The 23-year-old striker joined from League Two club Macclesfield for an undisclosed fee on Wednesday, signing a three-year contract.
He is keen to get to grips with Championship football the second time around, having started his career at Nottingham Forest.
"I'm excited by the move and I can't wait to get started," Sinclair told the club's official website. "I didn't know anything about it until Tuesday night but once I travelled down here I just wanted to sign straight away.
"It's obviously a great move for me. It took me time to adjust to League Two football but I was enjoying a good start to the season at Macclesfield.
"My game is about pace, power and goals and I am confident I can fit in at Peterborough. I watched them in the play-offs last season and I was very impressed. I love their style of football and I will be playing for a manager with a great reputation in football."