Sigh, by way of definition, is Strong's #584, and it means "to groan," "to mourn," and "to moan." Its rather interesting first use is found in Exodus 2:23-25:
Now it happened in the process of time that the king of Egypt died. Then the children of Israel groaned because of their bondage, and they cried out; and their cry came up to God because of the bondage. So God heard their groaning, and God remembered His covenant with Abraham, with Isaac, and with Jacob. And God looked upon the children of Israel, and God acknowledged them. (Emphasis ours throughout.)
Note from the context that our God is a covenant-keeping God. He remembers His covenant and acknowledges those who hear Him and those who sigh among His people. In the Exodus story, He moved to redeem them from their bondage in Egypt, making a distinction between them and their oppressors (Exodus 8:22; 11:7).
Cry is Strong's #602 (a fairly rare word, used only four times in Scripture), and it also means "to groan," but it has another meaning as well: "to shriek." This word contains a great deal of emotional meaning. It involves a person's innermost feelings.
But, sighing and crying involves a lot more than emotion. For us to rightly understand what God requires of us, today, it is necessary to explain the thinking, the reason, that is behind "sighing and crying." Sound reason underlies the emotion expressed by sighing and crying, which needs elaboration before proceeding further.
Neuroscientists used to talk about compartments in the brain. Sometimes in the popular press there is an occasional assertion that one section of the brain is for sight, another one for hearing, another one for mathematical skills, and yet another for artistic skills. The faculty of reason is supposed to reside in the prefrontal cortex, and emotion comes from another area. This idea is called the "localization thesis." It is a simplistic view that has pretty much fallen by the wayside by neuroscientists who have come to know more about how our brains function. One critic of this thesis says:
. . . functions [of the brain], like properties, are distributed (they require a whole system or mechanism to be realized [or actuated]). . . . A danger inherent in the localization thesis may be illuminated by analogy to an internal combustion engine. In describing an engine, one might be tempted to say, "the opening of the intake valve is caused by the movement of the rocker arm." Except that the rocker is, in turn, set in motion by the camshaft, the camshaft by the crankshaft, the crank by a connecting rod, the rod by the piston. But of course, the piston won't move unless the intake valve opens to let the air-fuel mixture in. This logic is finally circular because, really, it is the entire mechanism that "causes" the opening of the intake valve; any less holistic view truncates the causal picture and issues in statements that are, at best, partially true. Given that the human brain is more complexly interconnected than a motor by untold orders of magnitude, it is a dubious undertaking to say that any localized organic structure [any section of the brain] is the sufficient cause and exclusive locus of something like "reason" or "emotion." . . .
[For instance] the amygdala is said to be the seat of emotion, the prefrontal cortex of reason. Yet when I get angry, for example, I generally do so for a reason; typically I judge myself or another wronged. To cleanly separate emotion from reason-giving makes a hash of human experience. . . . (Matthew B. Crawford, "The Limits of Neuro-Talk," The New Atlantis, Number 19, Winter 2008, pp. 65-78)
Emotion and reason are not separate entities. They do not occur in discrete areas of the brain, and it is far better to understand them to be two sides of the same coin. One needs both sides; one cannot have a coin with a single side. It is an impossibility.
Therefore, sighing and crying are not just emotions or feelings. They are not just matters of the heart but also matters of the head. These expressed feelings have reason—thought—firmly attached to them.