Ch 9 Complex Numbers Tut 9B APQ Solutions (RVHS)

Complex Numbers Tutorial 9B: Polar and Exponential Forms Solutions Additional Practice Questions 1 2 2 2z z       arg arg arg( ) 4 4 arg( ) 4 2 3 4 iz i z z                   2 2 2 2 2 wz w z w     2 5arg 6 52arg( ) arg( ) 6 3 5arg( ) 2 4 6 7 3 ( )3 2 cos sin3 3 1 3 z w z w w pv w i i                                          

2 (i) 33 32 cos sin cos sin4 4 6 6w i i             Let 1 3 3cos sin4 4w i    2 cos sin6 6w i      3 1 22 w w w 3 1 22 2(1)(1) 2  w w w     3 1 2 1 2 arg( ) arg 2 arg(2) arg( ) 3arg( ) w w w w w     30 3 4 6 4             2 cos sin4 4w i        (ii)   2 πarg arg 4 π π2 (cos i sin )4 4 nn n n n n w w nw n w n nw        When n is a muliple of 4, let n = 4k, where k is an integer. 4 4 2 (cos π i sin π) ( 1) 2 n k k k w k k    (because cos πk =1 when k even, 1 when k odd and sin πk =0 for all k.

3 a = 1 + i 3 =    3 i 2  e  1 + a + a2 + a3 + … + a9 = a a   1 1 10 = )3i1(1 21 10 3 i                 e = 3i 21 3 2 i10         e = 3i )i(21 2 3 2 110   = 3i i 3512513   3i 3i = i 3171512 //

4(a) 3 i 2 cos isin 6 6z          2 cos i sin6 6 n n n nz         * 2i Im( ) 2i 2 sin 0 6 nn n n nz z z         sin 06 n  ,6 n k k     The set of values of n is  : 6 ,n n k k    (b)    3 1 i 1 3z w     Note that OWPZ is a parallelogram, 1 3 2arg( ) tan 1 3w     2 3 6 2WOZ       OZ= 2z  OW= 1 i 3 2w     Since 2WOZ   and OZ OW , OWPZ is a square. (shown) 5arg( ) arg( ) 6 4 12z w z POZ         Also, 1 3 1arg( ) tan 3 1 z w     15 3 1 tan12 3 1        2 3 15 3 1 4 2 3tan 2 312 3 1 23 1          (deduced) Im P W i Z Re O

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6(i) (ii) or (iii) 7(a) Since is a root, Hence the equation becomes . Comparing coefficient of , . For , Hence the other 2 roots are .             44 2 3 44 3 2 4 2 3 i 3 4 i 3 6 i 3 4 i 3 i 3 18 9 i 4 3 12 3 z k k k k k k k k k                    4 3 2 real 4 3 12 3 0 4 3 3 0 0 (rej) or 3 z k k k k k        3 i 3z   3 i 3z       arg 3 i 3 4 7 15arg 3 i 3 , , ,... 4 4 4 3 7 15 , , ...4 4 4 4 3 1 7 15 , , ...4 4 4 4 Least 5 n n n n n                                 55 6 36