Question

How do you convert `-7-5i` to polar form?

Answer 1

Please see the explanation.

Explanation:

The radius of polar form is the magnitude of the Cartesian:

`r = |-7 - 5i|`

`|-7 - 5i| = sqrt((-7)^2 + (-5)^2)`

`|-7 - 5i| = sqrt(49 + 25)`

`r = sqrt(74)`

The angle, `theta`, of the polar form is found by using the inverse tangent on the imaginary part divided by the real part but, because the signs indicate that the number is in the 3rd quadrant, we must add `pi` to the value returned by the inverse tangent:

`theta = tan^-1((-5)/-7) + pi`

`theta ~~ 3.76`

The polar form is `(sqrt(74), 3.76)`
The exponential form is `sqrt(74)e^(3.76i)`