If "Sigh-yin" is the correct pronunciation of the word "Saiyan", then do you also say "Fureeza" instead of "Freeza", and "Ceru" instead of "Cell"? It's a damn race. Just because the guy who created the fictional race is Japanese doesn't mean a damn thing.

Saiya-jin = Sigh-yaw-jeen
Saiyan = Say-yin