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sin(theta) is the y-coordinate of the intersection of a line forming an angle with the positive x-axis and the unit circle (i.e., circle of radius one, centered at the origin). The unit circle has its highest y-value at the point (0,1). Drawing the unit circle on a graph makes this obvious.

Assume that there were a higher y-value than one possible. y>1 --> y2>1. Because x2>=0 for all real x, y2>1 --> x2+y2>1, which is impossible for a point on the unit circle (since for all points on the unit circle (x,y), x2+y2=1).

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