Plane trigonometry, by S.L. Loney.

SIMPLE TRIGONOMETRICAL SERIES. 285
Ex. 4. A-1A...A, is a regular polygon of n sides inscribed in a circle,
whose centre is 0, and P is any point on the arc AxAl such that the angle
POA1 is 0; find the sum of the lengths of the lines joining P to the angular
points of the polygon.
Each of the angles A OAp, A20A3,... AOA1 is 2, so that the angles
POA1POAOA2,... are respectively
2w 4w, +-, 0+-,...
d' 0+ n' 0+ 4 '
I n n
Hence, if r be the radius of the circle, we have
PA = 2r sin 1 = 2r sin,
2 2
PA2=2r sin 2 2r sin +
POA. ( 27r
PA3= 2r sin- 2 = 2r sin +-).............................................
Hence the required sum
=2r [sin+snsin + s in + (2- +......to n terms].f 0 n-1. n 7r
sin + - sin 2'= 2r (Art. 241)
sin
= 2rcosec 7n. sin F + - -]
2n L2 22n
=2rcosec - cos ( - 2)
2n \2 2-n
EXAMPLES. XLIV.
Sum the series:
1. cos 0 + cos 30 + cos 50 +... to n terms.
A 7A
2. cos + cos 2A + cos +... to n terms.
2 2
sin a sina in 2a + sin 3a +... - sin na n +1
3. cos a +.= tan a.
cos a+cos2a+... +cos na 2