# Distance Between Any Two Points on a Unit Circle

As part of a larger investigation, I am required to be able to calculate the distance between any two points on a unit circle. I have tried to use cosine law but I can't determine any specific manner in which I can calculate theta if the angle between the two points and the positive axis is always given.

Is there any manner in which I can do this?

Thanks

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– Jeff E Apr 21 '12 at 0:21
I already read that but I don't think it really helped. – n0shadow Apr 21 '12 at 0:33
Did you learn dot product? – FiniteA Apr 21 '12 at 1:16
Yes I did...I'll try to look into that. – n0shadow Apr 21 '12 at 14:43

If the arc distance between the two points is $\theta$, the length of the chord between them is $2\sin\frac{\theta}{2}$:

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@ProgramFun: Sounds like you should just subtract the two values you know. (You may end up with a negative value of $2\sin\frac\theta2$, but just take the absolute value of that). Unless the "angles between each point and the positive $x$-axis" are not always measured in the same direction, that is. (And if they aren't, then you need more data). – Henning Makholm Apr 21 '12 at 14:43
Thanks for that, I had just figured that out myself right before your post and was spending time testing it. Is there any name for the formula you suggested? – n0shadow Apr 21 '12 at 14:57
@ProgramFun: I don't think it has a fancy distinguished name, but you won't go much wrong by calling it, say "the chord-length formula". Or even the "chord formula", if it's alright by you that attempts to google it will return lots of noise about musical notation. – Henning Makholm Apr 21 '12 at 15:45
That diagram rates a +1! :-) – robjohn Apr 21 '12 at 17:06

Hint

• Points on the unit circle centered at $(0,0)$ on the argand plane are of the form $(\cos \theta, \sin \theta)$, with $0 \leq \theta \lt 2\pi$.

• Can you use distance formula now to calculate the requires to distance?

With some knowledge in complex numbers, you'd realise that, if $z_1$ and $z_2$ are two complex numbers, the amplitude, $|z_1-z_2|$ is the distance between the two of them.

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