Plane trigonometry, by S.L. Loney.

282 TRIGONOMETRY.
and
2cos a +(n- 1)/3} sin s + =n —)sin + -S {a+(n- )/3}.
By adding together these n lines, we have
2S x sin- = sin la + (n -D) 3A -sin a -},
the other terms on the right-hand sides cancelling one
another.
Hence, by Art. 94, we have
3 n -I ' S). n/3
2S x sin- =2 cos a +- sin,
2 ( 2 ) 2
{O n - 1 3}.n
cos a+ 2- sin -
i.e. S=
sin
243. Both the expressions for S in Arts. 241 and 242
vanish when sin - is zero, i.e. when - is equal to any
2 2
multiple of 7r,
n/3
i.e. when = p7
where p is any integer,
27 -i.e. when =p=.-.
Hence the sum of the sines (or cosines) of n angles,
which are in arithmetical progression, vanishes when
the common difference of the angles is any multiple
27r
of
n
Exs. cos a + cos (a+ -) +cos a+ -) +... to n terms= 0,
\ n) \ n!