How do you find the stationary points of the function ##y=sin(x)##?

The stationary points of a function ##y = f(x) ## are all those points where ##{dy}/dx = 0##.

In the case ##y = sin(x)##, we know that ##{dy}/dx = cos(x)##, and so the problem then becomes solving ##cos(x) = 0##. Looking at a graph like the one below:

we see that ##cos(x) = 0## has solutions at ## x = +-pi/2, +-{3pi}/2, +-{5pi}/2, …## In general, ##{(2n+1)pi}/2, n in ZZ## will be a solution of ##cos(x) = 0##, and hence of ##{dy}/dx = 0##.

So, these points are the stationary points of ##y = sin(x)##.