A 3000 watt inverter will output 120 VAC 25A & draw 260-280 A from 12 VDC batteries.
A 1 HP motor (about 900 watts at full power) will draw (85 A) from 12 VDC batteries.
Pure sine wave is only needed for noise sensitive devices such as audio amplifiers
The inductance of AC motors will smooth the current from modified sine wave inverters.
Pure sine wave inverters cost 3 to 4 times as much as modified sine wave inverters.
Be aware that some inverters contain two inverter boards (dual double outlets)
which can't be paralleled for a single load, to get the advertized surge power.
So always check the reviews to see if others have found problems.
Most inverters have a 1 year warranty, some have a 2 year warrenty.
A 12 VDC, 80 AH deep cycle battery will run a 1 HP motor at full load for an hour.
More capacity can be obtained by using parallel connected batteries.
Very large heavy duty (Jumper Cable) wires are required to reduce voltage drop.
Two 2.5 ft cables will have 1/4 the voltage drop of two 10 ft cables.
Two 10 ft Awg 2/0 cables will drop 0.52 volts at 330 A (4000 watts)
Two 10 ft Awg 3/0 cables will drop 0.41 volts at 330 A (4000 watts)
Two 10 ft Awg 4/0 cables will drop 0.325 volts at 330 A (4000 watts)
200 A fuses & fuse holders will increase the voltage drop & will cost about $10
The input current will be about 11 times the output current amps shown as ( A)
If a wireless remote is used, the inverter will draw some power when it's off.
Amazon.com has the lowest prices. Other web sites are more expensive.
Just google for the "model number Amazon.com"

I have a Cobra 1200 watt Modified Sine Wave inverter that Jeff bought me in 2006.
It starts & runs my Sears Craftsman 1/2 HP 5 Amp air compressor motor with no problem.
It also will run my Sears 14" chain saw rated at 2.5 HP peak (2 HP = 1800 watts).
We used the compressor on our Alaska trip to air up flat tires & provide AC power from
our Durango which had a 240A alternator & a heavy duty battery with extra side terminals.
The Auto Zone 1000A Durlast battery had an 8 year warranty, but it quit in 3 years.
So Auto Zone replaced it with a new 8 year warranty in 2009. It was still good
after 4 more years of travel trailer usage when I sold the Durango in 2013.

Here are some good high power inverters that I would recommend:

Cobra CPI-2590 12 VDC Power Inverter (Made in USA)
115 VAC Modified Sine Wave 2500 watts (21.7 A)
5000 watts peak power (43.5 A)
http://www.amazon.com Prime Price: $202.95
http://www.donrowe.com Sale Price: $229.95

Power Bright PW3500-12 Power Inverter (made in China)
110 VAC Modified Sine Wave 3500 watts (32 A)
7000 watts peak power (64 A)
Free wireless remote control and pro series cables
http://www.amazon.com Prime Price: $264.71
http://www.donrowe.com Sale Price: $309.00
http://www.lowes.com Sale Price: $378.57

Aims PWRINV500012W Power Inverter
110 VAC Modified Sine Wave 5000 watts (45 A)
Surge: 10000 watts peak power (90 A)
http://www.invertersrus.com Sales Price $399.00

AIMS PICOGLF30W12V120V Power Inverter/Charger
3000 Watt 12V Pure Sine Wave Inverter (26 A)
with charger 115 VAC (15 A) to 12 VDC (100 A}
Surge 9000 Watt peak power for 20 seconds (78 A)
http://www.amazon.com Prime Price: $749.86
http://www.ebay.com Sale Price: $934.94