The usual algebraic function for solving cubic equations involves inverting the cubic power function P3(x) = x^3. This can be defined as the function which can be developed in power series around t=1 as P1/3(t) = 1 + (t-1)/3 - (t-1)^2/9 + ..., and which can be analytically continued, with a branch cut from 0 to -infinity. The roots of the polynomial x^3-t are then x = P1/3(t), ωx and ω^2x.
However, we can use instead the Chebyshev polynomial (here normalized to -2 to 2) C3(x) = x^3-3x, which can be inverted as an algebraic function developed in power series around t=2 as C1/3(t) = 2 + (t-2)/9 - (2/243)(t-2)^2 + ..., and which can be analytically continued, with a branch cut from -2 to -infinity. The roots of the polynomial x^3-3x-t are then C1/3(t), -C1/3(-t), and C1/3(-t)-C1/3(t).
If we want to solve x^3+ax+b, then if a=0 we take ordinary cube roots. Otherwise, we apply the above three functions to -(-3/a)^(3/2)b and divide by sqrt(-3/a). That is, one of the roots will be C1/3(-(-3/a)^(3/2)b)/sqrt(-3/a), another will be -C1/3((-3/a)^(3/2)b))/sqrt(-3b), and the third will be minus the sum of these two.
Just as P1/3(t) can be computed via transcendental functions as exp(log(t)/3), C1/3(t) can be computed by 2cosh(arccosh(t/2)/3) or 2cos(arccos(t/2)/3). This no more makes the Chebychev cube root solution a solution in transcendental functions than logarithms makes the ordinary cube root solution a solution in transcendental functions. Moreover, most cases solving a solvable polynomial in Chebyshev radicals leads to a neater solution than solving it in ordinary radicals.