is it possible for a sinusoidal curve to be defined by more than one equation? For example, could you model a transformed sine function with a cosine function?

Answer 1

Yes, you can model any sinusoidal curve by either sin or cos equation. A sin curve can be turned into a cos curve by shifting it to the right or left and visa versa. Your amplitude, frequency and vertical shift would all be the same. If you get to pick which equation to use I think it is easier to use the one that doesn't have a phase shift. But, sometimes the questions will ask for both!

Answer 2

Yes, it is possible. If you think about transformations of graphs and compare the graphs of sin(x) and cos(x), cos(x) is exactly sin(x) with a horizontal shift of π/2 to the left. The horizontal shift of 'a' steps to the left is expressed as f(x+a), therefore, the relationship of cos(x) and sin(x) is expressed as cos(x)=sin(x+π/2).