The
AIM-7, while indeed a weapon in widespread use, is a less effective air-to-air weapon than its cousin, the IR-homing
AIM-9 Sidewinder. The reason for this is simple: the
Sparrow, while RADAR-guided as
DrSeudo notes, is a
passive RADAR seeker. This means the missile itself does not transmit
electromagnetic waves; it merely receives, interprets and homes on them.
This is important, because it means that in order for a Sparrow to hunt a target, another object must be continuously illuminating that target with its radar. So, if the weapon is launched from a fighter at a target, the fighter must keep its radar trained on the target for enough dwell time per second that the missile has enough reflected energy to home onto.
Naturally, this presents difficulties. First, just keeping the radar on the target is a challenge, especially if the target knows you're there; most fighters keep their radar behind their nose assemblies. For reasons of practicality (power, stealth, complexity, etc.) the 'cone' of signal from that radar is not very wide. Thus, the guiding fighter must keep its nose towards the enemy fairly constantly. This is much easier said than done; plus, if the target attempts to evade, it will have a much easier time evading a radar lock for a few seconds than it would evading the more maneuverable missile itself.
Finally, if the target had been unaware of the attacker's presence, a continuous radar paint would certainly tend to make them aware! If you are attempting to sneak a target while under EMCON in order to get a 'surprise shot' off, lighting up a radar on them is a bad plan. Plus, the type of radar transmission you must use sometimes will tell them exactly what they're up against, giving them a chance to employ targeted countermeasures.
There are, of course, applications where much of this is negated. The RIM-7 Sea Sparrow is a perfect example. Not only will a ship have a much larger, much more capable radar, but the entire purpose of the missile is to perform 'last chance' air defense. The odds that its target will not know about the radar emitter or its location are vanishingly small.