where to learn computing logic (ie. gate logic)?

Computational logic, at the level you refer to, is a branch of electronics.

Any library has books on digital electronics, and this is what you need.

You will start with simple gate construction, then move up to adders, counters etc.

And the more complex circuits tend to be multiples of the smaller adders etc you will build.

Something like Electronic Work Bench would help too - but is way overspecced for what you need.

The cosine function is a level of abstraction above logic gates. It is done in software building on more simple machine instructions in the processor (such as add, subtract, shift, AND, OR,...)
The processor is again a level of abstraction above the logic gates with the processor's ALU, Control Unit and registers each made up of many thousands of logic gates - even for a simple processor.

If you want to do more complex development, I'd suggest getting into microcontrollers such as the PIC, AVR or Arudino. These let you build complex systems in software and are very cheap (couple of pounds for a PIC chip)