# Closed solutions for FindRoot, no solutions for Reduce or Solve [duplicate]

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[b][t]*Derivative[a][t]) +
Derivative[a][t]*Derivative[b][t]]/(Derivative[a][t]^2 +
Derivative[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: 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?

• 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 Oct 24, 2014 at 10:42
• @Artes thank you - I haven't seen any of these before - I will take a look now Oct 24, 2014 at 10:43
• @Artes thank you - Reals - silly me! BTW, what does C mean? Oct 24, 2014 at 10:46
• 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. Oct 24, 2014 at 10:49
• Highlight C and press F1. Oct 24, 2014 at 10:54