What are the applications of cosine functions?

Answer 1

For school trigonometry, it is used to find the angles of a triangle when only the lengths of the sides are known.

Later on, in mechanics, you may come across the decomposition of forces along and across the line of motion. In calculus you may come across it in second order differential equations.

Answer 2

As diverse as those of any trig function, from simple triangle measurements to complex harmonic analysis (sound etc), electrical principles, and umpteen other practical uses.