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Question: How to convert real roots into expressions without imaginary units

Question: How to convert real roots into expressions without imaginary units

lcz 1029 Maple 2020
roots trigonometric
March 18 2021
1

I easily found the roots of the following equations,

s:=solve(x^3+x^2-2*x-1,x)

I accidentally discovered that  2*cos((2*Pi)/7) is also the root of this equation. 

simplify(eval(x^3+x^2-2*x-1,x=2*cos((2*Pi)/7)))

0

It’s easy to know that 2*cos((2*Pi)/7) is equivalent to the third root found above

s1:=fsolve(x^3+x^2-2*x-1,x);
2*cos((2*Pi)/7.)

         s := -1.801937736, -0.4450418679, 1.246979604

                          1.246979604

 

Maybe I don’t like the use of imaginary units, and I prefer  2*cos((2*Pi)/7) .

In the case that I don’t know that  2*cos((2*Pi)/7) is the solution of the equation, can I make a certain transformation without using the imaginary unit to represent the real number. For example, trigonometric functions, exponential functions, etc.

I tried to use the following functions, all failed.

s:=solve(x^3+x^2-2*x-1,x);
convert(s[3],cos);
identify(s[3])

 

What is interesting is the following phenomenon. Even if Zeta function does not look great:

s1:=fsolve(x^3+x^2-2*x-1,x):
identify(s1[3])

 

 

For this example, can all roots be transformed into trigonometric expressions by maple.

 

 
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