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Reduce, Solve and FindRoot all appear to struggle with this:

a[t_] := 1 + Cos[t] + Cos[2 t]
b[t_] := Sin[t] + Sin[2 t]
c[t_] := Sqrt[(Cos[t] + 2*Cos[2*t])^2 + (-Sin[t] - 2*Sin[2*t])^2]
d[t_] := Abs[-(Derivative[1][b][t]*Derivative[2][a][t]) + 
Derivative[1][a][t]*Derivative[2][b][t]]/(Derivative[1][a][t]^2 + 
Derivative[1][b][t]^2)^(3/2)
Solve[c[t] == 2/d[t], t]
FindRoot[c[t] == 2/d[t], {t, 2 Pi/3}]

(*out:
{}
{t -> 2.0944}
*)

Expected result is $2\pi/3$, the real roots being periodic:

enter image description here

but Mathematica appears to have difficulty with Solve and Reduce and it doesn't give a closed solution for the FindRoot. Is there anything I can do to yeild the expected result?

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    $\begingroup$ Solve[c[t] == 2/d[t] && 0 < t < Pi, t]. You should read Can Reduce really not solve for x here?, these are also on topic How do I solve this equation? and Solve symbolically a transcendental trigonometric equation and plot its solutions $\endgroup$
    – Artes
    Oct 24, 2014 at 10:42
  • $\begingroup$ @Artes thank you - I haven't seen any of these before - I will take a look now $\endgroup$
    – martin
    Oct 24, 2014 at 10:43
  • $\begingroup$ @Artes thank you - Reals - silly me! BTW, what does C[1] mean? $\endgroup$
    – martin
    Oct 24, 2014 at 10:46
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    $\begingroup$ This is a symbolic constant, don't use C symbol in Mathematica for your own variables. I recommend to read carefully all the above posts, then you will find adequate details concerning your problem. $\endgroup$
    – Artes
    Oct 24, 2014 at 10:49
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    $\begingroup$ Highlight C and press F1. $\endgroup$
    – Artes
    Oct 24, 2014 at 10:54

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