Editing Cubic equation (section)

==Trigonometric and hyperbolic solutions==<!-- linked from redirect [[Chebyshev cube root]] -->

===Trigonometric solution for three real roots===
When a cubic equation with real coefficients has three real roots, the formulas expressing these roots in terms of radicals involve complex numbers. [[Galois theory]] allows proving that when the three roots are real, and none is rational (''[[casus irreducibilis]]''), one cannot express the roots in terms of real radicals. Nevertheless, purely real expressions of the solutions may be obtained using [[trigonometry|trigonometric functions]], specifically in terms of [[cosine]]s and [[arccosine]]s.<ref>{{cite journal |last=Zucker |first=I.J. |date=July 2008 |title=The cubic equation — a new look at the irreducible case |journal=[[Mathematical Gazette]] |volume=92 |pages=264–268|doi=10.1017/S0025557200183135 |s2cid=125986006 }}</ref> More precisely, the roots of the [[#Depressed cubic|depressed cubic]]
<math display="block">t^3 + pt + q = 0</math>
are<ref name="crc">{{cite book
 |title=CRC Standard Mathematical Tables
 |editor-first=Samuel
 |editor-last=Shelbey
 |year=1975
 |publisher=CRC Press
 |isbn=0-87819-622-6
 |url=https://archive.org/details/crcstandardmathe00selb
}}</ref> 
<math display="block">t_k = 2\,\sqrt{-\frac{p}{3}}\,\cos\left[\,\frac{1}{3} \arccos\left(\frac{3q}{2p}\sqrt{\frac{-3}{p}}\,\right) - \frac{2\pi k}{3}\,\right] \qquad \text{for } k=0,1,2.</math>

This formula is due to [[François Viète]].<ref name=Nickalls>{{cite journal |last=Nickalls |first=R.W.D. |date=July 2006 |title=Viète, Descartes, and the cubic equation |journal=[[Mathematical Gazette]] |volume=90 |issue=518 |pages=203–208 |doi=10.1017/S0025557200179598 |s2cid=124980170 |url=https://www.nickalls.org/dick/papers/maths/descartes2006.pdf}}</ref> It is purely real when the equation has three real roots (that is <math>4p^3 + 27q^2 < 0</math>). Otherwise, it is still correct but involves complex cosines and arccosines when there is only one real root, and it is nonsensical (division by zero) when {{math|''p'' {{=}} 0}}.

This formula can be straightforwardly transformed into a formula for the roots of a general cubic equation, using the back-substitution described in {{slink||Depressed cubic}}. 

The formula can be proved as follows: Starting from the equation {{math|''t''<sup>3</sup> + ''pt'' + ''q'' {{=}} 0}}, let us set {{nowrap|&nbsp;{{math|''t'' {{=}} ''u'' cos&thinsp;''θ''}}.}} The idea is to choose {{mvar|u}} to make the equation coincide with the identity
<math display="block">4\cos^3\theta - 3\cos\theta - \cos(3\theta) = 0.</math>
For this, choose <math>u = 2\,\sqrt{-\frac{p}{3}}\,,</math> and divide the equation by <math>\frac{u^3}{4}.</math> This gives 
<math display="block">4\cos^3\theta - 3\cos\theta - \frac{3q}{2p}\,\sqrt{ \frac{-3}{p} } = 0.</math>
Combining with the above identity, one gets 
<math display="block">\cos(3\theta) = \frac{3q}{2p}\sqrt{\frac{-3}{p}}\,,</math>
and the roots are thus 
<math display="block">t_k = 2\,\sqrt{-\frac{p}{3}}\,\cos\left[ \frac{1}{3} \arccos\left( \frac{3q}{2p}\sqrt{\frac{-3}{p}} \right) - \frac{2\pi k}{3} \right] \qquad \text{for } k=0,1,2.</math>

===Hyperbolic solution for one real root===

When there is only one real root (and {{math|''p'' ≠ 0}}), this root can be similarly represented using [[hyperbolic function]]s, as<ref>These are Formulas (80) and (83) of Weisstein, Eric W. 'Cubic Formula'. From MathWorld&mdash;A Wolfram Web Resource. https://mathworld.wolfram.com/CubicFormula.html, rewritten for having a coherent notation.</ref><ref>Holmes, G. C., "The use of hyperbolic cosines in solving cubic polynomials", ''[[Mathematical Gazette]]'' 86. November 2002, 473–477.</ref> 
<math display="block">\begin{align} t_0 &= -2\frac{|q|}{q}\sqrt{-\frac{p}{3}}\cosh\left[\frac{1}{3}\operatorname{arcosh}\left(\frac{-3|q|}{2p}\sqrt{\frac{-3}{p}}\right)\right] \qquad \text{if } ~ 4 p^3 + 27 q^2 > 0 ~\text{ and }~ p < 0,\\
t_0 & = -2\sqrt{\frac{p}{3}}\sinh\left[\frac{1}{3}\operatorname{arsinh}\left(\frac{3q}{2p}\sqrt{\frac{3}{p}}\right)\right] \qquad \text{if } ~ p > 0.\end{align}</math>
If {{math|''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 {{math|''p'' {{=}} ±3}}, the above values of {{math|''t''<sub>0</sub>}} are sometimes called the '''Chebyshev cube root.'''<ref>Abramowitz, Milton; Stegun, Irene A., eds. ''Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables'', Dover (1965), chap. 22 p. 773</ref> More precisely, the values involving cosines and hyperbolic cosines define, when {{math|''p'' {{=}} −3}}, the same [[analytic function]] denoted {{math|''C''<sub>1/3</sub>(''q'')}}, which is the proper Chebyshev cube root. The value involving hyperbolic sines is similarly denoted {{math|''S''<sub>1/3</sub>(''q'')}}, when {{math|''p'' {{=}} 3}}.