How would I sketch the graph sin(x) + sin(2x - π/2) in my exam?

Sketching trigonometric graphs is all about working from the graphs you already know and seeing how they behave when added/multiplied/substracted from each other to sketch more complicated functions.

You would start by sketching sinx on your graph, this is a standard graph with a period of 2π and has a distinct shape. (Would draw it out on whiteboard to illustrate). Then sketch out the other graph of sin(2x - π/2), using graph transformations or identifying that it is -cos2x.

To sketch the final graph you would observe what happens at critical points, to get a set of points you can put your graph through. Take a look at the roots and also the minima and maxima as these are the easiest to work with. Throughout the problem I can illustrate how you would do this on a whiteboard.