Why are angles defined as positive counter clockwise?


Solution 1

It is simply a matter of convention.

If you want a justification for such convention, I think that it is physical. In a Cartesian plane we usually identify the $x$ axis as horizontal and the $y$ axis as vertical referring to our world where the direction of the gravity is orthogonal (and vertical), with respect to the horizon. And we assign the positive direction of the $x$ axis from left to right and the the positive direction of the $y$ axis from down to up. So it is more simple to assign the positive direction to an angle if it rotate the positive $x$ semi-axis toward the positive $y$ semi-axis and this is the counter clockwise direction.

But: why the clock hands go in the clockwise direction? This is a mistery!

Solution 2

Positive angles are counterclockwise only in right-handed coordinate systems, where $y$ axis increases upwards, and $x$ axis right.

In a left-handed coordinate system, $y$ axis increases down, and $x$ axis right, and positive angles are indeed clockwise. Such coordinate systems are often used in e.g. computer graphics.

(Note that by rotating the coordinate system 180°, $x$ axis increases left, and $y$ down; I do not recall seeing this convention anywhere in practice, but I guess it would be just as natural to predominantly left-handed people. It is just that most humans are predominantly right-handed, and that does seem to permeate our culture in very subtle ways.)

The underlying reason why we use $\cos$ for $x$ axis, and $\sin$ for $y$ axis, comes from Euler, and complex numbers in particular: $$\begin{align} z &= r e^{i\varphi} \\ \vec{p} &= \left ( r \cos\varphi, r \sin\varphi \right ) = \left ( \operatorname{Re} z, \operatorname{Im} z \right ) \end{align}$$

So, if you want to consider clockwise angles positive, just use a left-handed coordinate system (where $x$ increases right, and $y$ downwards).

Do remember to state your preferred handedness, though; most mathematicians et cetera assume a right-handed coordinate system unless stated otherwise.


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Updated on August 02, 2022


  • littleO
    littleO over 1 year

    A rather peculiar question and off topic in every way but though. In almost every situation clockwise is considered to be positive but not when it comes to angles. Why is that? Euler's fault or ...

    • Praneet Srivastava
      Praneet Srivastava about 7 years
      I think you mean "angles" and not "angels"
    • Vim
      Vim about 7 years
      Probably because people prefer the first quadrant to the fourth one, so naturally define an angle to be positive when its "rotatable" side (the "fixed" side being the positive half of the x-axis) lands in the first, rather than the fourth quadrant.
    • hmakholm left over Monica
      hmakholm left over Monica about 7 years
      Acatually in pure mathematics clockwise is usually considered the negative direction.
  • Andrew D. Hwang
    Andrew D. Hwang about 7 years
    (+1) For what it's worth, I've seen it suggested that "clockwise" is the direction that the shadow of a sundial moves in the northern hemisphere.