Say I'm looking at the graph of y=sin(x). My book states that the domain is measured in radians, and the graph of pi has interval lengths of π/2, and has a period of 2π. I understand what this means, or so I thought, but what confuses me is when there is a more complicated equation of a sine (or cosine) graph that has interval lengths on the x-axis labeled as say for example "1, 2, 3, 4" instead of "π/2, π, 3π/2, 2π." I know that sometimes π is canceled out when I'm trying to get a graph from an equation, for example if I use Period=2π/w, if w was π, I would end up with a graph with a period of 2, but that period still in radians? Do we just use π because it's more exact than trying to keep track of decimals? When I use desmos graphing tool, the graph of y=sin(x) looks like it has a period of a little over 6, but is that because 2π (2x3.14) is literally just a bit over 6? I am so used to seeing something measured in radians always containing the π symbol.
And lastly, when a graph has a horizontal shift "blank" units to the left or right, does that mean "blank" interval lengths? For example, if I had to shift the graph of y=sin(x) to the right 3 units, would it be shifted to the right by 3π/2 because the interval lengths of the x-axis are π/2? Or would it just be shifted over by 3 "radians"? If it's shifted over by 3 radians, it seems like it'd take an awful long time to approximate that while drawing the graph.
I'm sorry if these are a lot of questions, and I know it seems like I shouldn't even be in trig but I'm determined to pass this class.