Abraham De Moivre was born in 1667, and was influenced by Sir Isaac Newton. He is best known for "De Moivre's Theorem", in the field of complex numbers.
De Moivre's Theorem states that for every positive integer, n, the following trigonometric relationship is true:
(cosx + i.sinx)^n = cos(nx) + i.sin(nx)
where i = √(-1) by convention. (In some literature "j" is used instead of "i", so as not to confuse it with the symbol for electrical current.)
The result is very useful, especially when trying to sum trigonometric functions such as
cos.x + cos.2x + cos.3x + ... + cos.nx
De Moivre's Theorem Proof
The theorem is derived from Euler's formula
e^(ix) = cosx + i.sinx,
where e = base of natural logarithms, 2.71828...
Raising both sides of the equation to the power of n,
[e^(ix)]^n = (cosx + i.sinx)^n
But since (e^x)^n = e^(x×n) then
e^(i.nx) = (cosx + i.sinx)^n
But by Euler's original formula,
e^(i.nx) = cos(nx) + i.sin(nx), and therefore
(cosx + i.sinx)^n = cos(nx) + i.sin(nx)
De Moivre's Theorem Examples and Applications
When considering De Moivre's Theorem it is important to remember that if two complex numbers are identical, then the real parts of those numbers must be the same, and the complex ("imaginary") parts of those numbers are the same. i.e.
if
a + i.b = c + i.d, then
a = c, and b = d.
With this in mind, the case for n = 2 can be examined:
(cosx + i.sinx)^2 = cos(2x) + i.sin(2x)
Expanding the left hand side of the equation gives
(cos²x - sin²x) + 2.i.sinx.cosx = cos(2x) + i.sin(2x)
Noting that the real and imaginary parts must equate, then
cos²x - sin²x = cos(2x), and 2.i.sinx.cosx = i.sin(2x)
The formula for cos(2x) and sin(2x) are true, as these are well-known trigonometric identities.
By using the Binomial Theorem, the algebraic expansions for cos(3x), cos(4x), sin(3x) etc can all be easily obtained. The article "De Moivres Theorem Examples Cos 3x, Sin3x, Cos4x and Sin4x" describes these in detail.
Trigonometric Series Summation Using De Moivre's Theorem
A further use of De Moivre's Theorem is the summation of trigonometric series. For example, to find the sum of the series
cos(x) + cos(2x) + cos(3x) + ... + cos(nx)
then the series may be re-written as the real part of the series
(cosx + i.sinx) + (cos2x + i.sin2x) + (cos3x + i.sin3x) + ... + (cos(nx) + i.sin(nx))
This series may in turn be re-written as (using De Moivre's Theorem)
(cosx + i.sinx) + (cosx + i.sinx)^2 + (cosx + i.sinx)^3 + ... (cosx + i.sinx)^n
This series is simply a geometric progression, where
Common ratio (r) = (cosx + i.sinx),
First term (a) = (cosx + i.sinx)
Number of terms = n - 1
So the sum = Real part of (cosx + i.sinx)×((cosx + i.sinx)^n - 1) / ((cosx + i.sinx) - 1)
The rest of this is simply algebraic manipulation, but the steps will be described: (The full proof can be found at "Sum of Trigonometric Series Using De Moivres Theorem").
- Multiply above and below by the complex conjugate, ((cosx - 1 - i.sinx))
- Expand and simply the denominator
- Expand and simplify the numerator
- Gather real and complex terms
- State the result in terms of sinθ and cosθ
De Moivre's Theorem Summary
De Moivre's Theorem is an extension of Euler's formula. It has several uses, such as the production of formulae for cos(nx) and sin(nx), or the evaluation of the sums of trigonometric series.
De Moivre's Theorem References
There are many good online resources for complex numbers generally, including De Moivre's Theorem in particular. The book " Calculus with Complex Numbers " is a good beginners guide. The article "Trigonometric Identities and Manipulating Trig Functions" is useful for those without college-level algebra.
Martin Bell - Martin holds a B.Sc. degree in chemical engineering, and an M.Sc. degree in electronics and computing. He has spent more than 25 years ...
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