Question

How do you convert `3i` to polar form?

Answer 1

Use `z=r(costheta +isintheta)`

Explanation:

A complex number takes the form `z=a+bi`.
In this example, `a=0` and `b=3` because `z=0+3i`.

To find `r`, use the pythagorean theorem.
`r=sqrt(a^2 +b^2)`.
`r=sqrt(0^2 +3^2) = 3`

To find `theta`, think about `a` as a value along the x-axis, and `b` as a value along the y axis. In this case, `a` is zero. So, we have a point on the positive y axis, which implies `theta=pi/2`.

Substitute r and theta into `z=r(costheta+isintheta)`
`z=3(cos(pi/2) + isin(pi/2))`

Of course, `cos(pi/2) = 0`, so some teachers would give the answer as `z=3isin(pi/2)`.