Plane trigonometry, by S.L. Loney.

PROPERTIES OF THE CIRCLE.

427

Again xn+- 2 cos na= xn - - 2 cosn a+ -,
and is similarly divisible by
1 27\
x +- -2 cos a -
x n
Proceeding in this way we can shew that it is divisible by
x+ -- 2cos (a+ -,......x +- 2 cos (a+ - 2r,
x \ n x \
and hence obtain equation (2) of Art. 362.
364. De Moivre's Property of the Circle.

A geometrical meaning may be
(3) of Art. 362.
Let ABCD... be the angular
points of a polygon of n sides
which is inscribed in a circle of
radius a, so that, 0 being the
centre, we have
2wr
z A OB= z BOC= z COD=... -.
n
Let P be a point within, or
without, the circle such that

given to the equation
E D
F B
A
H Q

OP=x and zPOA=0.
Then
POB = 0 +-2, POC= +,...
n n
and we have
PA2 = OP2 + OA2- 20P. OA cos POA
= 2 - 2ax cos 0 + a2,
PB2= OP2 + OB2- 20P. OB cos POB
= x2- 2axcos (O +-) + a2,
PC2 = x - 2ax cos (o + + a,
\ n