Question

Consider the complex number z = 1+iV3.Which of the following complex numbers best approximates z4?Hint: z has a modulus of 2 and an argument of 60”.Choose 1 answer:А4- 6.9iB-4 - 6.918 – 13.97–8 – 13.94

Answer 1

A complex number z with modulus |z| and argument θ can be written as:

$z=|z|\cdot(\cos \theta+i\sin \theta)$

And the nth power of z can be written as:

$z^n=|z|^n\cdot\lbrack\cos (n\theta)+i\sin (n\theta)\rbrack$

Thus, using the given hint, we have:

$\begin{gathered} |z|=2 \\ \theta=60^{\circ} \end{gathered}$

So, the fourth power of z is given by:

$\begin{gathered} z^4=2^4\cdot\lbrack\cos (4\cdot60^{\circ})+i\sin (4\cdot60^{\circ})\rbrack \\ \\ z^4=16\cdot\lbrack\cos (240^{\circ})+i\sin (240^{\circ})\rbrack \end{gathered}$

Now, notice that:

$\begin{gathered} \cos (240^{\circ})=-\frac{1}{2} \\ \\ \sin (240^{\circ})=-\frac{\sqrt[]{3}}{2} \end{gathered}$

So, we obtain:

$z^4=16\mleft(-\frac{1}{2}-\frac{\sqrt[]{3}}{2}i\mright)=-\frac{16}{2}-\frac{16}{2}\sqrt[]{3}i=-8-8\sqrt[]{3}i\cong-8-13.9i$

Therefore, option D is correct.