Is it possible to ask Mathematica to give all the roots of the given function?

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 $$0? Preferably as a rational multiple of $$\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



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]}]]}]


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

Select[sol, #/Pi \[Element] Rationals &]

• 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?
– user67849
Commented Apr 15, 2021 at 0:47
• @quark - it means they're not rational multiples of Pi, and yes, N will get you the numeric result.
– ciao
Commented Apr 15, 2021 at 0:48
• Strictly speaking, this doesn't prove that the ArcTan[..] terms are not rational multiples of Pi. They may be. Commented Apr 15, 2021 at 2:54