Transcription
Ever wonder why we measure counterclockwise on the unit circle rather than clockwise? You may have heard this is just a convention, and that's basically true, but I always felt like that was sort of a cop-out. If it really is arbitrary, like a free choice, why not go clockwise instead?
And there's sort of a reason, just not the kind you might expect. And I should point out here, the clockwise version genuinely works. You can even keep the regular definition of X being cosine theta, Y being sine theta as a point on the unit circle. Spin it another way, you get the mirror image. Nothing really breaks here.
I've heard it said we start where things are positive to go counterclockwise, but that doesn't make a ton of sense because you can do the same thing clockwise and still end up with positive values. Both directions are perfectly consistent.
So, the real reason is probably closer to something like consistency. Consistency with stuff that had already been written down. By the time anyone drew a unit circle, three things were basically locked in. The Y axis points up. From right triangle trigonometry, thousands of years older, cosine of zero equals one. And the sign of a small positive angle is positive.
Now sweep clockwise from one zero. A tiny positive angle dips below the axis, which makes sine of 10° negative. And that disagrees with the triangle where that side is a positive length. So, counterclockwise wasn't necessarily picked out of nowhere. It's the one direction that doesn't fight the triangle we already had. The circle just sort of inherited its spin.
But it's not the same as a law. Flipping it doesn't necessarily break anything. If you flip the whole thing clockwise, sin squared plus cos squared still equals one. The derivative rules still work when you relabel them. E to the i theta is fine, too. If the direction were truly forced, something would break here, but nothing really does.
This is essentially like driving on the right side of the road. You wouldn't necessarily demand some deep reason for it. You'd look at the history.