### De Moivre's Theorem

We know that the trigonometrical form of a complex number    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  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    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