Chapter 6, Problem 146RE

a.

To determine

The roots in the trigonometric form by using the graph of the roots of the complex number.

Answer to Problem 146RE

  $z_0=4(\cos {60}^0+i\sin {60}^0),z_1=4(\cos {150}^0+i\sin {150}^0),$

  $z_2=4(\cos {240}^0+i\sin {240}^0)$ and $z_3=4(\cos {330}^0+i\sin {330}^0)$ .

Explanation of Solution

Given information:

Write each of the roots in trigonometric form.

  

Calculation:

From the given information, write each of the roots in trigonometric form using the graph of roots are as shown below:

Let’s note the roots:

  $z_i=r_1(\cos {\theta }_i+i\sin {\theta }_i)$ , where

  $i=0,1,2$ .

From the given graph,$\begin{array}{l}{r}_0=r_1=r_2=r_3=4\\ {\theta }_0={60}^0\\ {\theta }_1={150}^0\\ {\theta }_2={240}^0\\ {\theta }_3={330}^0\end{array}$

Now, write the roots:

  $\begin{array}{l}{z}_0=4(\cos {60}^0+i\sin {60}^0)\\ z_1=4(\cos {150}^0+i\sin {150}^0)\\ z_2=4(\cos {240}^0+i\sin {240}^0)\\ z_3=4(\cos {330}^0+i\sin {330}^0)\end{array}$

Thus, the roots in trigonometric forms are $z_0=4(\cos {60}^0+i\sin {60}^0),z_1=4(\cos {150}^0+i\sin {150}^0),$

  $z_2=4(\cos {240}^0+i\sin {240}^0)$ and $z_3=4(\cos {330}^0+i\sin {330}^0)$ .

b.

To determine

The complex number by using roots. Verify the results by using graphing utility.

Answer to Problem 146RE

  $-128-128\sqrt{3}i$ .

Explanation of Solution

Given information:

Identify the complex number whose roots are given.Use a graphing utility to verify your results.

  

Calculation:

From the given information, identify the complex number whose roots are given and by using graphing utility verify the result are as shown below:

From the part (a) found the roots:

  $\begin{array}{l}{z}_0=4(\cos {60}^0+i\sin {60}^0)\\ z_1=4(\cos {150}^0+i\sin {150}^0)\\ z_2=4(\cos {240}^0+i\sin {240}^0)\\ z_3=4(\cos {330}^0+i\sin {330}^0)\end{array}$

These equations verify the equation:

  $z^4=\omega$

Now find

  $\omega$ , by using the fact that any

  $z_i$ verifies the above equation:

  $\omega =z_0^4=z_1^4=z_2^4=z_3^4$

To find $\omega$ ,

  $\begin{array}{l}$ \omega =z_0^4={[4(\cos {60}^0+i\sin {60}^0)]}^4$=4^4[\cos (4\cdot {60}^0)+i\sin (4\cdot {60}^0)]\\ =256(\cos {240}^0+i\sin {240}^0)\\ =256\left(-\frac{1}{2}-\frac{\sqrt{3}}{2}i\right)\\ =-128-128\sqrt{3}i\\ \omega =-128-128\sqrt{3}i\end{array}$

Thus , the complex number is $-128-128\sqrt{3}i$ .