cock (n.2)youdaoicibaDictYouDict[cock 词源字典]
in various mechanical senses, such as cock of a faucet (early 15c.) is of uncertain connection with cock (n.1), but German has hahn "cock" in many of the same senses. The cock of an old matchlock firearm is 1560s, hence half-cocked (for which see half).[cock etymology, cock origin, 英语词源]
halfyoudaoicibaDictYouDict
Old English half, halb (Mercian), healf (W. Saxon) "side, part," not necessarily of equal division (original sense preserved in behalf), from Proto-Germanic *halbaz "something divided" (cognates: Old Saxon halba, Old Norse halfr, Old Frisian, Middle Dutch half, German halb, Gothic halbs "half"), perhaps from PIE (s)kel- (1) "to cut" (see shell (n.)). Noun, adjective, and adverb all were in Old English.

Used also in Old English phrases, as in modern German, to mean "one half unit less than," for example þridda healf "two and a half," literally "half third." The construction in two and a half, etc., is first recorded c. 1200. Of time, in half past ten, etc., first attested 1750; in Scottish, the half often is prefixed to the following hour (as in German, halb elf = "ten thirty"). To go off half-cocked in the figurative sense "speak or act too hastily" (1833) is in allusion to firearms going off prematurely; half-cocked in a literal sense "with the cock lifted to the first catch, at which position the trigger does not act" is recorded by 1750. In 1770 it was noted as a synonym for "drunk."