Question

why we take sin in cross product & cos in dot product?

Answer 1

Definition of cross product says that it is another vector, that is
perpendicular to each of given vectors and this 3 vectors have right

orientation, and length of third vector is equal product of lengths of
first two vectors and sine of angle between them.
Other words length of vector
cross-product is area of parallelogram built on given 2 vectors.
Since area
of parallelogram includes sine in its formula, then we take sine.

In dot
product we take projection& of the first vector on another one, and that is
why we used cosine.