Chapter 6.6, Problem 1E

To determine

To fill in the blanks:

The of the complex number $z=a+bi$ is $z=r\left(\cos \theta +i\sin \theta \right)$ where $r$ is the of $z$ and $\theta$ is an of $z$ .

Answer to Problem 1E

The trigonometric form of the complex number $z=a+bi$ is $z=r\left(\cos \theta +i\sin \theta \right)$ where $r$ absolute value of is $z$ and $\theta$ is an argument of $z$ .

Explanation of Solution

Given information:

The incomplete sentence is:

‘The of the complex number $z=a+bi$ is $z=r\left(\cos \theta +i\sin \theta \right)$ where $r$ is the of $z$ and $\theta$ is an of $z$ .’

If there is complex number is represented by equation $z=a+bi$ then its absolute value and argument are shown by $r$ and $\theta$ .

Absolute value of complex number is the mod value and argument is the angle in radians.

Therefore, the complete sentence is:

‘The trigonometric form of the complex number $z=a+bi$ is $z=r\left(\cos \theta +i\sin \theta \right)$ where $r$ absolute value of is $z$ and $\theta$ is an argument of $z$ .’