# How do you write the complex number in trigonometric form -7i?

Jan 6, 2018

The answer is $= 7 \left(\cos \left(- \frac{\pi}{2}\right) + i \sin \left(- \frac{\pi}{2}\right)\right) = 7 {e}^{- \frac{1}{2} i \pi}$

#### Explanation:

Any complex number $z = a + i b$ can be represented as

$z = r \left(\cos \theta + i \sin \theta\right)$

Where,

$r = | | z | | = \sqrt{{a}^{2} + {b}^{2}}$

$\cos \theta = \frac{a}{| | z | |}$

and

$\sin \theta = \frac{b}{|} | z | |$

Here, we have

$z = 0 - 7 i$

$| | z | | = \sqrt{{\left(0\right)}^{2} + {\left(- 7\right)}^{2}} = 7$

$z = 7 \left(\left(\frac{0}{7}\right) + \left(- \frac{7}{7}\right) i\right)$

$\cos \theta = 0$ and $\sin \theta = - 1$

Therefore,

$\theta = - \frac{\pi}{2}$ , $\left[\mod 2 \pi\right]$

So,

$z = 7 \left(\cos \left(- \frac{\pi}{2}\right) + i \sin \left(- \frac{\pi}{2}\right)\right)$