First, because sin is an odd function, just looking at the graph, we cannot distinguish between a horizontal or vertical flip. Moreover, as we shift a sin graph through multiples of 1/4 of its period, we obtain graphs that look like ±sin or ±cos; look at the following graphs obtained by shifting left by 2p/4 = p/2 each time:
These identities follow directly from the angle sum formula for sin:
sin(t + p/2) = sin(t)cos(p/2) + cos(t)sin(p/2)
= sin(t)·0 + cos(t)·1 = cos(t)
sin(t + p) = sin(t)cos(p) + cos(t)sin(p)
= sin(t)·(-1) + cos(t)·0 = -sin(t)
sin(t + 3p/2) = sin(t)cos(3p/2) + cos(t)sin(3p/2)
= sin(t)·0 + cos(t)·(-1) = -cos(t)
This means that there is no way to tell from the graph alone if we should view it as coming from a sin or cos function, and whether we should see it as flipped or translated. In particular, it is very difficult to give a simple, well-reasoned rule one could use to identify the horizontal translation (i.e., phase shift) from the graph alone.
Go back to Graphing and Trigonometric Functions
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