Question

Select the correct answer.For 2 =2cis , find in rectangular form.O A. 16731673 - 161ОВ.16V3 + 161OC. –32/3 – 321OD.-1673 - 161O E. -1673 + 161|

Answer 1

To raise a complex number to a power, we use the De Moivre's Theorem.

Given,

`z=2cis(\pi)/(6)`

We can expland and write in >>>

`\begin{gathered} 2(\cos \theta+i\sin \theta) \\ =2(\cos (\pi)/(6)+i\sin (\pi)/(6)) \end{gathered}`

De Moivre's theorem tells us >>>>

`\lbrack r(\cos \theta+i\sin \theta)\rbrack^n=r^n(\cos n\theta+i\sin n\theta)`

Let's raise the complex number given to the fifth power >>>>

`\begin{gathered} \lbrack2(\cos (\pi)/(6)+i\sin (\pi)/(6))\rbrack^5 \\ =2^5(\cos (5\pi)/(6)+i\sin (5\pi)/(6)) \\ =32(-(\sqrt3)/(2)+i(1)/(2)) \\ =-16\sqrt[]{3}+16i \end{gathered}`