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(Solved): (RI16/Promo/Q8) (a) The complex number z is given by z=1+cos\alpha _(+)isin\alpha _(), where 0<\a ...
(RI16/Promo/Q8) (a) The complex number
zis given by
z=1+cos\alpha _(+)isin\alpha _(), where
0<\alpha <(\pi )/(2). (i) Show that
zcan be expressed as
(2cos((\alpha )/(2)))(cos((\alpha )/(2))+isin((\alpha )/(2))). (ii) Find
zz^(**), where
z^(**)denotes the conjugate of
z. (iii) Given that
\alpha =(\pi )/(3), without using a calculator, find the values of
|z^(6)|and
argz^(6). Deduce the value of
z^(6).
3(b) The complex numbers
uand
vare such that
(i(u+v))/((u-v))is real. Show that
|u|=|v|. [3]
