Question

Express the complex number in trigonometric form. 4i

Answer 1

Question:   Express $4i$ in trigonometric form.

Answer:

A few ways to express this is:

$4(\cos(\frac{\pi}{2})+\sin(\frac{\pi}{2})i)$

$4cis(\frac{\pi}{2})$

$4e^{i\frac{\pi}{2}}$

Choose the notation your class is using.

Step-by-step explanation:

$r(\cos(\theta)+\sin(\theta)i)$ is trigonometric form.

Distributing $r$ to terms inside the ( ) gives:

$r\cos(\theta)+r\sin(\theta)i$.

Our number has no real part so we are just trying to find $r$ and $\theta$ such that $r\sin(\theta)=4$.

Since we want $\cos(\theta)=0$, we will have to $\sin(\theta)=1$ (or -1).

This happens at $\frac{\pi}{2}$.

This means $\theta=\frac{\pi}{2}$.

So while $\sin(\theta)=1$ this forces $r=4$

The trigonometric form is $4(\cos(\frac{\pi}{2})+\sin(\frac{\pi}{2})i)$.

Some people like to express this as $4cis(\frac{\pi}{2})$ which is way shorter.

Some people might also express this as $4e^{i\frac{\pi}{2}}$.