AngouriMath

Symbolics for .NET


Table of contents

AngouriMath

Navigation
Home Quick start Wiki Almanac Research What's new Try Use Angouri.org GitHub Report a bug Donate Contacts

← Back to list of members

SolveByUnifyingTrigonometricArguments​(AngouriMath.​Entity,​AngouriMath.​Entity.​Variable,​System.​Boolean)

 Method (no overloads)

Summary

An integrand whose trigonometric functions have different multiples of one
argument — sin(x)/cos(2x), cos(x)/(sin(x) tan(x/2)) — rewritten so that
every one of them is of the same argument, and handed on.

Remarks

Every trigonometric rule here reads one argument: a quotient of homogeneous
polynomials in sin(u) and cos(u), a function of tan(u), the
half-angle substitution in u. An integrand with x and 2x in it
is none of those and every one of them declined it — while the same integrand with
cos(2x) written as 1 - 2 sin(x)^2 came out. Nothing was missing but the
rewriting. Product-to-sum, beside this, goes the other way, and fires on a product of
two different arguments; this fires on anything else built from them.
 The common argument is g x with g the greatest common divisor of the
slopes, so x and x/2 share x/2 and 2x and 3x share
x. Each function of n g x is then a polynomial in sin(g x) and
cos(g x) through the Chebyshev recurrences, cos(n t) = T_n(cos t) and
sin(n t) = sin(t) U_(n-1)(cos t), which are identities and owe no condition;
a tangent, secant or cosecant is the quotient of those it stands for. Slopes are
rational and the multiples are capped, since a function of 50x beside one of
x is a degree-fifty polynomial nobody wants.
 It terminates: what it hands on has one argument, on which this rule declines.
 https://github.com/asc-community/AngouriMath/issues/718

























Angouri © 2019-2023 · Project's repo · Site's repo · Octicons · Transparency · 4378 pages online