De Moivre's formula

Derivation from Euler's formula

Although historically proven earlier, de Moivre's formula can easily be derived from Euler's formula

e i x = cos ⁡ x + i sin ⁡ x

and the exponential law for integer powers

( e i x ) n = e i n x .

Then, by Euler's formula,

e i n x = cos ⁡ n x + i sin ⁡ n x .

Proof by induction (for integer n)

The truth of de Moivre's theorem can be established by mathematical induction for natural numbers, and extended to all integers from there. For an integer n, call the following statement S(n):

( cos ⁡ x + i sin ⁡ x ) n = cos ⁡ n x + i sin ⁡ n x .

For n > 0, we proceed by mathematical induction. S(1) is clearly true. For our hypothesis, we assume S(k) is true for some natural k. That is, we assume

( cos ⁡ x + i sin ⁡ x ) k = cos ⁡ k x + i sin ⁡ k x .

Now, considering S(k + 1):

( cos ⁡ x + i sin ⁡ x ) k + 1 = ( cos ⁡ x + i sin ⁡ x ) k ( cos ⁡ x + i sin ⁡ x ) = ( cos ⁡ k x + i sin ⁡ k x ) ( cos ⁡ x + i sin ⁡ x ) by the induction hypothesis = cos ⁡ k x cos ⁡ x − sin ⁡ k x sin ⁡ x + i ( cos ⁡ k x sin ⁡ x + sin ⁡ k x cos ⁡ x ) = cos ( k + 1 ) x + i sin ( k + 1 ) x by the trigonometric identities

See angle sum and difference identities.

We deduce that S(k) implies S(k + 1). By the principle of mathematical induction it follows that the result is true for all natural numbers. Now, S(0) is clearly true since cos(0x) + i sin(0x) = 1 + 0i = 1. Finally, for the negative integer cases, we consider an exponent of −n for natural n.

( cos ⁡ x + i sin ⁡ x ) − n = ( ( cos ⁡ x + i sin ⁡ x ) n ) − 1 = ( cos ⁡ n x + i sin ⁡ n x ) − 1 = cos ⁡ ( − n x ) + i sin ⁡ ( − n x ) . ( ∗ )

The equation (*) is a result of the identity

z − 1 = z ¯ | z | 2 ,

for z = cos (nx) + i sin (nx). Hence, S(n) holds for all integers n.

Formulae for cosine and sine individually

Being an equality of complex numbers, one necessarily has equality both of the real parts and of the imaginary parts of both members of the equation. If x, and therefore also cos x and sin x, are real numbers, then the identity of these parts can be written using binomial coefficients. This formula was given by 16th century French mathematician François Viète:

sin ⁡ ( n x ) = ∑ k = 0 n ( n k ) ( cos ⁡ x ) k ( sin ⁡ x ) n − k sin ⁡ ( n − k ) π 2 cos ⁡ ( n x ) = ∑ k = 0 n ( n k ) ( cos ⁡ x ) k ( sin ⁡ x ) n − k cos ⁡ ( n − k ) π 2 .

In each of these two equations, the final trigonometric function equals one or minus one or zero, thus removing half the entries in each of the sums. These equations are in fact even valid for complex values of x, because both sides are entire (that is, holomorphic on the whole complex plane) functions of x, and two such functions that coincide on the real axis necessarily coincide everywhere. Here are the concrete instances of these equations for n = 2 and n = 3:

cos ⁡ 2 x = ( cos ⁡ x ) 2 + ( ( cos ⁡ x ) 2 − 1 ) = 2 ( cos ⁡ x ) 2 − 1 sin ⁡ 2 x = 2 ( sin ⁡ x ) ( cos ⁡ x ) cos ⁡ 3 x = ( cos ⁡ x ) 3 + 3 cos ⁡ x ( ( cos ⁡ x ) 2 − 1 ) = 4 ( cos ⁡ x ) 3 − 3 cos ⁡ x sin ⁡ 3 x = 3 ( cos ⁡ x ) 2 ( sin ⁡ x ) − ( sin ⁡ x ) 3 = 3 sin ⁡ x − 4 ( sin ⁡ x ) 3 .

