Is there a mathematical reason why rotation in the counterclockwise direction positive and clockwise rotation negative?

Question

This inquiry has recently come to me in my study of trigonometry and the unit circle. It was said right from the very start that counterclockwise rotation were positive while clockwise rotations are negative, and I was wondering if there was a mathematical reason for this or if it was just picked that way.

Answer 1

Draw the axes with the $y$ one going downwards, while the $x$ one goes as usual. Now clockwise rotation is positive.

The mathematics behind this behavior is called orientation.

Answer 2

There is no mathematical reason for this.

The reason for picking it this way (irritating people in other fields, as you can see from the comments!) is that rotating the [positive] $x$-axis onto the [positive] $y$-axis is about the simplest rotation you can think of, so we decide to call it "positive". And clearly rotating the positive $x$-axis onto the positive $y$-axis is, in the normal way we draw Cartesian coordinates, anticlockwise.

There is a bonus. The point $(1,0)$, rotated through an anticlockwise angle $\theta$, ends up at $(\cos\theta,\sin\theta)$, which it wouldn't have if we had defined "positive" rotations as going in the other direction.

Answer 3

Besides the right-handed system convention, sometimes we use clockwise sense. Of course the rotation of hour/minute/second hand of a clock. Also, whole-circle bearing is using clockwise direction (though polar coordinates take anti-clockwise as positive).

Interestingly the elliptic integral $\displaystyle \int_{0}^{t} \sqrt{a^2\cos^2 t+b^2\sin^2 t} dt=aE\left( t,\sqrt{1-\frac{b^2}{a^2}} \, \right)$, used in calculating the arc length of an ellipse, are based on $(x,y)=(a\sin t, b\cos t)$ which is clockwise sense.