Trigonometric derivatives and roots of unity

[tickle_monste]

sin x.
d(sin x)/dx = cos x.
d(cos x)/dx = -sin x.
d(-sin x)/dx = - cos x.
d(-cos x)/dx = sin x.

i.
i^2 = -1.
i^3 = -i
i^4 = 1
i^5 = i.

I know there is a relationship between trig, the complex numbers, and exponential functions. Is there a relationship between the pattern shown here?

 

[Integral]

Google Eulers relationship.

 

[tickle_monste]

Gracias.