1. ## Complex numbers

Any help on these two would be appreciated:

1) Express the complex number -2[sqrt)2+2[sqrt]2i in the trigonometric form r(cos[theta]+isin[theta]) or rcis(theta).

2) Express the complex number 12|cos(pi/3)+isin(pi/3)| in the form a+bi without trigonometric functions.

2. Hello, Mike!

I connect the two forms of the numbers with this diagram.
Code:
        |                 *
|              *  |
|       r   *     |
|        *        |y
|     *           |
|  * θ            |
- * - - - - - - - - * -
x
1) Express the complex number $\text{-}2\sqrt{2} +2\sqrt{2}i$ in the form $r(\cos\theta + i\sin\theta)$

Going from $x + iy$ to $r(\cos\theta + i\sin\theta)$, use: . $\begin{Bmatrix}r^2 & = & x^2+y^2\\ \tan\theta & = & \frac{y}{x}\end{Bmatrix}$

We are given: . $x \,=\,\text{-}2\sqrt{2},\;y \,=\,2\sqrt{2}$

We have: . $r^2 \:=\:(\text{-}2\sqrt{2})^2 + (2\sqrt{2})^2\:=\:8 + 8 \:=\:16\quad\Rightarrow\quad r \,=\,\pm4$

. . And: . $\tan\theta \:=\:\frac{2\sqrt{2}}{\text{-}2\sqrt{2}}\:=\:-1\quad\Rightarrow\quad\theta \:=\:\frac{3\pi}{4},\:\text{-}\frac{\pi}{4}$

We know the point in in Quadrant 2, so we'll use $\theta = \frac{3\pi}{4},\;r = 4$

. . Answer: . $4\left(\cos\frac{3\pi}{4} + i\sin\frac{3\pi}{4}\right)$

2) Express the complex number $12\left(\cos\frac{\pi}{3} + i \sin\frac{\pi}{3}\right)$ in the form $a+bi$

Going from $r(\cos\theta + i\sin\theta)$ to $x + iy$, use: . $\begin{Bmatrix}x \:=\:r\cos\theta \\ y \:=\:r\sin\theta\end{Bmatrix}$

We are given: . $r \,=\,12,\;\theta\,=\,\frac{\pi}{3}$

We have: . $\begin{array}{cc}x \:=\:12\cos\frac{\pi}{3}\:=\:12\left(\frac{1}{2}\r ight) \:=\:6 \\
y \:=\:12\sin\frac{\pi}{3} \:=\:12\left(\frac{\sqrt{3}}{2}\right)\:=\:6\sqrt{ 3} \end{array}$

. . Answer: . $6 + 6\sqrt{3}i$

3. Thanks, that should help me with the other ones I'm working on. Thanks a lot.