A slightly different "angle."
The practical use of the trigonometric functions goes back at least as far as the Greek-speaking mathematicians, who used it for practical things like architecture and astronomy. All sorts of angles may arise in the course of a practical application. In general, the trigonometric functions are irrational numbers, which means that for practical purposes, we must use approximations. In the old days, you had to look those up in cumbersome tables that were time consuming to use and easy to make mistakes with.
The so-called "special" angles happen to arise often in practical problems and to have easily memorable values. So, we ask students to memorize them and build problems using them. But there is nothing particularly peculiar about those angles in terms of pure mathematics. They are frequently practically useful and have trig values that are easy to rememeber, but they are not mathematically special.