You are trying to run a marathon without training.
Here's what I would do:
1) Look at a signals and system course online.
2) Look at the course's prerequisites.
3) Do you understand concepts in the pre-req? If not, goto 2)
4) Once you get to a course you understand reasonably, move on to the next course you don't understand. Do this (unwind the stack) until you reach the signals and systems course.
A reasonable chunk of an electrical engineer's education is required to get to the point you wish to reach, so there is a lot of effort involved.
This is great advice. I did EE and I don't think it was until about the 2nd or 3rd year that we got a treatment of systems and signals that was sufficient to understand DCT. You seem to be focusing on the cosine part but it turns out that the cosine part of the DCT is probably the easiest part to understand. It probably won't make much sense until 1) you understand something about discrete signals and 2) you understand why transforms from one domain to another are particularly useful for solving certain problems.
I recommend 'Linear Systems and Signals' by B.P. Lathi. It was the book I used. It doesn't cover DCT explicitly, but it does have a friendly, Knuth-like treatment of the history of the mathematics. It makes for a nice break between sections of heavily technical material. Here is an excerpt from the beginning of the book on the topic of complex numbers and why they are useful. It should give you an idea what the rest (and the general subject) is like:
"If we want to travel from City a to City b (both in Country X), the shortest route is through Country Y, although the journey begins and ends in Country X. We may, if we desire, perform this journey by an alternate route that lies exclusively in X, but this alternate route is longer. In mathematics, we have a simliar situation with real numbers (Country X) and complex numbers (Country Y). All real-world problems must start with real numbers, and all the final results must also be in real numbers. But the derivation of results is considerable simplified by using complex numbers as an intermediary. It is also possible to solve all real-world problems by an alternate method, using real numbers exclusively, but this would increase the work needlessly."
Here's what I would do:
1) Look at a signals and system course online.
2) Look at the course's prerequisites.
3) Do you understand concepts in the pre-req? If not, goto 2)
4) Once you get to a course you understand reasonably, move on to the next course you don't understand. Do this (unwind the stack) until you reach the signals and systems course.
A reasonable chunk of an electrical engineer's education is required to get to the point you wish to reach, so there is a lot of effort involved.
edit: that said, this link might also be helpful: http://www.redcedar.com/learndsp.htm