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How "pissed" come to mean "drunk" or "angry" in expressions such as: "I'm pissed" OR "I'm pissed off"? I
looked it up and all dictionaries just say it means so. So, does it have anything to do, even though figuratively, with "urine" —i.e. the literal meaning of "piss"???
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P.S.: Also for the expressions "Piss off!", meaning "Go away!" & "take the piss(out the someone)", meaning "make fun of someone": Do they have anything to do with the literal meaning of "piss" —"urine"— , even if figuratively????
————— **EDITED: If there is a relation, then what is it?

To answer the angry part of this. I believe some say that the phrase originates from "Pissing Blood", which is where someone gets so angry that they rupture a blood vessel which causes them to urinate.

It was the early 19th Century when the word came to mean drunk, but it is unclear why. I suspect it was just used by an author in a book or play and became a popular turn of phrase. (Charles Dickens and William Shakepear were both well known for making up many words and phrases which took off and are still used today.)

The application of piss or pissed to anger was first documented as an expression just after World War II. In two articles from American Speech in the same year, Fred Eikel, Jr. and Joseph W. Bishop, Jr. each documented the usage.

Here is Eikel, Fred. “An Aggie Vocabulary of Slang.” American Speech, vol. 21, no. 1, 1946, pp. 29–36. JSTOR, www.jstor.org/stable/487347, p. 33.

HE PISSED (or PEED) ME OFF. An expression used of a person who in any way disappointed the speaker.

This usage comes from Texas A&M, which at the time had a major military presence.

Here is Bishop, Joseph W. “American Army Speech in the European Theater.” American Speech, vol. 21, no. 4, 1946, pp. 241–252. JSTOR, www.jstor.org/stable/487320. p.249.

a. Pissed-off (or P'd off). This means, roughly, fed-up, irritated, depressed. I have no idea of its history. The British say browned off and it may be that the Americans who borrowed the phrase simply felt that 'browned' was not strong enough. The superlative is, for some reason, highly pissed off, which may also be a Briticism.

Bishop is an amateur recording slang from his time in service. His note - "I have no idea of its history" - carries over to explaining why this usage appeared, or what its precise relation to micturation is. The guess of an association with "brown off" is possible, as is an attraction to the vulgarity of "piss" or an affinity for another figurative usage of "piss."

From the etymology of piss (v.): etymonline

late 13c., from Old French pissier "urinate" (12c.), from Vulgar Latin
*pissiare, of imitative origin.

indeed reveals its literal association to urine. As @johnlowler noted in a comment on the variation in AmE and BrE, but I can find no facts as to the "what is the relation" (the when is easy). There are many phrases and phrasal verbs using piss, most dating to the middle ages.