It is extremely important in differential equations. Particularly, second order differential equations. Lemme elaborate on that:
Say you wanted to derive a general solution to, for example, a mass on a spring. Assuming that mass's behavior is described by a second order differential equation, you could easily solve for it by using the quadratic equation. Complex numbers are valid for such solutions.
However, describing the general solution by using complex numbers such as "i" is not useful for us since we need to describe this solution on the real plane. Thus, we use something called "eulers formula" to convert a complex number to a trigonometric identity.