Chapter 6.6, Problem 108E

a)

To determine

To find: The Square roots of $16\left(\cos 60°+i\sin 60°\right)$

Answer to Problem 108E

The square roots of $16\left(\cos 60°+i\sin 60°\right)$ are as follows:

  $\begin{array}{l}\Rightarrow 4\left(\cos \frac{\pi }{6}+i\sin \frac{\pi }{6}\right)\\ \Rightarrow 4\left(\cos \frac{7\pi }{6}+i\sin \frac{7\pi }{6}\right)\end{array}$

Explanation of Solution

Given information: The complex number $16\left(\cos 60°+i\sin 60°\right)$

Formula used: $n^{th}$ roots of the complex number: $\sqrt[n]{r}\left(\cos \frac{\frac{3\pi }{2}+2\pi k}{6}+i\sin \frac{\frac{3\pi }{2}+2\pi k}{6}\right);k=0,1,2,3,\mathrm{...},n-1$

Calculation: Apply the $n^{th}$ roots formula to obtain the roots of the given number.

  $16\left(\cos 60°+i\sin 60°\right)=16\left(\cos \frac{\pi }{3}+i\sin \frac{\pi }{3}\right)$

  $\begin{array}{l}\sqrt{16}\left(\cos \frac{\frac{\pi }{3}+2\pi k}{2}+i\sin \frac{\frac{\pi }{3}+2\pi k}{2}\right)=4\left(\cos \frac{\pi +6\pi k}{6}+i\sin \frac{\pi +6\pi k}{6}\right)\\ \therefore 4\left(\cos \frac{\pi +6\pi \left(0\right)}{6}+\frac{\pi +6\pi \left(0\right)}{6}\right)=4\left(\cos \frac{\pi }{6}+i\sin \frac{\pi }{6}\right)\\ \therefore 4\left(\cos \frac{\pi +6\pi \left(1\right)}{6}+\frac{\pi +6\pi \left(1\right)}{6}\right)=4\left(\cos \frac{7\pi }{6}+i\sin \frac{7\pi }{6}\right)\end{array}$

b)

To determine

To find: The Square roots of $16\left(\cos 60°+i\sin 60°\right)$

Answer to Problem 108E

The standard forms of Square roots of $16\left(\cos 60°+i\sin 60°\right)$ are as follows:

  $\begin{array}{l}\Rightarrow 2\sqrt{3}+2i\\ \Rightarrow -2\sqrt{3}-2i\end{array}$

Explanation of Solution

Given information: The complex number $16\left(\cos 60°+i\sin 60°\right)$

Formula used: $n^{th}$ roots of the complex number: $\sqrt[n]{r}\left(\cos \frac{\frac{3\pi }{2}+2\pi k}{6}+i\sin \frac{\frac{3\pi }{2}+2\pi k}{6}\right);k=0,1,2,3,\mathrm{...},n-1$

Calculation: Apply the $n^{th}$ roots formula to obtain the roots of the given number.

  $16\left(\cos 60°+i\sin 60°\right)=16\left(\cos \frac{\pi }{3}+i\sin \frac{\pi }{3}\right)$

  $\begin{array}{l}\sqrt{16}\left(\cos \frac{\frac{\pi }{3}+2\pi k}{2}+i\sin \frac{\frac{\pi }{3}+2\pi k}{2}\right)=4\left(\cos \frac{\pi +6\pi k}{6}+i\sin \frac{\pi +6\pi k}{6}\right)\\ \therefore 4\left(\cos \frac{\pi +6\pi \left(0\right)}{6}+\frac{\pi +6\pi \left(0\right)}{6}\right)=4\left(\cos \frac{\pi }{6}+i\sin \frac{\pi }{6}\right)\\ \therefore 4\left(\cos \frac{\pi +6\pi \left(1\right)}{6}+\frac{\pi +6\pi \left(1\right)}{6}\right)=4\left(\cos \frac{7\pi }{6}+i\sin \frac{7\pi }{6}\right)\end{array}$

Standard forms of the roots are as follows:

  $\begin{array}{l}\Rightarrow 4\left(\cos \frac{\pi }{6}+i\sin \frac{\pi }{6}\right)=2\sqrt{3}+2i\\ \Rightarrow 4\left(\cos \frac{7\pi }{6}+i\sin \frac{7\pi }{6}\right)=-2\sqrt{3}-2i\end{array}$

c)

To determine

To graph: The Square roots of $16\left(\cos 60°+i\sin 60°\right)$

Explanation of Solution

Given information: The Square roots of $16\left(\cos 60°+i\sin 60°\right)$

Graph: The required graph of roots are given below,