Applications of Trig functions in Economics?
The main property of trig functions is their cyclicality. Then one would think that they could be ideal in time series analysis, to model "fluctuations around a trend". I believe that the reasons they are not actually used in such a setting are 

1) They are deterministic functions, so they do not allow for fluctuations to be stochastic

2) If the researcher wants to create a model that produces up and down fluctuations (oscillations) around a trend, he would want to obtain that property from the behavioral and other assumptions of the model. If he were to use a trig function, he would a priori impose on the model the sought theoretical outcome.

Instead, one opts for difference-differential equations. There we obtain oscillations (damped or not) if some characteristic roots are complex -and then the trig functions appear, but as an alternative representation, not as buidling blocks.