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单词 TrigonometricCubicFormula
释义

trigonometric cubic formula


Given a cubic polynomial of the form f(X)=X3+aX2+bX+c=0, one may reduce f(x) via the substitution X(x-a/3) to obtain f~(x)=f(x-a/3) where the reduced polynomialMathworldPlanetmathPlanetmathPlanetmath may be represented

f~(x)=x3+qx+r(1)

The roots to (1) are given by Viéte in the following cases:

Case I The roots of f~(x) are real:
Define t-4q/3 and α=arccos(-4r/t3).Then the roots of f~(x) are

tcos(α/3),tcos(α/3+2π/3),tcos(α/3+4π/3)

Case II The roots of f~(x) are complex:
Keeping the definition of t from Case I, if -4q/30, then the real root of f~(x) is

tcosh(β/3)wherecosh(β)=(-4r/t3)

If -4q/3<0, then the real root of f~(x) is

tsinh(γ/3)wheresinh(γ)=(-4r/t3)

One may then inversePlanetmathPlanetmathPlanetmathPlanetmathPlanetmath transform the roots of f~(x) to obtain the roots of the desired cubic f(x)

We note there are no other cases for the possibilities of the roots of a cubic (i.e. there is no instance where one finds one complex and two real roots). This result is intuitively obvious after graphing cubic polynomials and taking into account that imaginaryPlanetmathPlanetmath roots may only occur in conjugate pairs.

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更新时间:2025/5/4 10:34:34