electromagnetism - Why is divergence and curl related to dot and cross product?

I've been reading Griffith's intro to electrodynamics and I've been a bit confused on his explanation of divergence and curl. I don't understand how divergence is the dot product of a gradient acting on a vector function and curl is the cross product of gradient acting on a vector function. Does it relate to the fact that one uses sine while other uses cosine? Just to clarify, I understand the concept of divergence and curl from a purely conceptual standpoint, it's just this mathematical definition that I can't wrap my head around.