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A lot of these problems in math with no exact solution could have an exact solution if we defined some new function. Another way, we wouldn't have exact solutions to the angles of a triangle from the length of the sides without the sine function. I wonder if any of these hypothetical functions have utility outside their specific problems.

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For anyone confused: There is no exact equation for the perimiter of a circle either. It looks like there is because we use Pi - but Pi is not exact, you need an infinite series to calculate it, just like you need an infinite series for the perimeter of a ellipse.

PI in itself is as exact as you desire it to be, the real world is unfortunately unable to handle infinite precision :)

Yes, but the point is the Pi is hiding an infinite series, just like the what you'd see for an ellipse length. The more interesting point for me is to phrase it as: why can't we write the lenght of an ellipse as some function of pi, since you're able to do that for most trig functions.

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