# Why do we start measuring angle from positive direction of X axis only?

Why do measure the angle from positive X axis in coordinate geometry? Further we also say clockwise rotation would produce negative angles? How can angles be negative? I mean, how much is $-30°$? Does that even make sense?

## 4 Answers

There is no particular reason we measure positive angles as counterclockwise from the positive $x$ axis, it was adopted as a convention in order to standardize the way we do it. Negative angles are also used in the way you ask, as a clockwise rotation.

In the Argand diagram, $e^{it}$ is a point on the unit circle when $t$ is real. It lies on the positive $x$-axis for $t=0$, and when $t$ increases it moves anti-clockwise about the unit circle, and when $t$ decreases it moves clockwise. This is the reason that in complex analysis, anti-clockwise is regarded as the positive direction.

Astronomy played a big part in the development of trigonometry and, in particular, angles. Consider a plane with the sun at the origin and the earth travelling, in the plane, in an almost circular path around the sun. Put your right hand on the origin with your thumb pointing north. Your fingers will curl in the direction that the earth is rotating and in the direction that the earth is travelling around the sun. This is called a right-handed system. Most, not all, of the sun-planet, planet-axis, and planet-moon systems in the solar system are right-handed systems. The bolts and lug nuts on most tires are right handed systems. Most bathroom faucet handles are right-handed systems. So it is convenient for counter clockwise to be the positive direction for an angle and for the counterclockwise direction to be the negative direction for an angle.

I have answered the first of your questions here elsewhere (sufficiently, I hope). I shall address the rest, which express a bafflement about using negative numbers to name angles.

I think the key thing to note here is first, what do we mean by angle? There are many ways to view the same thing, but some are better than others in some contexts. In elementary synthetic euclidean geometry, it may be sufficient to define an angle as a connected union of two rays intersecting only at their origins. However, in order to extend the trigonometric functions to all possible real numbers (and we want this, I hope you now know), we need a more encompassing idea of angle. The concept of a rotation or turn provides the answer. It is clear that we always need to fix some point about which we shall rotate, and that the locus of a point on some rotating object is a circle.

It is clear that in the plane, we rotate in at most two possible directions, from some initial configuration. If we wish to describe all such rotations in such a way that we want the original configuration to be identified with the real number $0$ (and we do want this for reasons I've explained elsewhere), then it is clear that all rotations in one direction would be positive while the oppositely directed rotations would be negative (whichever is which is our choice to make).