Neural networks usually don't use trigonometry or complex numbers.
> The signal each neuron outputs is calculated from this number, according to its activation function
The activation function is usually a sigmoid, which is also usually defined in terms of trigonometric functions
Which neural networks don’t use trigonometric functions or equivalent?
I don't immediately see how cordic would help you calculate e^(x) unless x is complex valued (which it is usually not).
It's literally been decades, but the last I learned about neural networks, a non-linear activation function was important for Turing completeness.
The nonlinearity (plus at least two layers) are required to solve nonlinear problems (like the famous XOR example).
However, the "decision point" of the ReLU (and it's everywhere-differentiable friends like leaky ReLU or ELU) provides a sufficient non-linearity - in essence, just as a sigmoid effectively results in a yes/no chooser with some stuff in the middle for training purposes, so does the ReLU "elbow point".
Sigmoid has a problem of 'vanishing gradients' in deep networks, as the sigmoid gradients of 0 - 0.25 in standard backpropagation means that a 'far away' layer will have tiny, useless gradients if there's a hundred sigmoid layers in between.
clamp(x/2, -1, 1) is basically a sigmoid.