Question

Let z=13+7i and w=3[cos(1.43)+isin(1.43)]
A) Convert z to polar form
B) Calculate zw using De Moivre's theorem
C) Calculate z/w using De Moivre's theorem

Answer 1

For \(z=x+iy=r\exp(i\theta)\), you have
\[\begin{cases}r=\sqrt{x^2+y^2}\\\theta=\tan^{-1}\dfrac{y}{x}\end{cases}\]
\[r=\sqrt{13^2+7^2}=\sqrt{218}\approx14.765\]
\[\theta=\tan^{-2}\frac{7}{13}\approx0.494\]

Answer 2

Then, for \(z=r\exp(i\theta)\) and \(w=s\exp(i\phi)\), you have
\[z\cdot w=rs\exp(i(\theta+\phi))~~~\text{and}~~~\frac{z}{w}=\frac{r}{s}\exp(i(\theta-\phi))\]

Answer 3

Thanks