对于集合{α1,α2,...,αn}和常数α0定义:μ=(sin(α1-α0)^2+sin(α2-α0)^2+...+sin(αn-α0)^2)/n为集合{α1,α2,...,αn}相对α0的正弦方差;求证:集合{π/2,5π/6,7π/6}相对任何常数α0正弦方差是一个与α0无关的定值.
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![对于集合{α1,α2,...,αn}和常数α0定义:μ=(sin(α1-α0)^2+sin(α2-α0)^2+...+sin(αn-α0)^2)/n为集合{α1,α2,...,αn}相对α0的正弦方差;求证:集合{π/2,5π/6,7π/6}相对任何常数α0正弦方差是一个与α0无关的定值.](/uploads/image/z/13917104-8-4.jpg?t=%E5%AF%B9%E4%BA%8E%E9%9B%86%E5%90%88%7B%CE%B11%2C%CE%B12%2C...%2C%CE%B1n%7D%E5%92%8C%E5%B8%B8%E6%95%B0%CE%B10%E5%AE%9A%E4%B9%89%3A%CE%BC%3D%28sin%28%CE%B11-%CE%B10%29%5E2%2Bsin%28%CE%B12-%CE%B10%29%5E2%2B...%2Bsin%28%CE%B1n-%CE%B10%29%5E2%29%2Fn%E4%B8%BA%E9%9B%86%E5%90%88%7B%CE%B11%2C%CE%B12%2C...%2C%CE%B1n%7D%E7%9B%B8%E5%AF%B9%CE%B10%E7%9A%84%E6%AD%A3%E5%BC%A6%E6%96%B9%E5%B7%AE%3B%E6%B1%82%E8%AF%81%3A%E9%9B%86%E5%90%88%7B%CF%80%2F2%2C5%CF%80%2F6%2C7%CF%80%2F6%7D%E7%9B%B8%E5%AF%B9%E4%BB%BB%E4%BD%95%E5%B8%B8%E6%95%B0%CE%B10%E6%AD%A3%E5%BC%A6%E6%96%B9%E5%B7%AE%E6%98%AF%E4%B8%80%E4%B8%AA%E4%B8%8E%CE%B10%E6%97%A0%E5%85%B3%E7%9A%84%E5%AE%9A%E5%80%BC.)
对于集合{α1,α2,...,αn}和常数α0定义:μ=(sin(α1-α0)^2+sin(α2-α0)^2+...+sin(αn-α0)^2)/n为集合{α1,α2,...,αn}相对α0的正弦方差;求证:集合{π/2,5π/6,7π/6}相对任何常数α0正弦方差是一个与α0无关的定值.
对于集合{α1,α2,...,αn}和常数α0定义:
μ=(sin(α1-α0)^2+sin(α2-α0)^2+...+sin(αn-α0)^2)/n为集合{α1,α2,...,αn}相对α0的正弦方差;
求证:集合{π/2,5π/6,7π/6}相对任何常数α0正弦方差是一个与α0无关的定值.
对于集合{α1,α2,...,αn}和常数α0定义:μ=(sin(α1-α0)^2+sin(α2-α0)^2+...+sin(αn-α0)^2)/n为集合{α1,α2,...,αn}相对α0的正弦方差;求证:集合{π/2,5π/6,7π/6}相对任何常数α0正弦方差是一个与α0无关的定值.
正弦方差=[sin²(π/2-α0)+sin²(5π/6-α0)+sin(7π/6-α0)]/3
只要证明分子是常数即可
sin²(π/2-α0)+sin²(5π/6-α0)+sin²(7π/6-α0)
=cos²α0+sin²[π-(5π/6-α0)]+sin²[π+(π/6-α0)]
=cos²α0+sin²(π/6+α0)+sin²(π/6-α0)
=cos²α0+(sinπ/6cosα0+cosπ/6sinα0)²+(sinπ/6cosα0-cosπ/6sinα0)²
=cos²α0+(1/2*cosα0+√3/2sinα0)²+(1/2*cosα0-√3/2sinα0)²
乘出来整理
=cos²α0+1/2*cos²α0+3/2*sin²α0
=3/2*cos²α0+3/2*sin²α0
=3/2*(cos²α0+sin²α0)
=3/2
所以这个正弦方差是一个与α0无关的定值.