Chapter 6, Problem 148RE

a.

To determine

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

Answer to Problem 148RE

  $z_0=3(\cos {45}^0+i\sin {45}^0),z_1=3(\cos {135}^0+i\sin {135}^0)$

  $z_2=3(\cos {225}^0+i\sin {225}^0)$ and $z_3=3(\cos {315}^0+i\sin {315}^0)$ .

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, there are three roots, then they are:

  $\frac{{360}^0}{n}=\frac{{360}^0}{4}={90}^0$

Starting from the $x-$ axis going counter clockwise, the roots are:

  $\begin{array}{l}{z}_0=3(\cos {45}^0+i\sin {45}^0)\\ z_1=3(\cos {135}^0+i\sin {135}^0)\\ z_2=3(\cos {225}^0+i\sin {225}^0)\\ z_3=3(\cos {315}^0+i\sin {315}^0)\end{array}$

Thus, the roots in trigonometric forms are $z_0=3(\cos {45}^0+i\sin {45}^0),z_1=3(\cos {135}^0+i\sin {135}^0)$

  $z_2=3(\cos {225}^0+i\sin {225}^0)$ and $z_3=3(\cos {315}^0+i\sin {315}^0)$ .

b.

To determine

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

Answer to Problem 148RE

  $-81$ .

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:

  $\begin{array}{l}{z}_0=3(\cos {45}^0+i\sin {45}^0)\\ z_1=3(\cos {135}^0+i\sin {135}^0)\\ z_2=3(\cos {225}^0+i\sin {225}^0)\\ z_3=3(\cos {315}^0+i\sin {315}^0)\end{array}$

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 $\omega$ ,

  $\begin{array}{l}\omega =z_0^4={[}^3\\ z_0^4=3^4[\cos (4\cdot {45}^0)+i\sin (4\cdot {45}^0)]\\ z_0^4=81(\cos {180}^0+i\sin {180}^0)\\ z_0^4=81\left(-1+0\right)\\ z_0^4=z=-81\end{array}$

Thus, the complex number is $-81$ .