.The horizontal axis is called the real axis and the vertical axis is called the imaginary axis.
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real axis
imaginary axis
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Example 1
Graph.a. 2 + 3ib. -1 – 2i
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I
R
(2, 3)2 + 3i
(-1, -2)
-1 - 2i
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In Algebra II you learned how to add, subtract, multiply, and divide complex numbers.In Pre-Calculus you will learn how to work with powers and roots of complex numbers.To do this you must write the complex numbers in trigonometric form (or polar form).
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On the next slide you will see how we change a rectangular (standard) complex number into a trigonometric (polar) complex number.
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θ
(a, b)
ab
r
I
R
cos ar
a cosr
sin br
b sinr
2 2a b
a + bi
r
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The trigonometric form of the complex number z = a + bi is
z = r(cos θ + i sin θ)where a = r cos θ, b = r sin θ,
In most cases 0 ≤ θ < 2π or 0° ≤ θ < 360°.
2 2 , and tan . br a ba
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There is a shortcut for writing a trigonometric complex number. The shortcut is
z = rcis θ = r(cos θ + i sin θ)
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Example 2
Write the complex number z = 6 – 6i in trigonometric (polar) form in radians.1. The point is in what quadrant?
4th quadrant
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226 6 r 6 2
6tan 16
2. Find r.
3. Find θ.
74
Remember θ is in the 4th quad.
7 76 2 cos sin4 4
z i 76 2cis
4
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Example 3Represent the complex number graphically and then find the rectangular (standard) form of the number. No rounding.
z = 6(cos 135° + isin 135°)
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135°
6
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Find a and b.
6cos135 a 262
6sin135 b 262
3 2 3 2 z i
3 2
3 2
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Now we will learn how to multiply and divide complex numbers in trigonometric (polar) form.
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Product of Complex Numbers
1 1 1 1 2 2 2 2Let cos sin and cos sin
be complex numbers.
z r i z r i
1 2 1 2 1 2 1 2
1 2 1 2
cos sin where
0 2 or 0 360
z z r r i
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Example 4
Find the product z1z2 of the complex numbers. Write your answer in standard form.
1 26 cos sin and 4 cos sin3 3 6 6
z i z i
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1 2 z z 6 4 cos sin3 6 3 6
i
24 cos sin2 2
i
24 0 1 i
24 i
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Quotient of Complex Numbers
1 1 1 1 2 2 2 2Let cos sin and cos sin
be complex numbers.
z r i z r i
1 11 2 1 2
2 2
1 2 1 2
cos sin
where 0 2 or 0 360
z r iz r
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Example 5
1
2
Find the quotient of the complex numbers.
Write your answer in standard form.
zz
1 26 cos40 sin 40 and 2 cos10 sin10 z i z i
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1
2
6 cos40 sin 402 cos10 sin10
iz
z i
6 cos 40 10 sin 40 102
i
3 cos 30 sin 30 i
3 132 2
i
3 3 32 2
i
END OF 1ST DAY
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2nd DayToday we will learn to:1. Raise complex numbers to a power.2. Find the roots of complex numbers.
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Multiply:(4 + 2i)10
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DeMoivre’s Theorem
If z = r(cos θ + isin θ) is a complex number and n is a positive integer, then
cos sin nnz r i
cos sin n nz r n i n
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Example 6
12Use DeMoivre's Theorem to find 1 3 .
Write your answer in standard form.
i
In what quadrant is this complex number?2nd Quadrant
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1. Change to polar form.
2. Find θ.
221 3 r 2
3tan 31
23
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12
12 2 21 3 2 cos sin3 3
i i
12 12 2 12 22 cos sin
3 3
i
4096 cos8 sin8 i
4096 1 0 i
4096
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The nth Root of a Complex Number
The complex number u = a + bi is an nth root of the complex number z if
z = un = (a + bi)n
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Finding the nth Root of a Complex Number
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For a positive integer n, the complex number z = r(cos θ + i sin θ)
has exactly n distinct nth roots given by
2 2cos sin
where 0,1,2,..., 1
n k kr in n
k nor
360 360cos sin
where 0,1,2,..., 1
n k kr in n
k n
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Example 7
Find all the fourth roots of 1.This means: x4 = 1.1. Change 1 into a polar complex number.
1 = cos(0π) + isin(0π)r = 1, n = 4 and k = 0, 1, 2, 3
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k = 0
0 2 0 0 2 01 cos sin
4 4
x i
1
1 cos0 sin 0 i
1 1 0 i
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k = 1 0 2 1 0 2 1
1 cos sin4 4
x i
i
1 cos sin2 2
i
1 0 i
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k = 2
0 2 2 0 2 21 cos sin
4 4
x i
1
1 cos sin i
1 1 0 i
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k = 3
0 6 0 61 cos sin4 4
x i
i
3 31 cos sin2 2
i
1 0 i
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Notice that the roots in example 2 all have a magnitude of 1 and are equally spaced around the unit circle. Also notice that the complex roots occur in conjugate pairs.The n distinct nth roots of 1 are called the nth roots of unity.
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Example 8
Find the three cube roots of z = -6 + 6i to the nearest thousandth.This means x3 = -6 + 6i.
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1. Change to polar complex form in degrees.
2. Now find the three cube roots.
72cis 135 z
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k = 0
6 135 072cis3
x
6 72cis 45
1.442 1.442 i
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k = 1
6 135 36072cis3
x
6 72cis 165
1.970 0.528 i
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k = 2
6 135 72072cis3
x
6 72cis 285
0.528 1.970 i