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I guess if it was tracing paper you could flip the paper over through y=x and look through the back.
To find the inverse graph of a function, draw the function and then reflect it with the line of symmetry being y=x. To John, I tried your method and it seems that it returns the inverse of the negative function. eg. y=sin x would become y=-arcsin x.
How about just turning your paper 90 degrees when going from a trig function to the arc or inverse trig function?
To restate what I just said in a slighty more organized form:
Sketching sine and cosine curves are relatively easy. You just need the vertical translation, the magnitude, the period, and the horizontal translation.