The asymptotic behaviour of the coefficients of cosine series is related to the behaviour at the origin of its sum function, in terms of slowly varying functions (SVF's), and regularly varying sequences. Our work is motivated by the study of the sine series with monotone coefficients of Aljan\u{c}ić, Bojanić and Tomić (1956), which is in terms of SVF's in the Karamata sense. The direction of our approach to the cosine series is motivated by the recent presentation (using SVF's in the Zygmund sense) of Samorodnitsky (2016). An obituary for the first author by the second author, with specific relevance to our subject matter, is attached as Section 7.