# Why are angles defined as positive counter clockwise?

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## 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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### littleO

Updated on August 02, 2022