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The Titans Sunday swapped the Patriots their second-round pick in 2010 for the 25th pick in the third and 89th overall.
The 6-foot-4, 246-pound Cook ran the fastest time of any tight end at the NFL Combine. Cook caught 37 passes for 573 yards and three touchdowns.
This is the second straight year the Titans have drafted a tight end in the third round after grabbing Craig Stevens of California.
They used the franchise tag on Bo Scaife, their leading receiver last season. Scaife has yet to sign his tender.
