余弦倍角公式和切比雪夫多项式

On Cosine Formula with Multiple Angles And Chebyshev Multinomial

在初等数学中,三角函数是一个十分有用的工具.余弦倍角公式是由余弦的幂整系数线性组合来表示倍角的余弦,这样就产生余弦的n倍角能否用余弦的幂次的整系数线性组合表示等问题,通过研究,发现cosnα都是关于2cosα的首项系数为1的、次数等于α的倍数的、系数符号正负相间的整系数多项式,还进一步得到cosnα的一些性质.应用此性质,可以得到一些求和公式及解决许多数学问题.进一步研究,发现此多项式可以转化为切比雪夫多项式,这一多项式及系数有一些有趣的性质.

In elementary mathematics, trigonometric function is an important tool. For example, cosine formula can be used in many fields. When we expand the cosine fomula, it is the linear combination of cosine. In detail, cos na is an integral coefficient polynomial of 2cos a, where the leading term coefficient is 1 and the order is the time of a. Based on this we obtained many properties, from which we can derive some sum formulas and resolve many mathematical problems. Furthermore, this polynomial can be converted into Chebyshev polynomial which has some interesting properties.