An Analysis Of The Relation Between Triangle Inequality And Algebraic Inequality

Inequality is one of the core contents of elementary mathematics, it is an excellent material to exercise students' algebraic operation and logical reasoning.Inequality is also an important tool to study "analysis" in higher mathematics, and it is the foundation and tool for further study of modern mathematics and even other disciplines. In the entire mathematical knowledge system occupies a place, is an important part of the basic theory of mathematics.Most of the univariate inequality of the problem is solved with the function point of view;On the n-element inequality problem is flexible, strong skills, is the hot topic at home and abroad research, it is quite difficult; However, the inequality issue of binary and ternary although flexible, but with respect to the univariate inequality,but more specific and systematic inequality, is also relatively more interesting, And a large number of ternary inequalities and triangles in the angular trigonometric function of the identity or inequality is very closely linked, various mathematics competitions at all levels in a large number of mathematical competition classical ternary algebraic inequalities triangle inequality can be found in its background,while using the existing triangle inequality to be a beautiful life made a large number of forms and have a certain degree of difficulty of mathematics contest questions.This article aims to reveal the intrinsic link between them, on the one hand by two, by a simple triangle inequality arising beautiful algebraic inequalities, there are two main ways: First, by using the classic "inscribed circle substitution", and then will be three inside or half-trigonometric function value associated with the algebraic expression, thus transforming the simple algebraic inequality is the triangle inequality; the second is based on trigonometric identities for the substitution based on the introduction of a variable, then the other three interior angles or half-width value associated with trigonometric substitution variables to be expressed, and thus the triangle inequality into simple algebraic inequalities. On the other hand, by the beautiful algebraic inequalities, we can also consider looking for its substitution by an equivalent triangle forms, one way this is mathematics competition problems.Theory and practice, we propose to give two specific research cases, one is a 2002 Iranian mathematical Olympiad inequality test questions to find a variant of the use of common triangular inequality, combined with triangular substitution for variational inquiry, A large number of new algebraic inequalities have been obtained,and many results coincide with the maths of previous years. The second is to find a variant of the Iranian Mathematical Olympiad algebraic inequality test in 1996, using the common triangular substitution for variational inquiry, and obtain a large number of new trigonometric inequalities. For the study of specific cases, we aim to more specifically reveal the close relationship between the trigonometric inequality and the algebraic inequality. This is a reference value for the problem solving and proposition of the competition mathematics and the research study.