Complex numbers

**mike1** · January 14th 2007, 04:22 PM

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.

**Soroban** · January 14th 2007, 06:16 PM

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

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

**mike1** · January 14th 2007, 08:56 PM

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

Thread: Complex numbers

LinkBack

Thread Tools

Display

Complex numbers

Similar Math Help Forum Discussions

raising complex numbers to complex exponents?

Finding real numbers in equations containing complex numbers

What is complex slope of a line in complex numbers

Pre-Calc Graphing with Complex numbers. Multiplying with complex numbers.

Complex Numbers- Imaginary numbers

Search Tags