Trigonometric functions fulfill the same roles as seasonal components in exponential smoothing: they are periodic and tie together time points one cycle apart.
The advantage of trigonometrics is that they are nice and smooth. If you just treat the, say, 12 seasonal indices in monthly exponential smoothing as free parameters, they do not hang together in any way. The seasonal index for January could be -10, for February +15, for March -5, for April +27, without rhyme or reason - this "models" discontinuous jumps between months, which usually does not make a lot of sense. Trigonometric functions impose a certain continuity between neighboring points.
Alternatives to trigonometric functions would be periodic splines, or periodic "bump functions".
One example for using trigonometric functions in time series modeling and forecasting is the tbats model. The "T" actually stands for "trigonometric". And there are people who work in the frequency domain, using Fourier transforms, which of course are also trigonometry.
Note that seasonal ARIMA will typically not use seasonal dummies, but seasonal differences. (Of course, you can run a regression with ARIMA errors or an ARIMAX model, where you account for the seasonality using trigonometric or other periodic functions. This requires a few degrees of freedom, but it doesn't throw away an entire year's worth of observations by seasonal differencing.)