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Applications

(A) Roots of a complex number

If p and q are integers relatively prime to each other, then (cos q + i sin q )p/q has exactly 'n' distinct roots (i.e. values) which can be arranged in Geometric progression.

Since cos q and sin q are periodic functions

\ cos q = cos (2pk + q) and sin q = sin (2pk + q)

where k is an integer.

Hence (cos q + i sin f )p/q= [ cos (2pk + q) + i sin (2pk + q) ]p/q
= Cos p + i sin p

where k = 0, 1, 2, 3, ...... (q - 1)

These are 'q' different values (roots) of

(cos q + i sin q )p/q

\ The q values of (cos q + i sin q )p/q are

a0, a0b, a0b2, a0b3 ........, a0bq - 1
which are obviously in G.P.

Working rule

1) Express (a + ib) in the polar form i.e. a + ib = r (cos q + i sin q)

2) Add 2p k to the angle q.

3) Apply De'Moivre's theorem.

4) The required q roots are obtained.

From (3) by putting k = 0, 1, 2, .......(q - 1)

Example 1 Simplify :

Solution

cos 3q + i sin 3q = (cos q + i sin q)3........ by De' Moivre's Theorem.

cos 4q - i sin 4q = (cos q + i sin q)-4
(cos 4q + i sin 4q) = (cos q + i sin q)4and

(cos 5q + i sin 5q) = (cos q + i sin q)5

\ The given expression

= (cos q + i sin q)12 - 20 - 12 + 20

= (cos q + i sin q)0

= 1

Example 2

(i) If 2 cos q = . Show that = 2 cos r q (ii) q

Solution
(i)     = 2 cos q

     \ x2 + 1 = 2 cos q x      i.e. x2 - 2 cos qx + 1 = 0

\

\ = cos q ± i sin q

\ xr=(cos q ± i sin q)r = cosrq ± i sinrq

Also, = cos rq ± i sin rq

Example 3 Express in the form a + i b

Solution

Let a + ib = b = 1 (1st Quad.)

\ r = = 2

and q = tan-1 (b / a) = tan-1 ( ) = 300 or

\ = r (cos 300 + i sin 300 )

\ ()7= [ 2 (cos 300 + i sin 300)]7

                     = 27[cos 7 ´ 300+ i sin 7 ´ 300]

                     = 27[cos 2100 + i sin 2100]

                     = 27 [cos (180 + 30) + i sin (180 + 30 ]

                     = 128 [ - cos 300- i sin 300]

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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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