Being a mechanical engineer - I would have thought your first choice would've been a relay! First thing to note about an electric motor - startup current is very high. As to how long this exists depends on how fast it spins up. If the motor seizes then we get high current until the fuse blows or smoke happens.

With a triac there's two main things that will kill it - voltage spikes and excessive i2t (change of current vs time). With a relay the failures are caused by arcing and mechanical wearout. The manufacturer specifies the expected life of the relay under certain conditions so wearout is reasonably easy to estimate.

If you really want to use a triac, lets assume the startup is 10 times the running current (I'm not sure of the startup characteristics of a single phase motor). It may be more than 10 times. So we need a triac rated for 30 amps so a bit of safety. Triacs also have loss - about 1-2V as a guess. So heat loss at 2.9A would be about 6W worst case. So for a triac such as a BTA41-600B you will need a heatsink as the temp rise (at a rough guess) of 50degrees C per watt will give you a case temperature of 300degrees! Even at a loss of 3W thats still 150degrees - ouch! To cope with voltage spikes you'll need a snubber circuit- something like a 100nF X2 capacitor and a 100R 2W resistor would be a start. To cope with i2t is the hard part. One would normally use a high speed semiconductor fuse which are a bit more expensive that your average 3AG slow blow. i2t is an issue when dealing with an overload. Relays normally just take the kick. There's heaps of app notes on the web regarding triac design - maybe a read of these will convince you that the humble relay is probably much easier.