I do not understand the pinout of the variable inductors from Murata. Here is an example:
I can see that the coil is on pins four and six, but what are the other pins for?
The full datasheet is here.
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Variable Inductor, yes, if only a single coil is present.
Turning the slug towards and away from the coil increases/decreases the inductance.
If multiple coils are present, then it is a variable transformer.
Turning the slug changes the inductance of the bigger coil (more than it changes the inductance of the smaller coil), and also changes the coupling factor between the two coils.
These are typically used in intermediate-frequency (IF) radio applications. Here the datasheet specifies "Frequency Range： 0.05 ～ 15MHz" with a corresponding min/max inductance of "0.1 ～ 52mH (for the main coil) [and] 1 ～ 7mH (for corner censor)."
The datasheet is calling the smaller coil the "corner censor", which could be used for a variety of things. Since this is a transformer, it will be a (smaller) mirror duplicate of whatever frequency is oscillating on the main coil and vice-versa. And the two are galvanically isolated from each other, allowing greater versatility.
These are primarily intended for radio circuitry (oscillators, tuners, filters, detectors, modulators) and likely won't work well for other applications because they have very little power-handling capability. Might find one in a metal detector or CRT display though.