It's a lot less surprising if you look at the functions as Taylor series, then you'd see that exponentiation is basically the same thing as cos + sin. This also explains "why e and not 2?", because the Taylor series for e, cos and sin are very nice by definition (e is defined to have trivial derivative, and radians are defined to make sin and cos have simple derivatives) while the Taylor series for exponentiation by 2 is not.