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SteamKing

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Since the sine function is related to a unit circle, think of the amplitude as the y coordinate on the unit circle when the radius makes an angle theta to the positive x-axis.

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Sine(pi) equals zero. Geometrically the sine of angle α equals the length of the opposite side divided by the length of the hypotenuse in a right triangle. The particular size of the triangle is unrelevant as long as it contains the angle. In an unit circle centered at the origin sine(α) equals the distance of a point on the circumference (at angle α) from the x-axis. It corresponds to the imaginary part of e^(iα) in complex analysis.

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HallsofIvy

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I am wondering what you think the x axis "means" and what

And, by the way, "sin(pi)" is 0, not 1.

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Stephen Tashi

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On an exaggerated graph, if you try to measure the slope of the tangent line to the graph of sin(x) in terms of the angle it makes with the x-axis, you'll get the wrong answer since you won't be using the same scale in both the vertical and horizontal directions. (This line of thinking also shows why the derivative of the function sin(x) where x is in degrees is a different than derivative of sin(x) where x is in radians.)

Exaggeration of vertical scales for an artistic effect is common. For example in most "realistic" computer renderings of terrain ( on earth or on other planets) the vertical scale in the picture is exaggerated. As another example, books for artists teach that the human figure is "7 heads" tall, but statistically this is an exaggeration. (People who draw fashion drawings for clothing advertisements are taught to draw figures that are 8 or 9 heads tall!)

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