Why NOT tanh(3x) for Hyperbolic solution of a cubic


Hello from Ireland,

Since Tanh(x) would seem to have the same range of values as Cos(x) ie
-1 to +1,
I can not understand why ALL cubics of the form:-

x^3 + px +q = 0

can not be solved using hyperbolic functions.

While Hyperbolic Sinh(3x) & Cosh(3x) lend themselves for solutions
where the magnitude of Q is > 1, I tried expanding Tanh(3x) to see if
it could likewise be used instead of the

Trignometric Cos(3x) to solve such a cubic as has to depend on Cos(3x)
for solution.

I am having difficulty in expanding Tanh(3x) in a way that would lend
itself towards that end.

It would need to be something like this that i found on the internet:-
TANH(3x) = (3*TANH(x) + TANH^3(x)) / (1 + 3*TANH^2(x))
but alas i dont know where to go from here with it

Can anyone help or even show me why this might not be even possible.


Thank You

.

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