Chapter 9.7, Problem 31E

To determine

To describe the transformation applied to the graph of the complex number.

Answer to Problem 31E

The point is rotated clockwise about the origin by an angle of $\theta$ .

Explanation of Solution

Given:

The $z$ is multiplied by $\cos \theta +i\sin \theta$ .

Calculation:

The point is rotated clockwise about the origin by an angle of $\theta$ because:

  $\arg (zz\text{'})=\arg (z)+\arg (z\text{'})=\arg (z)+\theta$

To determine

To describe the transformation applied to the graph of the complex number.

Answer to Problem 31E

The point is rotated clockwise about the origin by an angle of ${60}^{\circ }$ .

Explanation of Solution

Given:

The $z$ is multiplied by $\frac{1}{2}+\frac{\sqrt{3}}{2}i$ .

Calculation:

Since $\frac{1}{2}+\frac{\sqrt{3}}{2}i=\cos \frac{\pi }{3}+i\sin \frac{\pi }{3}$

The point is rotated clockwise about the origin by an angle of $\theta$ because:

  $\arg (zz\text{'})=\arg (z)+\arg (z\text{'})=\arg (z)+\theta$

So it can be concluded that the point is rotated clockwise about the origin by an angle of ${60}^{\circ }$