Why don't we use trigonometric functions for the output neurons? Why don't we use a trigonometric function, such as $\tan(x)$, where $x$ is an element of the interval $[0,pi/2)$, instead of the sigmoid function for the output neurons (in the case of classification)?
The main reason why the sigmoid function is used is because it 'does not blow up' since it is between 0 and 1 always. As for the relu it is used because it is computationally cheap and even resolves the problem of vanishing gradient(and hence used more often than sigmoid).
So a function like tan(x) will blow up for certain values of x. This can cause a problem of exploding gradients. So, I believe for this reason tan(x) cannot be a good non-linearity to be used.
As for any other function, it is more because of the results that we have gotten over the years and sigmoid and relu have been promising.