One reason for writing complex numbers inOne reason for writing complex numbers intrigonometric form is the convenience for multiplyingtrigonometric form is the convenience for multiplyingand dividing:and dividing:
• The The product product involves the product of the involves the product of the moduli moduli and the sum of the and the sum of the argumentsarguments
• The The quotientquotient involves the quotient of the involves the quotient of the moduli moduli and the difference of the and the difference of the argumentsarguments
2 2 2 2cosθ sin θz r i
1 2 1 2 1 2 1 2cos θ θ sin θ θz z r r i
1 11 2 1 2
2 2
cos θ θ sin θ θz r
iz r
2 0r
1 1 1 1cosθ sin θz r i Let
and
1.
. Then
2. ,
1
π π25 2 cos sin
4 4z i
2
π π14 cos sin
3 3z i
π π
350 2 cos sin 478.109 128.10912 12
i i
Express the product of the given complex numbers instandard form:
Product:
1 2 2 cos135 sin135z i
2 6 cos300 sin 300z i
2cos 165 sin 165 0.455 0.122
3i i
Express the quotient of the given complex numbers instandard form:
Quotient:
1
3 33 cos sin
4 4z i
2
1cos sin
3 6 6z i
1 2
3 11 11cos sin
3 12 12z z i
Express the product of the given complex numbers intrigonometric form:
1 5 cos 220 sin 220z i
2 2 cos115 sin115z i
1
2
5cos105 sin105
2
zi
z
Express the quotient of the given complex numbers intrigonometric form:
Find the product and quotient of the given complex numbersin two ways, (a) using standard forms, and (b) using trig. forms.
1 2 3z i 2 1 3z i The product:
1 2 2 3 1 3z z i i 22 2 3 3 3 3i i i
2 3 3 2 3 3 i 3.196 6.464i
The quotient:
1
2
2 3
1 3
z i
z i
1 3
1 3
i
i
2 3 3 2 3 3
4
i
1.799 0.116i
Find the product and quotient of the given complex numbersin two ways, (a) using standard forms, and (b) using trig. forms.
1 2 3z i 2 1 3z i Next, find the trigonometric forms:
1 13 cos 0.983 sin 0.983z i 2 2 cos 3 sin 3z i
The product:
1 2 2 13 cos 0.983 3 sin 0.983 3z z i 3.196 6.464i The quotient:
1
2
13cos 0.983 3 sin 0.983 3
2
zi
z 1.799 0.116i
First, let’s look at a problem: cosθ sin θz r i
2 cosθ sin θ cosθ sin θz z z r i r i
2 cos θ θ sin θ θr i 2 cos 2θ sin 2θr i
RealAxis
ImaginaryAxis
z
z2
Graphically:Graphically: 2θ
θrr 2
Now, let’s find the cube of z:3 2z z z
2cosθ sin θ cos 2θ sin 2θr i r i
3 cos θ 2θ sin θ 2θr i
3 cos3θ sin 3θr i
4 4 cos 4θ sin 4θz r i
5 5 cos5θ sin5θz r i
And the pattern continues for higher powers:
De Moivre’s TheoremDe Moivre’s TheoremThis pattern is generalized to give:
Let
and let n be a positive integer. Then
cosθ sin θz r i
cos sinnnz r i
cos θ sin θnr n i n
31 3i
πθ
3
Find using De Moivre’s Theorem.
Begin witha graph:
Modulus r = 2
Argument
2 cos sin3 3
z i
31 3i
31 3 8i
Find using De Moivre’s Theorem.
Verify with your calculator!!! Verify with your calculator!!!
2 cos sin3 3
z i
3 32 cos 3 sin 3
3 3z i
8 cos sini 8 1 0i 8
8
2 2
2 2i
Find using De Moivre’s Theorem.
Convert to trig. form:3 3
cos sin4 4
z i
8 3 3cos 8 sin 8
4 4z i
cos6 sin 6i 1 0i 1
31 3 8i
8
2 21
2 2i
1 3i2 2
2 2i
The complex number
is a third root of –8
The complex number
is an eighth root of 1
15 15243 cos sin 243
2 2i i
53 3
3 cos sin2 2
i
Use De Moivre’s Theorem to find the indicated powerof the given complex number. Write your answer instandard form.
10 101296 cos sin 648 648 3
3 3i i
45 5
6 cos sin6 6
i
Use De Moivre’s Theorem to find the indicated powerof the given complex number. Write your answer instandard form.
20 205 cos18.546 sin18.546 5 0.954 0.299i i
203 4i
Use De Moivre’s Theorem to find the indicated powerof the given complex number. Write your answer instandard form.
205 cos0.927 sin 0.927i
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