complementary / complimentary are similar-sounding terms with different meanings (referred to as homophones). To better understand the differences, see below for definitions, pronunciation guides, and example sentences using each term. 👇
complementary: (adjective) of words or propositions so related that each is the negation of the other.
complimentary: (adjective) conveying or resembling a compliment. (adjective) costing nothing.
complementary: k·AA·m·p·l·ah·m·EH·n·t·r·iy
complimentary: k·AA·m·p·l·ah·m·EH·n·t·er·iy
(Examples are Wikipedia snippets under the CC ShareAlike 3.0 license. Definitions drawn from WordNet.)