Both are useful, and are easy to use once you learn the relevant formulas; but I personally prefer the factored form a sin(b(t - c)) + d, because the shift is explicitly given as c, rather than having to be extracted by dividing. I've taught from books that do it either way, and live with it.This is the 2nd time recently I saw y= asin(b(t-c))+d [= asin(bt-bc)+d = asin(bt-C)+d, where C=bc]
Isn't y = asin(bt-c)+d easier to work with?
One period of the sin graph starts when bt-c=0 or t = c/b
That period ends when bt-c =2pi or t = (2pi+c)/b
And the period is therefore 2pi/b