Let's look back to how each person obtained Super Saiyan. Goku saw his best friends die at the hands of Frieza while simultaneously out of power, as he spent most of his strength trying to destroy him with a spirit bomb. He was cornered and, in the words of Master Roshi "kept digging, and finally hit the motherload."
Vegeta was training at some planet/moon and had the snot beat out of him by flying celestial objects. He was about to get smashed by an even larger meteoroid when he "kept digging, and finally hit the motherload."
So while it's not anger that causes Super Saiyan power to manifest, it's actually "the deep-seated need for more power in order to live".
Same goes for Future Trunks, who saw Gohan get killed by the androids. He was the only man left to stop them, and soon it would next be him to die, if he didn't keep digging and hit the motherload.
You should ask yourselves why Goten and Present Trunks have Super Saiyan. They obtained it in peacetime! That shouldn't happen.