The right hand side of the formula for cos nx is in fact the value T_n(cos x) of the Chebyshev polynomial T_n at cos x.

Failure for non-integer powers, and generalization

De Moivre's formula does not hold for non-integer powers. The derivation of de Moivre's formula above involves a complex number raised to the integer power n. If a complex number is raised to a non-integer power, the result is multiple-valued (see failure of power and logarithm identities). For example, when n = 1/2, de Moivre's formula gives the following results:

for x = 0 the formula gives 1^1⁄₂ = 1, and for x = 2π the formula gives 1^1⁄₂ = −1.

This assigns two different values for the same expression 1^1⁄₂, so the formula is not consistent in this case.

On the other hand, the values 1 and −1 are both square roots of 1. More generally, if z and w are complex numbers, then

( cos ⁡ z + i sin ⁡ z ) w

is multi-valued while

cos ⁡ w z + i sin ⁡ w z

is not. However, it is always the case that

cos ⁡ w z + i sin ⁡ w z

is one value of

( cos ⁡ z + i sin ⁡ z ) w .

Roots of complex numbers

A modest extension of the version of de Moivre's formula given in this article can be used to find the nth roots of a complex number (equivalently, the power of 1/n).

If z is a complex number, written in polar form as

z = r ( cos ⁡ x + i sin ⁡ x ) ,

then the n nth roots of z are given by

r 1 n ( cos ⁡ x + 2 π k n + i sin ⁡ x + 2 π k n )

where k varies over the integer values from 0 to n − 1.

This formula is also sometimes known as de Moivre's formula.

Hyperbolic trigonometry

Since cosh x + sinh x = e^x, an analog to de Moivre's formula also applies to the hyperbolic trigonometry. For all n ∈ ℤ,

( cosh ⁡ x + sinh ⁡ x ) n = cosh ⁡ n x + sinh ⁡ n x .

Also, if n ∈ ℚ, then one value of (cosh x + sinh x)ⁿ will be cosh nx + sinh nx.

Quaternions

To find the roots of a quaternion there is an analogous form of de Moivre's formula. A quaternion in the form

d + a i ^ + b j ^ + c k ^

can be represented in the form

q = k ( cos ⁡ θ + ε sin ⁡ θ ) for  0 ≤ θ < 2 π .

In this representation,

k = d 2 + a 2 + b 2 + c 2 ,

and the trigonometric functions are defined as

cos ⁡ θ = d k and sin ⁡ θ = ± a 2 + b 2 + c 2 k .

In the case that a² + b² + c² ≠ 0,

ε = ± a i ^ + b j ^ + c k ^ a 2 + b 2 + c 2 ,

that is, the unit vector. This leads to the variation of De Moivre's formula:

q n = k n ( cos ⁡ n θ + ε sin ⁡ n θ ) .

Example

To find the cube roots of

Q = 1 + i ^ + j ^ + k ^ ,

write the quaternion in the form

Q = 2 ( cos ⁡ π 3 + ε sin ⁡ π 3 ) where  ε = i ^ + j ^ + k ^ 3 .

Then the cube roots are given by:

Q 3 = 2 3 ( cos ⁡ θ + ε sin ⁡ θ ) for  θ = π 9 , 7 π 9 , 13 π 9 .

2×2 matrices

Consider the following matrix A = ( cos ⁡ ϕ sin ⁡ ϕ − sin ⁡ ϕ cos ⁡ ϕ ) . Then ( cos ⁡ ϕ sin ⁡ ϕ − sin ⁡ ϕ cos ⁡ ϕ ) n = ( cos ⁡ n ϕ sin ⁡ n ϕ − sin ⁡ n ϕ cos ⁡ n ϕ ) . This fact (although it can be proven in the very same way as for complex numbers) is a direct consequence of the fact that the space of matrices of type ( a b − b a ) is isomorphic to the space of complex numbers.