The forward arrives at Belle Vue in a swap deal that sees Keegan Hirst go the other way.
Both players join their new clubs on loan until the end of the season.
Tangata follows the arrivals of Morgan Escare and Chris Green, who both joined Trinity in the space of five hours on Monday.
The 28-year-old started the season at Widnes Vikings but left the club following their financial troubles earlier this year.
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Since 2013 he has spent the majority of time at Halifax, helping the club earn three top-four finishes in the last four seasons.
“I’m really looking forward to getting started here at Wakefield. I’m excited to give my all for the team in the remaining five games," said Tangata.
“I’ve met all the boys at training and they are a great bunch so I’m just raring to go from now until the end of the season.”
Chris Chester feels the Tangata signing represents another piece of "good business" for Wakefield, who have seriously bolstered their options heading into the final stretch of the season.
He said: "It’s another good bit of business by us to get Adam Tangata signed until the end of the season.
“Along with Chris Green, he’ll add a lot to the pack in these remaining games and i’m looking forward to seeing him out there."