That's a strange question!

First and foremost, mathematics isn't actually required to be useful in the real world to have a "purpose". The equation derived in this article is intrinsically beautiful.

But of all the branches of mathematics, trig is one of the most useful - you use it almost anytime you need numerical answers out of geometry - for example, if you're building a building, or any sort of machine or vehicle, or if you're rendering some 2D or 3D scene.

Electrical engineering also makes heavy use of trigonometry because Euler's Identity connects rotation and exponentiation: https://en.wikipedia.org/wiki/Euler%27s_identity

In fact, your typical electrical power signal comes in the form of a sine wave.

It is my belief that almost all human numerical fields use trigonometry to some extent. Here are some uses for it in economics: https://economics.stackexchange.com/questions/19172/applications-of-trig-functions-in-economics