Sinclair scored 62 goals in 167 appearances for the Bhoys.
Speaking to the club's website, Sinclair said: "I am so delighted to get the deal done. I have been waiting to finally get down here to sign and I can't wait to get started and to finally meet the lads.
"I have had a fantastic three-and-half-years with Celtic, we did the treble-treble, I got the PFA player of the year, scored goals and was top goalscorer, but now it is a new chapter for me.
"I am coming here to improve and to help the team push for promotion."