Chapter 6, Problem 106RE

(a)

To determine

To find: The trigonometric form of the complex number.

Answer to Problem 106RE

The trigonometric form of the complex number is $\frac{\left[\sqrt{2}\left(\cos \frac{5\pi }{4}+i\sin \frac{5\pi }{4}\right)\right]}{\left[2\sqrt{2}\left(\cos \frac{5\pi }{4}+i\sin \frac{5\pi }{4}\right)\right]}$

Explanation of Solution

Given information:

The given complex number is,

  $\frac{-1-i}{-2-2i}$

The absolute value of complex number is calculated as,

  $\begin{array}{c}{r}_1=\left|-1-i\right|\\ =\sqrt{{\left(-1\right)}^2+{\left(-1\right)}^2}\\ =\sqrt{2}\end{array}$

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

Calculate the angle $\theta$ :

  $\begin{array}{l}\tan {\theta }_1=\frac{-1}{-1}\\ {\theta }_1=\frac{5\pi }{4}\\ \tan {\theta }_2=\frac{-2}{-2}\\ {\theta }_2=\frac{5\pi }{4}\end{array}$

The trigonometric form of the complex numberis calculated as,

  $\frac{-1-i}{-2-2i}=\frac{\left[\sqrt{2}\left(\cos \frac{5\pi }{4}+i\sin \frac{5\pi }{4}\right)\right]}{\left[2\sqrt{2}\left(\cos \frac{5\pi }{4}+i\sin \frac{5\pi }{4}\right)\right]}$

Therefore, the trigonometric form of the complex numberis $\frac{\left[\sqrt{2}\left(\cos \frac{5\pi }{4}+i\sin \frac{5\pi }{4}\right)\right]}{\left[2\sqrt{2}\left(\cos \frac{5\pi }{4}+i\sin \frac{5\pi }{4}\right)\right]}$ .

(b)

To determine

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

Answer to Problem 106RE

The result of operation one the complex number is $\frac{1}{2}$ .

Explanation of Solution

Given information:

The given complex number is,

  $\frac{-1-i}{-2-2i}$

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

  $\begin{array}{c}\frac{-1-i}{-2-2i}=\frac{\left[\sqrt{2}\left(\cos \frac{5\pi }{4}+i\sin \frac{5\pi }{4}\right)\right]}{\left[2\sqrt{2}\left(\cos \frac{5\pi }{4}+i\sin \frac{5\pi }{4}\right)\right]}\\ =\frac{1}{2}\end{array}$

Therefore, the result of operation one the complex number is $\frac{1}{2}$ .

(c)

To determine

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

Answer to Problem 106RE

The result of operation one the complex number is $\frac{1}{2}$ .

Explanation of Solution

Given information:

The given complex number is,

  $\frac{-1-i}{-2-2i}$

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

  $\begin{array}{c}\frac{-1-i}{-2-2i}=\frac{\left(-1-i\right)\left(-2+2i\right)}{\left(-2-2i\right)\left(-2+2i\right)}\\ =\frac{2-2i+2i+2}{4+4}\\ =\frac{1}{2}\end{array}$

Therefore, the result of operation one the complex number is $\frac{1}{2}$ .