Trigonometric Form

[chelseafry19]

I have no idea how to rewrite things in trigonometric form

"Rewrite in trigonometric form:"

-2 + sqrt(3i)

 

[soroban]

Hello, chelseafry19!

$\displaystyle \text{Rewrite in trigonometric form: }\;z \;=\;-2 + \sqrt{3}\,i$

Graph the complex number.

        *         |
        : *       |
      _ :   *     |
     /3 :     *
        :       * @
  - - - + - - - - + - - - -
            -2    |

\(\displaystyle \text{The hypotenuse is: }\:r^2 \:=\-2)^2 + (\sqrt{3})^2 \:=\:4+3 \:=\:7 \quad\Rightarrow\quad r \:=\:\sqrt{7}\)

$\displaystyle \text{Note that: }\;\tan\theta \:=\:\frac{\sqrt{3}}{-2} \:=\:-\frac{\sqrt{3}}{2}$

. . $\displaystyle \text{Hence: }\:\theta \:=\:\arctan\left(-\frac{\sqrt{3}}{2}\right)\:\text{where }\theta\text{ is in Quadrant 2.}$


$\displaystyle \text{Therefore: }\;z \;=\;\sqrt{7}\,\bigg(\cos\theta + i\sin\theta\bigg)$

 

[chelseafry19]

Thank you!