Chapter 6, Problem 104CR

To determine

To calculate:

The value of the following complex number in the trigonometric from without using calculator:

  $-9$

Answer to Problem 104CR

The trigonometric form of the complex number is $z=9\left(\cos 0+i\sin 0\right)$ .

Explanation of Solution

Given information:

  $-9$

Formula used:

The trigonometric form of the complex number $z=a+bi$ is given by the following expression:

  $z=r\left(\cos \theta +i\sin \theta \right)$

The modulus or absolute value of complex number $z=a+bi$ is given by the following expression:

  $r=\sqrt{a^2+b^2}$

The argument of complex number $z=a+bi$ is given by the following expression:

  $\theta ={\tan }^{-1}\left(\frac{b}{a}\right)$

Calculation:

The given complex number is:

  $-9$

Compare it with $z=a+bi$ :

  $\begin{array}{l}a=-9\\ b=0\end{array}$

The modulus is calculated as:

  $\begin{array}{l}r=\sqrt{a^2+b^2}\\ r=\sqrt{{\left(-9\right)}^2+{\left(0\right)}^2}\\ r=9\end{array}$

The argument is calculated as:

  $\begin{array}{l}\theta ={\tan }^{-1}\left(\frac{b}{a}\right)\\ \theta ={\tan }^{-1}\left(\frac{0}{9}\right)\\ \theta ={\tan }^{-1}\left(0\right)\\ \theta =0\end{array}$

The trigonometric form of the complex number is calculated as:

  $\begin{array}{l}z=r\left(\cos \theta +i\sin \theta \right)\\ z=9\left(\cos 0+i\sin 0\right)\end{array}$