Question

What is the trigonometric form of `(2-12i)`?

Answer 1

`2sqrt(37)(cos(arctan(-6))+isin(arctan(-6))`
`color(white)("XXXXXXXXXX")~~12.166(0.1644-i*0.9864)`

Explanation:

For a complex number in the form: `a+bi`
its trigonometric form is:
`color(white)("XXX")|z|(cos(theta)+i*sin(theta))`
where
`color(white)("XXX")|z| = sqrt(a^2+b^2)`
and
`color(white)("XXX")theta = {(arctan(b/a),"if " a+bi " in QI or QIV"),(pi+arctan(b/a), "if "a+bi " in QII or QIII"):}`

Given `(2-12i)`
`color(white)("XXX")|z| =sqrt(2^2+(-12)^2) =4sqrt(37)`

and since `(2-12i)` is in QII
`color(white)("XXX")theta = arctan(-6)`