Question

How do you divide `( 4i+1) / (-8i +5 )` in trigonometric form?

Answer 1

`0.315i - 0.303`

Explanation:

First, convert both the numerator and denominator into polar form.

Converting `4i+1` to polar:

`r = sqrt(4^2 + 1^2) = sqrt17`
`theta = tan^-1(4/1) = 75.96^@`

`4i + 1 = sqrt17 color(white)"-" angle color(white)"."75.96^@`

Converting `-8i + 5` to polar:

`r = sqrt((-8)^2 + 5^2) = sqrt89`
`theta = tan^-1(-8/5) = -57.99^@`

`-8i + 5 = sqrt89 color(white)"-" angle color(white)"."-57.99^@`

Dividing in polar form:

`(sqrt17 color(white)"-" angle color(white)"."75.96^@) / ( sqrt89 color(white)"-" angle color(white)"."-57.99^@) = sqrt17/sqrt89 color(white)"-" angle color(white)"-" (75.96-(-57.99))^@`

`= sqrt17/sqrt89 color(white)"-" angle color(white)"." 133.95^@`

Now to convert back to rectangular:

`Re = r cos theta = sqrt17 / sqrt89 cos133.95^@ = -0.303`

`Im = ri sin theta = sqrt17/sqrt89 i color(white)"." sin133.95^@ = 0.315i`

So the final answer is `0.315i - 0.303`