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8.2 De - Moivre's Theorem

A French Mathematician De' Moivre (1967 - 1974) laid down an important theorem which gives us the powers of complex numbers.

If 'n' is any real number, then one of the value of (cos q + i sin q)n is cos nq + i sin nq

For we shall consider the following three cases.

Case (I) n be a positive integer

Consider (cos q + i sin q)n = (cos q + i sin q) (cos q + i sin q) ........n times.

= cos (q + q + ....n times) + i sin (q + q + .....n times)

= cos nq + i sin nq

Take n = 1 , 2, 3, ..... and verify

Case (II) n be a negative number

Let n = - m, m being positive integer

Case III Let n be a fraction, \ n = p /q

Now (cos q/q + i sin q / q )q = cos q + i sin q .......[ by (I) and (II)]
\ (cos q + i sin q)n = (cos q + i sin q) p/q

= (cos pq + i sin pq)1/q

= cos ( p/q q) + i sin ( p/ q) q = cos n q + i sin n q

Hence the theorem is proved for all real numbers.

Remarks

(1) (cos q1 + i sin q2) (cos q2 + i sin q1) = cos (q1 + q2) + i sin (q1 + q2)

This can be generalized.

(2) (cos q - i sin q) n= [ cos ( - q) + i sin (- q)]n = cos (- nq) + i sin (- nq)

= cos n q - i sin n q

(3) = cos q - i sin q and = (cos q + i sin q)

(4) = (cos q1 - q2) + i sin ( q1 - q2)

(5) [sin q + i cos q ]n = [ cos (900 - q) + i sin (900 - q)]n

= cos n(900 - q) + i sin n (900 - q)

(6) [ cos q + i sin f ]n ¹cos nq + i sin nf

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Index

8.1 Geometry of complex numbers
8.2 De - Moivres's theorem
8.3 Roots of complex numbers
8.4 Cirsular functions of complex angles & hyperbolic function
Supplementary Problems

Chapter 9

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