Chapter 6.6, Problem 28E

To determine

To write: the standard form of the complex number and represent it graphically.

Answer to Problem 28E

The standard form of the given complex number is $-2-2i$ .

Explanation of Solution

Given information:

The given Trigonometric complex number is

  $\sqrt{8}\left(\cos 225°+i\sin 225°\right)$

Properties used:

The standard form of a complex number is $x+iy$ .

Calculation:

The trigonometric form of the complex number is

  $\sqrt{8}\left(\cos 225°+i\sin 225°\right)=2\sqrt{2}\left(\cos \left(180°+45°\right)+i\sin \left(180°+45°\right)\right)$

As sine and cosine are negative in the third quadrant, so

  $=2\sqrt{2}\left(-\cos \left(45°\right)-i\sin \left(45°\right)\right)$

Now,

  $\begin{array}{l}\cos 45°=\frac{1}{\sqrt{2}}\\ \sin 45°=\frac{1}{\sqrt{2}}\end{array}$

Therefore,

  $\begin{array}{c}2\sqrt{2}\left(-\cos \left(45°\right)-i\sin \left(45°\right)\right)=2\sqrt{2}\left(-\frac{1}{\sqrt{2}}-i\frac{1}{\sqrt{2}}\right)\\ =2\left(-1-i\right)\\ =-2-2i\end{array}$

Thus, the standard form of the given complex number is $-2-2i$ .

Represent $-2-2i$ graphically