If the the hot water was used last but one needs to wash their hands on Shabbath, if they turn on the cold water but hot water comes out first, are they in violation of Shabbath?
2If you are talking about standard home plumbing (in the USA at least) with either two handles- two lines- or a single handle- cartridge that turns from one line to the other- either way, if you really turn only cold or the one handle all the way to cold, then any hot/warm water that comes out was sitting in the faucet already and will not cause the hot water heater to turn on, etc.– DamilaJan 7 at 0:52
@Damila Some single-handle faucets can be damaged or poorly designed/installed, and use a bit of warm water even when they are turned as far towards 'cold' as possible. But this doesn't usually happen.– FredJan 9 at 2:12
@Fred True, for that replace the cartridge. Regardless, that problem would be not rdependent to the question of hot water being used recently. So “May one use a single handled faucet with a damaged cartridge on Shabbat?” would be a different halachic question I guess.– DamilaJan 9 at 2:19
The actual problem with using hot water from the tap is that more cold water will replace the water you used in the hot water heater, and the new cold water will become "cooked" on Shabbos. In the case of water that is already in the pipes, if you turn on the cold water, no hot water is leaving the hot water heater (since you only turned the cold water tap), so no cold water will enter the hot water heater to replace it, hence there is no issue with using the tap on Shabbos, even if there is still hot water in the pipes.
Aside from the problem you describe, one may wash one's hands with hot water on Shabbos (Shulchan Aruch OC 326:1) unless it was heated in violation of Shabbos. If a non-Jew heated up the water on Shabbos for themselves, R' Moshe (Igros Moshe OC I, 126:2) says it is permissible for a Jew to use it, though the 'Aruch HaShulchan (OC 326:2, "אבל בחמין שהוחמו בשבת אפילו הוחם על ידי עכו"ם לצורך עצמו אסור אפילו לטבול ידו בהן") says it is forbidden to use it.– FredJan 9 at 2:07
So, depending on when, how, and by whom
the hot water was used last, these factors might come into play as well.– FredJan 9 at 2:13