# How do you write the complex number z = −8 in trigonometric form?

Feb 26, 2016

$8 \left(\cos \left(\frac{\pi}{2}\right) + i \sin \left(\frac{\pi}{2}\right)\right)$
$z = - 8 + 0 i$
In general $a + b i \Rightarrow r \left(\cos \left(\theta\right) + i \sin \left(\theta\right)\right)$
with $r = \sqrt{{a}^{2} + {b}^{2}}$
and $\theta = \left\{\begin{matrix}\arctan \left(\frac{b}{a}\right) \mathmr{if} a > 0 \\ \frac{\pi}{2} \mathmr{if} a = 0 \mathmr{and} b > 0 \\ \pi + \arctan \left(\frac{b}{a}\right) \mathmr{if} a < 0 \\ - \frac{\pi}{2} \mathmr{if} a = 0 \mathmr{and} b < 0 \\ \text{undefined} \mathmr{if} a = 0 \mathmr{and} b = 0\end{matrix}\right.$