Question

De Moivre's theorem solve and put in rectangular form ASAP

Answer 1

If a complex number is written in trigonometric form:

`z=r(\cos \theta+i\cdot\sin \theta)`

Then, the De Moivre's theorem says that:

`z^n=r^n(\cos (n\theta+i\cdot\sin (n\theta))`

For this particular case, we have that:

`\begin{gathered} r=\sqrt[]{3} \\ \theta=(5\pi)/(6) \end{gathered}`

Then, using the De Moivre's formula:

`\begin{gathered} \lbrack\sqrt[]{3}(\cos (5\pi)/(6)+i\cdot\sin (5\pi)/(6))\rbrack^(10)=\sqrt[]{3}^(10)(\cos 10\cdot(5\pi)/(6)+i\cdot\sin 10\cdot(5\pi)/(6)) \\ =3^(10/2)(\cos (50\pi)/(6)+i\cdot\sin (50\pi)/(6)) \\ =3^5(\cos (25\pi)/(3)+i\cdot\sin (25\pi)/(3)) \\ =243(\cos (8+(1)/(3))\pi+i\cdot\sin (8+(1)/(3))\pi) \\ =243(\cos (\pi)/(3)+i\cdot\sin (\pi)/(3)) \\ =243((1)/(2)+i\cdot\frac{\sqrt[]{3}}{2}) \\ =(243)/(2)+i\cdot\frac{243\sqrt[]{3}}{2} \end{gathered}`

Therefore:

`\lbrack\sqrt[]{3}(\cos (5\pi)/(6)+i\cdot\sin (5\pi)/(6))\rbrack^(10)=(243)/(2)+\frac{243\cdot\sqrt[]{3}}{2}i`