602.

To find the cosine and sine of the sum of any number of angles in terms of the cosines and sines of the angles.

Multiply out the product on the left-hand side of the above equation and equate the real and imaginary parts. Then ; observing that

...

cos (a+B+.... .+ §) = cos a cos ß...cos έ - Σ sin a sin ß cos y .. +Σ sin a sin ẞ sin y sin 8 cos e...cos έ — ...

+

sin (a +ß +...+ §) = Σ sina cosẞ...cos έ-Σsin a sin ẞsin ycos d...cos έ

...

where the symbol Σ denotes 'the sum of all terms of the form obtained by interchanging the letters a, ß, y...έ in the product to which it is attached.'

This conclusion is the same as that obtained in Art. 350. The two methods of proof should be carefully compared.

De Moivre's Theorem.

603. As in Art. 601 we have, for all real values of m and n, (cos me + i sin m✪) (cos n✪ + i sin n0) = cos (m + n) 0 + i sin (m + n) 0. That is, calling cos m✪ + i sin më, ƒ(m),

real,

ƒ(m) ׃(n) = f(m+n).

Hence, by the Index Theorem of Art. 463, m and n being

f(n) is one value of {ƒ(1)}",

i.e. cos no + i sin ne is one value of (cos + i sin 0)”,

for all positive and negative values of n.

This result is so important that we will give a proof of it without introducing the index theorem in its general form.

It is usually known as De Moivre's theorem.

604.

Whatever positive or negative value n may have, cos no + i sin ne is one of the values of (cos ◊ + i sin 0)". We have cos (0+ 6 + 4 + ...) + i sin (0 + 4 + 4 + ...)

=

(cos + i sin 0) (cos + i sin 4) (cos + i sin ↓)....

Now let ==&=..... and let there be n of these angles. Thus, if n is a positive integer,

P

Then (cos no + i sin në)2 = ( cos 20+ i sin ?0)"

= cos p✪ + i sin p✪ by (1) and (2)

(cos + i sin 0)

by (1) and (2)

..cos no + i sin ne is one of the values of

(cos

cos no + i sin n✪ is one of the values of (cos

+ i sin )": or

+ i sin 0)”.

605. To find q values of (cos +isin ) where p and q are prime to one another.

We have cos + i sin 0 = cos (0 + 2λπ) + i sin (0 + 2λπ) where λ has any integral value.

.. any one of the values of cos(0 + 2λπ) + i sin (0+2λπ) is

Giving to A any two values μ and μ', the expressions

cos 2 (0+2μπ) + i sin 2 (0 + 2μπ)

would be equal, only if cos(+ 2μπ) = cos(0 + 2μ'π)

and

q

i.e. if the angles 2 (6+ 2μ7) and 2 (0 + 2μ'π) were equi-cosinal and

equi-sinal;

9

ie. if 2 (0 + 2μπ) ~ 2 (0 + 2μ' ́π) were a multiple of 2′′ ;

i.e. if p (u') were a multiple of q.

If then Ρ and q are prime to one another, this condition becomes if up' is a multiple of q.'

Hence, if we give to λ the q values 0, 1, 2...n-1, we shall

obtain q different values of cos 2 (0 + 2λm) + i sin 2 (0 + 2λπ) : and,

if we give to A higher values, the values of this expression will simply recur.

Р

Hence q values of (cos + i sin 0) are given by the q values of

606. To find q values of the qth root of any complex quantity. Let a+bi be any complex quantity: and let p denote the positive value of √(a2+b2). Then p is the modulus of a + bi.

Since a/p and b/p are each numerically less than unity, and are such that the sum of their squares is equal to unity, therefore, there is one and only one angle between 0 and 2 whose cosine is a/p and whose sine is b/p. Let this angle be 0.

Now, since p is positive, there cannot be more than one positive qth root of p. And there is one positive qth root of P,

whose value can be approximately determined by the Binomial Theorem.

1

Let på denote this positive value of p.

Then, by the proposition of the last article, q values of the qth root of cos 0 + i sin ✪ are given by the

θ + 2λπ

q values of

θ + 2λπ

COS

+ i sin

Hence q values of the qth root of a + bi are given by putting successively equal to 0, 1, 2,...q – 1 in the expression

607. There is one point in the above result that requires special attention.

In the theory of indices, we use the symbol a as the equivalent of /(a2) or (a)o.

Now, by the theory of fractions, a2 = ar mate to infer that

(a2)="/(a)?

Is it then legiti

By the above proposition (a) has q values; and, by Art. 599, it cannot have more than q values. Similarly "/(ar) has rq values and no more.

Hence (a) and "/(a) have not the same number of values: and, therefore, cannot be said to have the same meaning.

rq

The q values of (a) are, however, all contained amongst the values of "1/(a”): as the student may show for himself.

The roots of +1 − 1, + i, and – i.

608. In De Moivre's theorem, Art. 605, let p = 1.

Now put 0 = 0. Thus cos 0= 1, and sin 0 = 0.

Hence the q qth roots of unity are given by the q

values of

q values of

Hence the q qth roots of i are given by the

Hence the q qth roots of -i are given by the q values of

609. Again, in Art. 606, put 0=0. Then a + bi reduces to a, i.e. p, a real positive quantity.

where the expression in the bracket represents the qth roots of unity.

Again put = π. Then a + bi reduces to a, i.e. —p, a real negative quantity.

where the expression in the bracket represents the qth roots of minus unity.

We may illustrate the important result of Art. 606 by showing that the ratio of any two qth roots of a quantity is a qth root of unity.

Thus giving to any two values and μ', the ratio of the two corresponding qth roots of a+bi is