Degrees or radians for angle in complex equation

Question

If you were asked to convert a complex equation to polar form :

$r\cos(\theta) + ir\sin(\theta)$

Does the angle have to be in radians for the value of theta or can the angle be in degrees too ?

In which form would the angle have to be in radians ?

Answer 1

No, the angle can also be in degrees.

To find $\theta$ from the Cartesian representation, you can form a right triangle with the opposite side as the $y$-coordinate, and the adjacent side as the $x$-coordinate.

Then we use the formula $\theta = \tan^{-1} \frac{y}{x}$, and this value can be expressed either in degrees or radians. This is since $x,y$ are dimensionless, so $\tan^{-1} \frac{y}{x}$ also has to be dimensionless.

Answer 2

The magnitude of the complex number is given only by $r$ and $\theta$ is only its argument, which represents the angle by which it is inclined to the real axis in the Argand plane.
So you may use $\theta$ either in degrees or in radians, because we're just finding $\sin$ and $\cos$ of that angle.

Answer 3

Generally speaking, when you are doing calculus, only use radians, otherwise derivatives require an annoying conversion factor.

(In the case at hand, it doesn't make a difference, but it is unnatural to use degrees.)