Tips:

A. Calculate a small angle, you'll find poligon's cords miserably.

B. Calculate a VERY small angle, you'll find the arc is superior than sine/tangent lines.

Translate those calculation results into several meters.

No, you wont.

Calculate an angle less than 90 degrees. This could be quite large, almost 90, or really really tiny, like 0.00000000000000001 degrees.

You will find that you are able to calculate the points for a polygon.

You will find that the sine is always smaller than the arc length and the tangent is always larger.

This is trivial to understand.

Get 2 such sections and stick them together back to back:

We see the sine is a straight line going from the top to the bottom.

This means it MUST be smaller than the arc length as it is the shortest path between the 2 points.

We see the tangent is longer path, going out to a point then turning and coming back in.

And we see the arc between these, where it doesn't go straight down, but it also doesn't go out as far as the tangent. Instead it cuts the corner.

As it cuts the corner, it must be shorter than the tangent.

We can see it even better if we show another shortcut path here:

We see the path in blue cuts out a large chunk of the tangent path opting for a straight path between 2 points on the tangent.

This makes it shorter than the tangent. We also see it better approximates the curve, yet the curve still cuts that, making the arc shorter than that path which is shorter than the tangent.

So if you do it properly you will always see that the arc is between the sine and the tangent.

There is no way out of that.