Made or done to celebrate a victory. `Triumphal` attaches to the ceremony, the object, the procession — the public machinery of celebrating a win. A triumphal arch is built so that a victorious army can march through it; a triumphal procession, entry, or return is the parade itself. The word points outward at the event, not inward at anyone's feelings. That is the line students most need: `triumphal` is about the CELEBRATION, `triumphant` is about the PERSON or their manner — a triumphant general grins on the triumphal arch. The word keeps its Roman clothing. In ancient Rome a "triumph" was a specific state honour: a winning general was voted the right to parade through the city with his soldiers, his loot and his prisoners. So `triumphal` sounds grand, old and slightly formal even today, and it is most at home in history, architecture and journalism about big ceremonies. Writers do stretch it to smaller things for effect — "a triumphal lap of the office with the signed contract" — and the stretch is felt as a mild joke, borrowing Roman grandeur for something small.