Question

Express the complex number in trigonometric form. 3 - 3i radians pls

Answer 1

$\bf \begin{array}{llll} 3&-&3i\\ \uparrow &&\uparrow \\ a&&b \end{array}\qquad \begin{cases} r=\sqrt{a^2+b^2}\\ \theta=tan^{-1}\left( \frac{b}{a} \right) \end{cases}\\\\ -----------------------------\\\\ r=\sqrt{3^2+3^2}\implies r=\sqrt{18}\implies r=3\sqrt{2} \\\\\\ tan(\theta)=\cfrac{3}{3}\implies tan(\theta)=1\implies \measuredangle \theta=tan^{-1}(1) \\\\\\ \measuredangle \theta=\frac{\pi }{4}\impliedby \textit{reference angle}$

now, b is -3, and a is 3
or in rectangular if you wish, y = -3 and x = 3....  on what quadrant is "y" negative and "x" positive?  well, the 4th quadrant
so, using our reference angle, that'd be $\bf \cfrac{7\pi }{4}\quad or \quad -\cfrac{\pi }{4}$

so, let's use the first one

$\bf (3,3i)\implies \begin{array}{llll} r[cos(\theta)+i\ sin(\theta)]\\\\ 3\sqrt{2}\left[ cos\left( \frac{7\pi }{4} \right)+i\ sin\left( \frac{7\pi }{4} \right) \right] \end{array}$