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在中学数学教学中常常遇到一类与三角形三内角有关的不等式,例如,在△ABC中, sinA/2sinB/2sinC/2≤1/8, sinA/2sinB/2sinC/2≤1/8, tg~2A/2+tg~2B/2+tg~2C/2≥1。文[1]中给出了证明这一类三角函数不等式的逐步凋整原理,并在例2中罗列了许多此类不等式。本文的目的是要说明,这一类三角函数不等式在有关三角形的几何极值问题中有着重要的应用。为使读者参阅方便和下文引用的需要,这里先将文[1]中的若干有关的不等式重写于下:
In middle school mathematics teaching, there is often encountered a type of inequality associated with triangular inner angles. For example, in ΔABC, sinA/2sinB/2sinC/2 ≤ 1/8, sinA/2sinB/2sinC/2 ≤ 1/8, tg ~2A/2+tg~2B/2+tg~2C/2≥1. In [1], the gradual withering principle of proving this type of trigonometric inequality is given, and in Example 2 many such inequalities are listed. The purpose of this paper is to show that this type of trigonometric inequality has important applications in the geometric extremum problem of triangles. For the reader to refer to the convenience and needs cited below, here are some of the related inequalities in [1] are rewritten below: