The Chemistry Maths Book, Second Edition

234 Chapter 8Complex numbers

This is de Moivre’s formulafor positive integers n.

3

The formula is also valid for other

values of n. For example, by equations (8.27) and (8.23),

so that

(cos 1 θ 1 + 1 i 1 sin 1 θ)

−n

1 = 1 cos(−nθ) 1 + 1 i 1 sin(−nθ) 1 = 1 cos 1 nθ 1 − 1 i 1 sin 1 nθ (8.28)

de Moivre’s formula can be used to derive many of the formulas of trigonometry (see

Section 3.4). For example, equation (8.27) withn 1 = 12 is

(cos 1 θ 1 + 1 i 1 sin 1 θ)

2

1 = 1 cos 12 θ 1 + 1 i 1 sin 12 θ

Expansion of the left side of this gives

(cos

2

1 θ 1 − 1 sin

2

1 θ) 1 + 1 i(2 1 sin 1 θ 1 cos 1 θ) 1 = 1 cos 12 θ 1 + 1 i 1 sin 12 θ

A single equation between two complex numbers is equivalent to two equations

between real numbers; two complex numbers are equal only when the real parts are

equal and the imaginary parts are equal. Therefore,

cos

2

1 θ 1 − 1 sin

2

1 θ 1 = 1 cos 12 θ,2 1 sin 1 θ 1 cos 1 θ 1 = 1 sin 12 θ

(see equations (3.23) and (3.24)). Similar formulas, expressing sin 1 nθand cos 1 nθin

terms of powers ofsin 1 θandcos 1 θ, are obtained in this way for any positive integer n;

the expression on the left side of equation (8.27) is expanded by means of the binomial

formula, equation (7.14), and its real and imaginary parts equated to the corresponding

terms on the right side of the equation.

The generalization and significance of de Moivre’s formula are discussed in

Section 8.5.

EXAMPLE 8.7Express cos 15 θand sin 15 θin terms of sin 1 θand cos 1 θ.

By the binomial expansion (7.14), or by using Pascal’s triangle,

(a 1 + 1 b)

5

1 = 1 a

5

1 + 15 a

4

b 1 + 110 a

3

b

3

1 + 110 a

2

b

3

1 + 15 ab

4

1 + 1 b

5

so that

(cos 1 θ 1 + 1 i 1 sin 1 θ)

5

1 = 1 (cos

5

1 θ 1 − 1101 cos

3

1 θ 1 sin

2

1 θ 1 + 151 cos 1 θ 1 sin

4

1 θ)

• 1 i(5 1 cos

4

1 θ 1 sin 1 θ 1 − 1101 cos

2

1 θ 1 sin

3

1 θ 1 + 1 sin

5

1 θ)

11

(cos sin )

cos sin

cos( ) sin(

θθ

θθ

θ

• 
  

=

• 
  

=−+

i

nin

ni

n

−−nθ)

3

Abraham de Moivre (1667–1754), fled to England in 1688 from the persecution of the French Huguenots.

The first form of the formula occurs in a Philosophical Transactionspaper of 1707. de Moivre was a friend of

Newton. In his later years, Newton would tell visitors who came to him with questions on mathematics to ‘go to

Mr. de Moivre, he knows these things better than I do’.