Just a nit - sin(1) is transcendental ( by the Lindemann Weirstrass Theorem). All real transcendental numbers are also irrational, but the converse is not true ie. the square root of 2 is irrational but not transcendental.

So I guess your teacher was just looking for a plain old irrational number and going all transcendental on him is "a bit too clever" as you put it :)