CONSTRUCTION    OF    TABLES.  131

Hence

(cos  x-\-i  sin  x)    «  ^  k  cos  ( —  x)  +  ^  sin  (—  x)  p

771  TTl

=  cos  —  (— cc  +  2A;7r)  +  *'sin  — (— a:  +  2A;7r),

(k  =  0,  1,  2, n  —  1)

'■  (x-{- 2kir)  ^  +  *' sin  -^ {x-\-2  Jctt)  j> ,

(k  =  0,l,2, 71-1).        (4)

Consequently  if  ti  is  a  positive  or  negative  integer  or  fraction

(cos  x-jri sin  af)"  =  cos  [n(x-\^2  kir)^  -{-  i  sin  [w  (cc  +  2  A^tt)],

(7c  =  0,  1,  2, n-1).        (5)

Example  :  Find  the  three  cube  roots  of  —  1.
We  have  —  1  =  cos  180°  +  i  sin  180°

,     ,,,             180°  +  2A:7r  ,    .   .    180°  +  2A;7r,,       ^  ,   „,
.  •.  (—  1)^  =  cos +  I  sm {k  =  0, 1,  2).
For  the  three  cube  roots  of  —  1  we  find  therefore

cos  60°  +  i  sin  60°,        cos  180°  +  i  sin  180°,        cos  300°  +  i  sin  300°,

l  +  iVs  ,  l-iVs

-^— '  -1'  -^-

By  aid  of  De  Moivre's  Theorem  we  may  express  sin  716  and

cos  nO,  when  n  is  an  integer,  in  terms  of  sin  0  and  cos  6.
Thus

cos  n6  +  i  sin  tiO  =  (cos  0-\-i  sin  Oy

71  {71  —  1  I

=  COS"  0-{-i7i  cos"-^  ^  sin  ^  +  P  -^ — ^  cos"-^  0  sin^  0

_^.3^(n-l)(.-2)^^^^_3^^.^3^^

Or,  sincei^=  — 1,  i^=  —  i,  {*=-}-l,

cos  w  ^  4"  *'  sin  w  ^  =  cos"  6-{-i7i  cos"~^  ^  sin  ^

2  |o


132  TRIGONOMETRY.
