AngouriMath
SolveARationalFunctionOfASineOverASymbolicQuadratic(AngouriMath.Entity,AngouriMath.Entity.Variable,System.Boolean)
Method (no overloads)
Summary
A rational function of a sine, or of a cosine, over a quadratic in it with a symbol
among the coefficients:sin(x)/(a + b sin(x) + c sin(x)^2) , and the same with
the other function to even powers, a cosecant or a secant beside. Written in
s = sin(x) it is a rational function of s , divided down and split over
the written factors as any is, and the piece over the quadratic is taken by the two
roots: withq = sqrt(b^2 - 4 a c) and r = (-b ± q)/(2c) ,
(d + e s)/(a + b s + c s^2) is (d + e r_1)/(q (s - r_1)) - (d + e r_2)/(q (s - r_2)) ,
and1/(sin(x) - r) is -2 atan((r tan(x/2) - 1)/sqrt(r^2 - 1))/sqrt(r^2 - 1) for any complex r but ±1 ; 1/(cos(x) - r) is
-2 atan(tan(x/2)/σ)/((1 + r) σ) with σ = sqrt((r - 1)/(r + 1)) .
among the coefficients:
the other function to even powers, a cosecant or a secant beside. Written in
the written factors as any is, and the piece over the quadratic is taken by the two
roots: with
and
Remarks
in every coefficient, which nothing factors, and the substitution
The two closed forms are each checked by differentiation in the summary of the
biquadratic rule's kind:
quadratic rule writes for a symbolic root, has no value for the conjugate pair the
ordinary case gives. On each interval between the poles of
standing property of the half-angle.
written declines.
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