Sinclair, who has been manager there since February 2013, saw his improving team ripped apart in the summer and they've struggled to match the results of last season.
He tendered his resignation after this week's big defeat and said: "It's a sad day for me because I feel a big part of this club and it has been a pleasure working here, but I just feel I've taken the team as far as I possibly can.
"I wanted to build on last season and finish in the top ten this year, but it has been very difficult because of the unexpected departure of so many players in the summer.
"To virtually start again from scratch was a difficult task. I tried to get together a group as could as I could and then look to improve as we went along, but it has come to the stage now where I think it is better for the club that they look at somebody else with new ideas and contacts to bring players to the football club."
Ade Akinbiyi, who was first team coach, has also gone with assistant manager Gus Williams appointed as Sinclair's successor.
Chairman Dave Titchiner said: "It has been a very sad day for everyone involved at the club. To have someone of Frank's celebrity status here, along with Ade, has been fantastic for the club and put us on the map in many ways by creating national interest. We thank them for all the work and effort they have put in."