Definition of sigh:
(v) :
to inhale a larger quantity of air than usual, and immediately expel it; to make a deep single audible respiration, especially as the result or involuntary expression of fatigue, exhaustion, grief, sorrow, frustration, or the like; hence, to lament; to grieve
(n) :
a deep and prolonged audible inspiration or respiration of air, as when fatigued, frustrated, or grieved; the act of sighing; figuratively, a manifestation of grief; a lament