# Cubic equation - Wikipedia - URL: <https://en.wikipedia.org/wiki/Cubic_equation#Trigonometric_and_hyperbolic_solutions> - Description: - Type: #link - Highlights - When there is only one real root (and p ≠ 0), this root can be similarly represented using hyperbolic functions, as[28][29] ��� 0 = − 2 | ��� | ��� − ��� 3 cosh ⁡ [ 1 3 arcosh ⁡ ( − 3 | ��� | 2 ��� − 3 ��� ) ] if    4 ��� 3 + 27 ��� 2 > 0    and    ��� < 0 , ��� 0 = − 2 ��� 3 sinh ⁡ [ 1 3 arsinh ⁡ ( 3 ��� 2 ��� 3 ��� ) ] if    ��� > 0. - If p ≠ 0 and the inequalities on the right are not satisfied (the case of three real roots), the formulas remain valid but involve complex quantities. - When p = ±3, the above values of t0 are sometimes called the Chebyshev cube root.[30] More precisely, the values involving cosines and hyperbolic cosines define, when p = −3, the same analytic function denoted C1/3(q), which is the proper Chebyshev cube root. The value involving hyperbolic sines is similarly denoted S1/3(q), when p = 3.