Chapter 9, Problem 25RE

To determine

Towrite:the given complex number in polar equation and express the arguments in degrees.

Answer to Problem 25RE

  $r=-1-i=\sqrt{2}\left(\cos {225}^{\circ }+i\sin {225}^{\circ }\right)$ .

Explanation of Solution

Given:

The given polarequation is $r=-1-i$ .

Calculation:

As it is known that $x=r\cos \theta y=r\sin \theta$ .

Therefore, $r=\sqrt{x^2+y^2}\Rightarrow \theta ={\tan }^{-1}\left(\frac{y}{x}\right)$

Also, $a+ib=r\left(\cos \theta +i\sin \theta \right)$

  $\Rightarrow -1-i=\sqrt{{\left(-1\right)}^2+{\left(-1\right)}^2}=\sqrt{2}$ .

Therefore, $\theta ={\tan }^{-1}\left(\frac{-1}{-1}\right)={\tan }^{-1}1={45}^{\circ }$

Since $r=-1-i$ is in third quadrant of the complex plane, therefore, $\theta ={180}^{\circ }+{45}^{\circ }={225}^{\circ }$

Therefore, $r=-1-i=\sqrt{2}\left(\cos {225}^{\circ }+i\sin {225}^{\circ }\right)$