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Yesterday, I came to the realization that I don’t know the actual methodology behind how I can calculate the length of any arc on a circle nor how I can even locate the start and end point of any circular arcs.

I’m given the arc length of a circle formula s=r$\theta$ where r is the radius of the circle and $\theta$ is the angle in radians.

But this led me to ask, of course we have a protractor and the modern individual can just use that to find angles etc. and then we can just plug in the angle value it into the arc length formula. but how really (given that angles are dependent on arc lengths) can we calculate the angles analytically?

This then opened up a further question (how can we even calculate the arc length of any circle analytically)?

This made me think “of course it’s dependent on endpoints of the arc somehow” but when I asked people “given that we start from the rightmost point on a circle centered at origin and measure the arc length counterclockwise, say for example that I want to find the endpoint of the arc length of 2.4 radians or even something as simple as pi/2 radians how would we go about doing this?”

Everyone just threw me the trig functions : “rcos$\theta$ and rsin$\theta$ are the parameterizations for any point of any circle centered end at the origin. Use the same formula for arc length but in terms of $\theta$ and then plug into cos and sin your desired theta using a calculator and you will find the endpoint”.

This is just circular reasoning and reuse of the initial formula for arc length and does not answer my question at all. I don’t understand the methodology and am blindly using formulas I don’t understand the workings behind. Perhaps my question is coming off as confusing but if anyone could actually explain or help me, it would be one of the greatest burdens lifted off my back.

Extras: These are two questions that I drafted that I think are important ones that I should have answers to but cannot answer myself. Please correct me if I’m wrong and if they’re trivial or even nonsensical.: I have two questions:

  1. Can we define sin and cos and all other trig functions in terms of circular (with radius r and center (h,k)) arcs measured counterclockwise wise from (h+r,k) instead of angles?
  2. If s is the length of any arc on a circle, do we have formulas to calculate s without any reliance at all on angles?

Thank you very much!

Matt Harper
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    There is an integral expression for arc length, and you can take that as a definition for trigonometric functions. – Trebor Dec 21 '24 at 19:35
  • Defining the length of a circular arc is indeed quite difficult and requires the concept of a limit, which you learn in calculus. – Deane Jan 23 '25 at 03:31
  • The conceptual idea of arc length is not difficult. But getting from the concept to the tools (like integrals) requires some effort. See this related answer: https://math.stackexchange.com/a/3072835/72031 – Paramanand Singh Jan 23 '25 at 03:35
  • For your second question, remember when you calculate $s$, the number $\pi$ is going to show up (related to the circumference of the circle) and your answer will be a multiple of $\pi$ depending on the angle of the arc length. So, if you calculate $s$ in any similar way that others have in human history, the angle is necessarily going to show up somewhere. – D_S Jan 23 '25 at 04:06

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  1. Yes, and I was taught how to do it that way, it seems you have a general grasp on how to pair (h,k) with (x,y), however, ensure you use the radius for the numerator of cosecant and secant.
  2. Yes, but I strongly advise against it. It involves integrals, and unless you are in Calculus, I wouldn't try it. It also involves cords on a circle, which are generally just awful.
user1546553
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