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x(n+1)=a*sin[x(n)]

and generate a list of x0,x1,x2 etc. up to x(n) like NestList does, starting from x0=1.0

So I want to make a list of these iterations, where

*a*is a constant of my choice, and n is the number of iterations. My first guess was to try

iteration[a_,n_]:=NestList[a*Sin,1.0,n]

but of course it doesn't work as Mathematica doesnt recognize a*Sin as a function. So I tried defining a function acos[a_,x_]:=a*Sin[x] first but not only does it look inefficient to define 2 functions for something this simple, it also introduces another argument, and I still can't find a way to get it to work.

Thanks!