How to use Ultrasonic Sensor's trig pin, and why it is needed?
Yes, you do need it.

The purpose of the TRIG pin is to cause the module to send out a "Ping".  It's what makes the whole thing work.  As soon as the TRIG pin goes HIGH then LOW (for a period of not less than 10µs) the internal clocks start ticking.  8 cycles of 40KHz audio are sent out of the transmitter and it starts counting how long it takes for the echo to arrive.

If you don't use the TRIG pin the module won't be able to do anything at all.