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((\pi)/(2))+\sin((\pi)/(2))i)`

`4cis((\pi)/(2))`

`4e^{i(\pi)/(2)}`

Choose the notation your class is using.

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
`(\pi)/(2)`.

This means
`\theta=(\pi)/(2)`.

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

The trigonometric form is
`4(\cos((\pi)/(2))+\sin((\pi)/(2))i)`.

Some people like to express this as
`4cis((\pi)/(2))` which is way shorter.

Some people might also express this as
`4e^{i(\pi)/(2)}`.