Chapter 6, Problem 103RE

(a)

To determine

To find: The trigonometric form of the complex number.

Answer to Problem 103RE

The trigonometric form of the complex number is $\left[2\sqrt{2}\left(\cos \frac{7\pi }{4}+i\sin \frac{7\pi }{4}\right)\right]\left[3\sqrt{2}\left(\cos \frac{\pi }{4}+i\sin \frac{\pi }{4}\right)\right]$

Explanation of Solution

Given information:

The given complex number is,

  $\left(2-2i\right)\left(3+3i\right)$

The absolute value of complex number is calculated as,

  $\begin{array}{c}{r}_1=\left|2-2i\right|\\ =\sqrt{{\left(2\right)}^2+{\left(-2\right)}^2}\\ =2\sqrt{2}\end{array}$

  $\begin{array}{c}{r}_2=\left|3+3i\right|\\ =\sqrt{{\left(3\right)}^2+{\left(3\right)}^2}\\ =3\sqrt{2}\end{array}$

Calculate the angle $\theta$ :

  $\begin{array}{l}\tan {\theta }_1=\frac{-2}{2}\\ {\theta }_1=\frac{7\pi }{4}\\ \tan {\theta }_2=\frac{3}{3}\\ {\theta }_2=\frac{\pi }{4}\end{array}$

The trigonometric form of the complex numberis calculated as,

  $\left(2-2i\right)\left(3+3i\right)=\left[2\sqrt{2}\left(\cos \frac{7\pi }{4}+i\sin \frac{7\pi }{4}\right)\right]\left[3\sqrt{2}\left(\cos \frac{\pi }{4}+i\sin \frac{\pi }{4}\right)\right]$

Therefore, the trigonometric form of the complex numberis $\left[2\sqrt{2}\left(\cos \frac{7\pi }{4}+i\sin \frac{7\pi }{4}\right)\right]\left[3\sqrt{2}\left(\cos \frac{\pi }{4}+i\sin \frac{\pi }{4}\right)\right]$ .

(b)

To determine

To find: The result of operation one the complex number.

Answer to Problem 103RE

The result of operation one the complex number is $12$ .

Explanation of Solution

Given information:

The given complex number is,

  $\left(2-2i\right)\left(3+3i\right)$

The result of operation one the complex numberis calculated as,

  $\begin{array}{c}\left(2-2i\right)\left(3+3i\right)=\left[2\sqrt{2}\left(\cos \frac{7\pi }{4}+i\sin \frac{7\pi }{4}\right)\right]\left[3\sqrt{2}\left(\cos \frac{\pi }{4}+i\sin \frac{\pi }{4}\right)\right]\\ =12\left(\cos \left(\frac{7\pi }{4}+\frac{\pi }{4}\right)+i\sin \left(\frac{7\pi }{4}+\frac{\pi }{4}\right)\right)\\ =12\left(\cos 2\pi +i\sin 2\pi \right)\\ =12\end{array}$

Therefore, the result of operation one the complex number is $12$ .

(c)

To determine

To find: The result of operation one the complex number in the standard form.

Answer to Problem 103RE

The result of operation one the complex number is $12$ .

Explanation of Solution

Given information:

The given complex number is,

  $\left(2-2i\right)\left(3+3i\right)$

The result of operation one the complex number is calculated as,

  $\begin{array}{c}\left(2-2i\right)\left(3+3i\right)=6+6i-6i+6\\ =12\end{array}$

Therefore, the result of operation one the complex number is $12$ .