max corrects with a triumphal (-phant?) smile.

triumphal, triumphant (adjs.)
Triumphal means “in honor of a triumph,” “like a triumph,” as in We had a triumphal celebration. Triumphant means “victorious,” “elated over a success,” as in She was wearing a triumphant smile.
- The Columbia Guide to Standard American English. 1993.

well now, there's a fine line -- I still don't know which is better to use in these two cases.