If we plot the following, we can see there are seven roots

Plot[{x, 3 Pi (1 - Sin[x])}, {x, 0, 22}]

We can easily solve for these roots

Reduce[x == 3 Pi (1 - Sin[x]), x, Reals]

The results are

enter image description here

Why is Mathematica showing the $3 \pi$ root twice? It is not a double root!

Solve does the same thing.

This appears to be a bug and I thought those sorts of questions were welcome here.

  • 4
    $\begingroup$ Showing the multiplicity of a root is what Solve normally does, at least for algebraic equations. Like Solve[x^2 == 0, x]. Seems unlikely it's a double root, though. $\endgroup$
    – Goofy
    Commented Mar 12 at 13:21
  • 1
    $\begingroup$ The behavior is different in "12.0.0 for Mac OS X x86 (64-bit) (April 7, 2019)":Reduce doesn't return a 3*Pi as a double root. It's possible that this is a bug (or was a bug if that point is actually a double root). $\endgroup$
    – march
    Commented Mar 12 at 15:35
  • 1
    $\begingroup$ Indeed: The Taylor expansion of $3\pi(1-\sin(x)) - x$ about $x=3\pi$ is $(3\pi-1)(x-3\pi) -\frac{1}{2}\pi(x-3\pi)^3 +\cdots$, indicating that $3\pi$ is not a double root. So we might be able to characterize this as a bug, depending on the intention of Reduce and when it returns multiple of the same answers. $\endgroup$
    – march
    Commented Mar 12 at 15:40
  • 3
    $\begingroup$ This has been reported as a bug. $\endgroup$ Commented Mar 12 at 20:02
  • 4
    $\begingroup$ As for the questionj of whether this post is appropriate for MSE, I would think the Reduce result is sufficiently confusing as to make it a reasonable query. It's not entirely obvious, at least at a glance, that it is "just" a bug. $\endgroup$ Commented Mar 12 at 20:04

1 Answer 1


This is a bug. A subdivision method happens to subdivide very close to a root, and the same root gets found in both branches (due to numerical tolerance). The code was not checking for that, it will in the next version.

  • $\begingroup$ Thank you! ${}{}$ $\endgroup$
    – Moo
    Commented Mar 12 at 18:21
  • 2
    $\begingroup$ Where? In Solve or Reduce? $\endgroup$ Commented Mar 12 at 18:29

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