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Both $x = \cos t$, $y = \sin t$ and $x = \sin t$, $y = \cos t$ describe a circle

So why is the first parameterization so commonly used in mathematics, and not the second?

Jason
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5 Answers5

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Because we usually start the circle at the right. There, the coordinates are $(1, 0)$, which is $(\cos(0), \sin(0))$.

If we started it at the top, we would probably use $(\sin(t), \cos(t))$.

marty cohen
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    But why do we usually start the circle at the right? Does that have any natural motivation? (if asked to draw a circle, I would likely start near the top). – Erick Wong Nov 20 '15 at 20:51
  • Tradition. For a somewhat relevant discussion, look up how Knuth discusses the various ways of drawing trees in TAOCP. – marty cohen Nov 20 '15 at 20:58
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    Are you saying that the tradition predates the parametrization of the circle by cosine and sine? I find that interesting and somewhat surprising! – Erick Wong Nov 20 '15 at 21:01
  • I don't know. (See - I can answer any question.) – marty cohen Nov 20 '15 at 21:01
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Because $e^{it} = \cos t+i\sin t$ (but not only because of that).

Michael Hardy
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Because of convension.

It is customary to view angles as lying with the vertex at the origin and the right leg along the positive $x$-axis. That means that small angles will have their left leg in the first quadrant, and right angles will have the left leg along the positive $y$-axis. Because of this, the cosine of the angle happens to be the $x$-coordinate of the point on the left leg that is distance $1$ from the vertex, while the sine of the angle is the $y$-coordinate.

Arthur
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Assuming it is reasonable to view tan $\theta$ as more "natural" than cot $\theta$, then one advantage to making $x$ the cosine coordinate is it gives a direct correspondence between slopes and tangents. But I much prefer the motivation in Michael Hardy's answer.

Erick Wong
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Using $t$ is the most ‘natural’ parametrisation because $t$ is the polar angle of the point $(x,y)$ in a standard polar coordinates system.

Bernard
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