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I have this equation

$$ 36 \cos \frac{3 x}{4} \cos \frac{27 x}{20} \left(\cos \frac{3 x}{5} +2 \cos \frac{21 x}{10} \right)=0 $$

Is it possible to ask Mathematica to give all the roots of the function on the domain $0<x<4\pi$? Preferably as a rational multiple of $\pi$? (the plot of the function is attached)

36 Cos[(3 x)/4] Cos[(27 x)/20] (Cos[(3 x)/5] + 2 Cos[(21 x)/10]) == 0

enter image description here

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sol = x /. Solve[36 Cos[(3 x)/4] Cos[(27 x)/20] (Cos[(3 x)/5] + 2 Cos[(21 x)/10]) == 0&& 0 < x < 4 Pi]

Plot[36 Cos[(3 x)/4] Cos[(27 x)/20] (Cos[(3 x)/5] + 
    2 Cos[(21 x)/10]), {x, 0, 4 Pi}, 
 Epilog -> {Red, 
   Point[Transpose[{sol, ConstantArray[0, Length@sol]}]]}]

enter image description here

enter image description here

If you desire only roots that are rational multiples of Pi, you can use (among many other ways):

Select[sol, #/Pi \[Element] Rationals &]
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  • $\begingroup$ Thanks, Then, does this mean that those $ArcTan[...]$ are definitely not multiple of $\pi$? Is it correct to use $N[ArcTan ...]$ to know their numerical value? $\endgroup$
    – user67849
    Commented Apr 15, 2021 at 0:47
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    $\begingroup$ @quark - it means they're not rational multiples of Pi, and yes, N will get you the numeric result. $\endgroup$
    – ciao
    Commented Apr 15, 2021 at 0:48
  • $\begingroup$ Strictly speaking, this doesn't prove that the ArcTan[..] terms are not rational multiples of Pi. They may be. $\endgroup$
    – user10181864
    Commented Apr 15, 2021 at 2:54

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