The Half Bow is Jin's first bow, lent to him by Sensei Ishikawa in The Tale of Sensei Ishikawa. It draws fast and carries plenty of arrows, which makes it the bow for mid-range fights and quick silent kills.
Draw longer for a flatter, longer shot, but Jin lets go on his own if you hold full draw too long. You can shoot well from a crouch. It can't punch through Mongol helmets, so armoured heads need the Longbow.
Level II: 150 Supplies, 25 Bamboo. III: 300 Supplies, 50 Bamboo, 15 Yew. IV: 450 Supplies, 100 Bamboo, 30 Yew, 2 Wax Wood. V: 600 Supplies, 150 Bamboo, 75 Yew, 4 Wax Wood. VI (New Game+): 850 Supplies, 200 Bamboo, 100 Yew, 5 Wax Wood.
Burning Fang is the default. Storming Falcon, Hidden Serpent and Midnight Shark come from Ishikawa Tales 4, 8 and 9. Crane's Strike and Piercing Shadows cost 5 flowers each from merchants.