Chapter 6.6, Problem 75E

Solution

To find: The parametric equations that represent ${\left(\sqrt{2}+i\right)}^n$ for $n=t$ . Also, draw and label an accurate spiral representing ${\left(\sqrt{2}+i\right)}^n$ for $n=0,1,2,3,4.$

The parametric equations are $x={\left(\sqrt{3}\right)}^t\cos 0.62t$ and $y={\left(\sqrt{3}\right)}^t\sin 0.62t$ , and graph is shown in figure (1).

Given information:

The complex number is ${\left(\sqrt{2}+i\right)}^n$ .

Formula used:

The polar form of a complex number $z=a+bi$ is given by,

  $z=r\left(\cos \theta +i\sin \theta \right)$

Here, $r=\sqrt{a^2+b^2}$ and $\tan \theta =\frac{b}{a}$ .

By the De Moivre’s Theorem,

If $z=r\left(\cos \theta +i\sin \theta \right)$ and $n$ is a positive integer then,

  $\begin{array}{c}{z}^n={\left[r\left(\cos \theta +i\sin \theta \right)\right]}^n\\ =r^n\left(\cos n\theta +i\sin n\theta \right)\end{array}$

The $n\text{th}$ roots of a complex number $z=r\left(\cos \theta +i\sin \theta \right)$ which are $n$ distinct complex numbers are given by,

  $\sqrt[n]{r}\left(\cos \frac{\theta +2\pi k}{n}+i\sin \frac{\theta +2\pi k}{n}\right)$

Here, $k=0,1,2,\mathrm{...},n-1$

Calculation:

From the given complex number,

  $\begin{array}{c}a=\sqrt{2}\\ b=1\end{array}$

Substitute $\sqrt{2}$ for $a$ and $1$ for $b$ in equation $r=\sqrt{a^2+b^2}$ to find modulus $r$ .

  $\begin{array}{c}r=\sqrt{{\left(\sqrt{2}\right)}^2+{\left(1\right)}^2}\\ =\sqrt{2+1}\\ =\sqrt{3}\end{array}$

Substitute $\sqrt{2}$ for $a$ and $1$ for $b$ in equation $\tan \theta =\frac{b}{a}$ to find argument.

  $\begin{array}{c}\tan \theta =\frac{1}{\sqrt{2}}\\ \theta ={\tan }^{-1}\left(\frac{1}{\sqrt{2}}\right)\\ \approx 0.62\end{array}$

Substitute $\sqrt{3}$ for $r$ and $0.62$ for $\theta$ in equation $z=r\left(\cos \theta +i\sin \theta \right)$ .

  $z=\sqrt{3}\left(\cos 0.62+i\sin 0.62\right)$

From the above complex number,

  $\begin{array}{c}r=\sqrt{3}\\ \theta =0.62\\ n=t\end{array}$

Substitute $\sqrt{3}$ for $r$ , $0.62$ for $\theta$ , and $t$ for $n$ , in expression $z=r^n\left(\cos n\theta +i\sin n\theta \right)$ to find first root.

  $z={\left(\sqrt{3}\right)}^t\left(\cos 0.62t+i\sin 0.62t\right)$

Plot the parametric equations $x={\left(\sqrt{3}\right)}^t\cos 0.62t$ and $y={\left(\sqrt{3}\right)}^t\sin 0.62t$ on viewing window of $[-8,5]\text{by}[-2,6]$ as follows.

  

Figure (1)

Therefore, the parametric equations are $x={\left(\sqrt{3}\right)}^t\cos 0.62t$ and $y={\left(\sqrt{3}\right)}^t\sin 0.62t$ , and graph is shown in figure (1).