There are numerous synonyms for the phrase "in an arrogant way", each with their own unique nuances and connotations. Some possible alternatives might include "haughtily", "superciliously", "proudly", "cockily", or "condescendingly". Depending on the specific context and tone of the situation, other options could include "smugly", "snobbishly", "disdainfully", or "patronizingly". Ultimately, the choice of synonym will depend on the desired effect and impact on the reader or listener, with each word conveying a slightly different shade of meaning and attitude.