For example if Falkenrath Noble was tapped and another creature dies then would his triggered ability still trigger or only if it was not tapped?
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Yes. Whether an permanent is tapped or not has no effect on its abilities. This is the case for abilities of all kinds (static, triggered and activated).
Three things change when a creature is tapped:
Do note that the first two items can indirectly prevent abilities from triggering ("Whenever an untapped creature dies, ...") or from functioning ("..., tap it. If you do, ...").