Etymology
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Words related to insinuate

*en 
Proto-Indo-European root meaning "in."

It forms all or part of: and; atoll; dysentery; embargo; embarrass; embryo; empire; employ; en- (1) "in; into;" en- (2) "near, at, in, on, within;" enclave; endo-; enema; engine; enoptomancy; enter; enteric; enteritis; entero-; entice; ento-; entrails; envoy; envy; episode; esoteric; imbroglio; immolate; immure; impede; impend; impetus; important; impostor; impresario; impromptu; in; in- (2) "into, in, on, upon;" inchoate; incite; increase; inculcate; incumbent; industry; indigence; inflict; ingenuous; ingest; inly; inmost; inn; innate; inner; innuendo; inoculate; insignia; instant; intaglio; inter-; interim; interior; intern; internal; intestine; intimate (adj.) "closely acquainted, very familiar;" intra-; intricate; intrinsic; intro-; introduce; introduction; introit; introspect; invert; mesentery.

It is the hypothetical source of/evidence for its existence is provided by: Sanskrit antara- "interior;" Greek en "in," eis "into," endon "within;" Latin in "in, into," intro "inward," intra "inside, within;" Old Irish in, Welsh yn, Old Church Slavonic on-, Old English in "in, into," inne "within, inside."
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sinus (n.)
"hollow curve or cavity in the body," early 15c., from Medieval Latin sinus, from Latin sinus "bend, fold, curve, a bent surface; a bay, bight, gulf; a fold in land;" also "fold of the toga about the breast," hence "bosom," and figuratively "love, affection, intimacy; interior, inmost part;" of unknown origin.
insinuating (adj.)
"wheedling, ingratiating," 1590s, present-participle adjective from insinuate (v.). Related: Insinuatingly.
insinuation (n.)

1520s, "act of making an indirect suggestion;" 1530s, "that which is indirectly suggested," from French insinuation (16c.) or directly from Latin insinuationem (nominative insinuatio) "entrance through a narrow way; an ingratiating oneself," noun of action from past-participle stem of insinuare "creep in, intrude, wind one's way into" (see insinuate).

sinuate (adj.)
1680s, from Latin sinuatus, past participle of sinuare (see insinuate).