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Trigonometry: De Moivre's Theorem

Abraham De Moivre (1667-1754) was born in France, but fled to England in 1688, after being imprisoned for his religious beliefs. A brilliant mathematician, he was unable to gain a university appointment (because he was born in France) or escape his life of poverty, gaining only a meagre income as a private tutor. He was friends with Sir Isaac Newton and Edmund Halley (1656 - 1742), was elected to the Royal Society in England, and to the Academies of Paris and Berlin, yet in spite of the support of the great Leibniz (1646 - 1716), and of Jacques Bernoulli, he never gained a university appointment and died in relative poverty. In spite of this, he made many discoveries in mathematics, some of which are attributed to others (For instance, Stirling's Formula for factorial approximations was known earlier by De Moivre).

This page deals with the proof of De Moivre's Theorem, etc. It has formula to compute the cosine and sine directly, but these require further algebraic manipulation, to reduce the sines in the cosine formula, and the cosines in the sine formula. This work is extended later beginning with the cosine, where formulae are also derived to calculate directly the coefficients of a given power in a given expansion for cos nx and sin nx. Before that, some pages, mentioned below, deal with the Chebyshev method.

See also

Trigonometry Contents

Page Contents

  1. De Moivre's Formula for Multiple Angles
  2. Explanation
  3. Proof by Induction
    1. Nonnegative n
    2. Negative n


De Moivre's Formula for Multiple Angles

The following is nowadays called De Moivre's formula, clearly known to De Moivre, but never explicitly expressed in this form by him.
demoireFormula.gif [1.1]
where i=√(-1)

Picking out the real parts from both sides and equating them, for cosine, we get:
cosnx.gif [1.2]
Where k, a nonnegative integer, is 2·p, for the pth term, p=0, 1, 2..., and k≤n

For the sine, we find:
sinnx.gif [1.3]
Where k, a nonnegative integer, is 1+2·p, for the pth term, p=0, 1, 2..., and k≤n

We can use this formula to find multiple angles of cosine and sine by associating the real parts of the right-hand side, expanded with the Binomial Theorem, with cosine and the imaginary parts with the sine. We give examples on a following page.

On this page, we make a limited claim, that the formula is true for all integers n. Actually it is true in a much wider context, for complex numbers.

Explanation

Leonhard Euler (1707-1783), was inspired by De Moivre, to formulate:
eixIscosAndIsine.gif [2.1]

Because
eTox.gif [2.2]

We can substitute ix for x (where i=√-1 )
eToIx.gif [2.3]
Replacing i2 by -1, and i4 by 1, etc:
eToIxSimplified.gif [2.4]

Grouping the real and imaginary parts:
eToIxRearranged.gif [2.5]
We note the first series is the expansion of cos x, and the second i·sin x, so:
eixIscosAndIsine.gif
(which is what Euler claimed)

By raising eix to the power n, we note, the two forms depending on how we group the indexes:
demoivreExplained.gif

Which, assuming the expansions above for eix , sin x, cos x, etc (Proved in calculus using Maclaurin's Theorem), we have proved De Moivre's Law:
demoireFormula.gif

Proof by Induction

Nonnegative n

We wish to show by mathematical induction that
demoireFormula.gif[1.1, repeated]
Where n is a natural number

When n=1, then:
cos x+i·sin x=(cos x+i·sin x)
Which is true, so the theorem is true for n=1.

For n=k, where k is a nonnegative integer, by the formula, we have
cos kx+i·sin kx=(cos x+i·sin x)k [3.1].

Assuming that it is true for n=k, then when n=k+1:
cos (k+1)x+i·sin (k+1)x=
(cos kx+i·sin kx)(cos x+i·sin x) [3.2]
by multiplying Equation 3.1 by (cos x+i·sin x)

Multiplying out Equation 3.2 we find:
=cos kx·cos x+i2·sin kx·sin x+i·sin kx·cos x+i·cos kx·sin x
=cos kx·cos x-sin kx·sin x+i·sin kx·cos x+i·cos kx·sin x [because i2=-1]
=cos (k+1)x+i·sin (k+1)
by the compound angle formula.

So if the formula is true for n=k, it is also true for n=k+1. As it true for n=1, then it is true for all n


Negative n

Assuming (or claiming) that the formula is true for nonnegative n, let us write n as -m, where m is a positive integer:negative1.gif [4.1]

So:
negative2.gif [4.2]

Applying De Moivre's formula to the right-hand side:
negative3.gif [4.3]

Multiplying top and bottom by cos (mx)-i·sin(mx), noting (a−b)(a+b)=a2+b2 :
negative4.gif [4.4]
And because sin2θ+cos2θ=1, we have:
negative5.gif [4.5]

Substituting n back into the formula, that is, substituting n for m:
negative6.gif [4.6]

Taking the minuses outside, and noting that cos(nx)=cos(nx), and sin (nx)=sin(nx), we have:
negative7.gif [4.7]

which is De Moivre's formula for negative n, so the formula works for all integers .



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