Inverse transformation of arctan2

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Similar to this and this I'm trying to recover the original 2D field after the arctan2 operation.

Here's a MWE in python:

import numpy as np
theta = np.random.uniform(low=-3*np.pi, high=3*np.pi, size=400)
theta = theta.reshape(20,20)

y     = np.sin(theta)
x     = np.cos(theta)

phi = np.arctan2(y, x)

# theta = f(phi)?

how to find f? Thanks.