De Moivre's Theorem

 


We know that the trigonometrical form of a complex number   z   is given by  

          z = r(cos theta + i sin theta)

 where ,    r =  /z/ , theta = arg z 

The product of two complex number is a complex number , As such 

 n  th  power of (a+ib)  is also a complex number .

But the methods of ordinary Algebra do not provide us with any precious method for computing

 n  th  power of (a+ib)   where  n  may be an integer or fraction .

De Moivre's Theorem  helps us to compute the value of

 n  th  power of (a+ib)  by changing it in a trigonometrical form .

The general enunciation of  De Moivre's Theorem :-  

For all values of   n   and   theta   , real or complex ;

cos n theta + i sin n theta      is a value of

n th power of    cos theta + i sin theta 

The theorem holds for real and non real complex values of    theta    and     n 

The expression     cos theta + i sin theta   is some times abbreviated to

cos theta

So , De Moivre's Theorem  is ;

n th power of    cos theta + i sin theta  

= cos n theta + i sin n theta