Why are the trigonometric functions so important?

Too long for a comment...

Consider that this may be, at least in part, be due to a historical bias: since trigonometric functions were available since the earliest days of mathematics, mathematicians tried to frame their results in familiar terms instead of in other less popular functions. This effect compounds itself, as more and more results are found that involve trigonometric functions.

So perhaps it's not that sin and cos have fundamental properties that make them pervade modern mathematics, but rather that the mathematics we have developed as humans is one which is historically biased towards geometrical descriptions. In another world where drum acoustics is paramount, perhaps expanding functions in Fourier-Bessel Series is the norm.

That's easy. $\sin$ and $\cos$ are solutions to the differential equation $\frac{d^2y}{dx^2}=-y$. What's more, they're a basis for the solutions, and a simple one at that (having value and derivative at the origin equal to 0 and 1).

Since that's such a simple equation, it's no surprise they appear often - much like their sibling $\exp$, the solution to $\frac{dy}{dx}=y$.

The equation represents a restoring force, and this is why they are so important in things that repeat, such as circles (and of course, the numerous physical applications, such as harmonic oscillators, waves etc.) This is perhaps why they are more common the hyperbolic functions - which solve $\frac{d^2y}{dx^2}=y$, but don't add much over the simpler exponential.

All of mathematics is beautifully integrated. If there were no consistency in this one whole tree of so many branches, it would have fallen apart long ago.

Sin/cos are circular function relations that are so fundamental to rotation of a line segment or vector and must essential for any periodic event or a phenomenon.