.. tan a tan 6+ tan y = tan a. tan ẞ. tan y.

.. tan a. tan y + tan a. tan y + tan ẞ. tan y =

174. COR. 3. If a + B + y = (i + 1) π

= 1.

[cos a± √ I sin a] may

175. PROP. The n values of [cos a ±√

be obtained from the formula

m

m

COS (2 i π + a) ± √ sin (2 ίπ + α),

n

n

by substituting for i the integers from 0 to n

For, by (art. 159), it was shown that

1.*

But there are n values of [cos a±√1 sin a]”.

* In this and the following propositions, when a quantity raised to a fractional power is inclosed by the ordinary parenthesis (), the meaning is restricted to the direct or arithmetic value of the function. If any whatever of the values which that function is capable of assuming is intended, the brackets [] are used.

L

Also, if 2 ia be substituted for a, i being an integer, the first member of the equation remains unchanged, and the formula becomes

m

[cosa sin a]"cos- (2i π+a) ±√=sin(2in+a).

n

Therefore, by substituting for i the integers 0, 1, &c. n − 1,

m

COS

{2 (n − 1) π + a} ± √ —I sin m {2 (n−1) π+a).

n

n

Thus may it be shown that the succeeding terms recur.

176. COR. 1. If the sign of a be changed the result is

m
n

m

[cosasin a] = cos(2i-a)±√1 sin

or,

m

m

COS

m

n

[cosa sin a]" = cos(2in—a)=√1 sinTM (2 i π—a).

the values of [cos a VI sin al I sin a] are to be found in both formulæ by substituting for i the integers from 0 to n − 1, and it does not follow that the contemporary values of i are the To find these quantities, suppose i and k to be con

same.

=0

.. by equating real and imaginary quantities and making a =

m

But is supposed to be in its lowest terms, therefore

n

ik is a multiple of n, but i + k is positive, and less than

177. COR. 2.

Similarly it may be shown that all the

m

√ sina]" may be found in the formulæ ± - I

values of [cos a +

by substituting for i the integers from 0 to n 1.

m

Hence it follows that the values of [1] obtained by using the positive sign, and substituting 0, 1, 2. . . n - 1, for i are the same as when the negative sign is used, and 0, n-1, n-2, .... 3. 2. 1 are put for i; and conversely.

For by making a = 0 in De Moivre's formula (art. 159)

all the values of [1] can be derived from the expression

by giving the integral values from 0 to m 1.

m

Firstly, if m be even it is evident that will be one of the

2

values of i. Hence, by the nature of equations, the roots