Question

How do you divide `(3+4i) / (9-4i)` in trigonometric form?

Answer 1

`0.51(cos(1.35)+isin(1.35))`

Explanation:

Let the quotient be `q=3+4i`
and the divisor be `d=9-4i`

In trigonometric form:
`color(white)("XXX")q=r_q(cos(theta_q)+isin(theta_q))`
with
`color(white)("XXX")r_q=sqrt(3^2+4^2)=5`
and
`color(white)("XXX")theta_q= arctan(4/3)~~0.927295`

Similaryly, in trigonometric form:
`color(white)("XXX")d=r_d(cos(theta_d)+isin(theta_d))`
with
`color(white)("XXX")r_d=sqrt(9^2+(-4)^2) =sqrt(97)~~9.848858`
and
`color(white)("XXX")theta_d=arctan(9/(-4))~~-0.41822`

Using trigonometric form:
`color(white)("XXX")q/d = (r_q)/(r_d)(cos(theta_q-theta_d)+isin(theta_q-theta_d))`

`color(white)("XXX")=5/sqrt(97)(cos(0.927295-(-041822))+isin(0.927295-(-0.41822))`

`color(white)("XXX")~~0.51(cos(1.35)+isin(1.35))`