Nagelsmann, whose sacking in March 2023 led to Tuchel's arrival, was tipped to return to Bayern, but has now extended his contract with Germany until 2026
The club also announced the signing of 21-year-old Brazilian midfielder Matheus Fernandes from Palmeiras for a fee of seven million euros, which could reach 10 million with add-ons.
Trincao, 20, will join on July 1 and sign a contract until June 2025 containing a release clause of 500 million euros.
The forward made his top-flight debut for Braga last season and has made four league starts this campaign, scoring twice.
Defensive midfielder Fernandes, who has been capped at Under-17 and Under-20 level, will also join Barca on July 1 on a five-year deal, with a release clause of 300 million euros.