Chapter 6, Problem 145RE

a.

To determine

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

Answer to Problem 145RE

  $z_0=4(\cos {60}^0+i\sin {60}^0),z_1=4(\cos {180}^0+i\sin {180}^0)$ and $z_2=4(\cos {300}^0+i\sin {300}^0)$ .

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:

  $z_i=r_1$

  $i=0,1,2$

(cosθi+isinθi) , where .

From the given graph,$\begin{array}{l}{r}_0=r_1=r_2=4\\ {\theta }_0={60}^0\\ {\theta }_1={180}^0\\ {\theta }_2={300}^0\end{array}$

Now, write the roots:

  $\begin{array}{l}{z}_0=4(\cos {60}^0+i\sin {60}^0)\\ z_1=4(\cos {180}^0+i\sin {180}^0)\\ z_2=4(\cos {300}^0+i\sin {300}^0)\end{array}$

Thus, the roots in trigonometric forms are $z_0=4(\cos {60}^0+i\sin {60}^0),z_1=4(\cos {180}^0+i\sin {180}^0)$ and $z_2=4(\cos {300}^0+i\sin {300}^0)$ .

b.

To determine

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

Answer to Problem 145RE

  $-64$ .

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:

  $\begin{array}{l}{z}_0=4(\cos {60}^0+i\sin {60}^0)\\ z_1=4(\cos {180}^0+i\sin {180}^0)\\ z_2=4(\cos {300}^0+i\sin {300}^0)\end{array}$

These equations verify the equation:

  $z^3=\omega$

Now find

  $\omega$ , by using the fact that any

  $z_i$ verifies the above equation:

  $\omega =z_0^3=z_1^3=z_2^3$

To find $\omega$ ,

  $\begin{array}{l}\omega =z_1^3={[}^4\\ =4^3{(}^{-}=64{(}^{-}=-64\\ \omega =-64\end{array}$

Thus , the complex number is $-64$ .