Given #z# is a complex number such that #Arg(z) = theta# and #|z| = r# , write an expression for each of #Arg(4z^3)# and #|4z^3|# in terms of r and/or θ. How should I approach this?

2 Answers
Jun 9, 2018

Contd.........

Explanation:

A complex no. #z# is such that #Arg(z)=theta and |z|=r#.

Clearly, #z=r(costheta+isintheta)#.

By De Moivre's Theorem, #z^n=r^n(cosntheta+isinntheta)#.

#:. z^3=r^3(cos3theta+isin3theta)#.

#:. 4z^3=4r^3(cos3theta+isin3theta)#.

Jun 9, 2018

#z = r e^(i theta)#

#4 z^3 = 4(r e^(i theta))^3 = 4 r^3 e^(i 3 theta) #

  • #"Arg"(4 z^3) = 3 theta#

  • #abs (4 z^3) = 4 r^3 #

Explanation: