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The words relief and amused might or might not be helpful, depending on what context is available at the time. If the "sigh of relief" line is the first line in a scene, then the phrase of relief ...
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#3: Attribution notice added
Source: https://writers.stackexchange.com/a/27979 License name: CC BY-SA 3.0 License URL: https://creativecommons.org/licenses/by-sa/3.0/
#2: Initial revision
The words _relief_ and _amused_ might or might not be helpful, depending on what context is available at the time. If the "sigh of relief" line is the first line in a scene, then the phrase _of relief_ is a useful, efficient way to distinguish a sigh of relief from a despondent sigh or an annoyed sigh. Without it, the reader doesn't have a clear picture of the mood and will have to catch up as you dribble out more information. This could be jarring if your reader's initial guess is wrong. Yes, some people would say that it violates the "show, don't tell" mantra. But if using a two-word modifier to tell instead of a contrived sentence to show gets to the action faster, I'd tell. On the other hand, suppose the "amused eyebrow" line is in the middle of a dialogue in which the other character has just said something amusing. Then the reader will naturally picture an amused eyebrow arch instead of a confused or a quizzical one. Though the action alone might have been ambiguous, the action in context was clear. Describing the arch as amusing would then be a bit clunky--an extra word to say what the reader was already picturing anyway. So I might modify your premise to this: You only need to directly state the emotion when (a) neither the action nor the context describes it clearly enough, and (b) the extra words required to show the emotion indirectly would disrupt the flow of the story.