All inductive loads - fridges, pumps, many power tools, motors and so on need pure sine wave, so the girlfriends hair dryer might not work on a modified sine wave even if under 1000w
A cheap way to get a pure sine wave inverter is to buy a UPS with dead batteries but check the battery voltage required, some are 12v, but some are 24v, 36v or even 48v
We picked up a 24v 1000w pure sine wave for £16 to go in our coach (so 24v is fine), it also has automatic cut over from shore power.