sin (n.) Look up sin at Dictionary.com
Old English synn "moral wrongdoing, injury, mischief, enmity, feud, guilt, crime, offense against God, misdeed," from Proto-Germanic *sun(d)jo- "sin" (cognates: Old Saxon sundia, Old Frisian sende, Middle Dutch sonde, Dutch zonde, German Sünde "sin, transgression, trespass, offense," extended forms), probably ultimately "it is true," i.e. "the sin is real" (compare Gothic sonjis, Old Norse sannr "true"), from PIE *snt-ya-, a collective form from *es-ont- "becoming," present participle of root *es- "to be" (see is).

The semantic development is via notion of "to be truly the one (who is guilty)," as in Old Norse phrase verð sannr at "be found guilty of," and the use of the phrase "it is being" in Hittite confessional formula. The same process probably yielded the Latin word sons (genitive sontis) "guilty, criminal" from present participle of sum, esse "to be, that which is." Some etymologists believe the Germanic word was an early borrowing directly from the Latin genitive. Also see sooth.

Sin-eater is attested from 1680s. To live in sin "cohabit without marriage" is from 1838; used earlier in a more general sense. Ice hockey slang sin bin "penalty box" is attested from 1950.
sin (v.) Look up sin at Dictionary.com
Old English syngian "to commit sin, transgress, err," from synn (see sin (n.)); the form influenced by the noun. Compare Old Saxon sundion, Old Frisian sendigia, Middle Dutch sondighen, Dutch zondigen, Old High German sunteon, German sündigen "to sin." Form altered from Middle English sunigen by influence of the noun.