Plane trigonometry, by S.L. Loney.

320 TRIGONOMETRY.
For we proved in Art. 267 that
+ yi = p [cos (2n7r + )+ - 1 sin (2n7r + 0)],
where p = + Jx2 + y2
and 0 is such that
cos 0 =x and sin 0 =
P P
Hence,( -. |i)9 _ g[ 2)r+i9r + + 2nr + (0
(x + yi) = p cos + ---- + 1 sin.
By giving n in succession the values 0, 1, 2,... - 1,
we obtain the q required roots.
272. Ex. 1. Find the values of
cos 3 + - sin 3
We have
(cos + ^ - sin ) = [cos (2n7+ +) + ^1 sin (2nhr + )],
where n is any integer,
= cos( r 12 ) '
4C+Oi-2sn 4 12r
Giving n in succession the values 0, I, 2, and 3 we have as our answers
the quantities
co + os - + in 7
-2 12' 12 12'
13I r 13- 197 197 -cos +/V-lsm-1-i, andcos -+-si 12
The student will note that the value n=4 will not give us an additional
value. For it gives
cos (27r + \ )+ sin (27r +,
i( 2 U2