De Moivre

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Chapter 40
De Moivres Theorem simple applications
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In mathematics, de Moivres formula, named after
Abraham de Moivre.
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The formula is important because it connects
complex numbers and trigonometry. The expression
"cos x i sin x" is sometimes abbreviated to
"cis x".
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By expanding the left hand side and then
comparing the real and imaginary parts under the
assumption that x is real, it is possible to
derive useful expressions for cos(nx) and sin(nx)
in terms of cos(x) and sin(x).
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Furthermore, one can use a generalization of this
formula to find explicit expressions for the n-th
roots of unity, that is, complex numbers z such
that zn 1.
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De Moivres theorem
For all values of n, the value, or one of the
values in the case where n is fractional, of
is
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Proofing of De Moivres Theorem
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Now, let us prove this important theorem in 3
parts.

1. When n is a positive integer
2. When n is a negative integer
3. When n is a fraction

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Case 1 if n is a positive integer
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Continuing this process, when n is a positive
integer,
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Case 2 if n is a negative integer
Let n-m where m is positive integer
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Case 3 if n is a fraction equal to p/q, p
and q are integers
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Raising the RHS to power q we have,
but,
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Hence, De Moivres Theorem applies when n is a
rational fraction.
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Proofing by mathematical induction
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The hypothesis of Mathematical Induction has been
satisfied , and we can conclude that
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e.g. 1
Let z 1 - i. Find .
Soln
First write z in polar form.
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Polar form
Applying de Moivres Theorem gives
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It can be verified directly that
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Properties of
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If
then
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Hence,
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Similarly,
if
Hence,
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We have,
Maximum value of cos? is 1, minimum value is -1.

Hence, normally
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What happen, if the value
of is more than 2 or less than -2 ?
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e.g. 2
Given that
Prove that
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e.g. 3
If , find
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Do take note of the following
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e.g. 4
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Applications of De Moivres theorem
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We will consider three applications of De
Moivres Theorem in this chapter.
1. Expansion of
.
2. Values of .
3. Expressions for in
terms of multiple angles.

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Certain trig identities can be derived using De
Moivres theorem. In particular, expression such
as
can be expressed in terms of
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e.g. 5
Use De Moivres Thorem to find an identity for
in terms of .
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e.g. 6
Find all complex cube roots of 27i.
Soln
We are looking for complex number z with the
property
Strategy
First we write 27i in polar form -
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Now suppose
Satisfies . Then, by De Moivres
Theorem,
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This means
where k is an integer.
Possibilities are k0, k1, k2
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In general to find the complex nth roots of a
non-zero complex number z.
1. Write z in polar form
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2. z will have n different nth roots (i.e. 3 cube
roots, 4 fourth roots, etc.)
3. All these roots will have the same modulus
the positive real nth roots of r) .
4. They will have different arguments
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5. The complex nth roots of z are given (in polar
form) by
etc
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e.g. 7
Find all the complex fourth roots of -16.
Soln
Modulus 16 Argument ?
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Fourth roots of 16 all have modulus
and possibilities for the arguments are
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Hence, fourth roots of -16 are
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e.g. 8
Given that and

find the value of m.

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e.g. 9
Solve , hence prove that

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e.g. 10
Find the cube roots of -1. show that they can be
denoted by and prove that
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e.g. 11
Solve the following equations, giving any complex
roots in the form
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e.g. 12
Prove that
Hence find
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e.g. 13
Show that
Use your result to solve the equation
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e.g. 14
Use De Moivres Theorem to find
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e.g. 15
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e.g. 16
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e.g. 17
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e.g. 18
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e.g. 19
Express in terms of multiple
angles and hence evaluate
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e.g. 20
Express in terms of
and hence evaluate in terms
of .
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The end