tan89.999=57.29577951*1000tan89.9999=57.29577951*10000tan89.99999=57.29577951*100000tan89.999999=57.29577951*1000000tan89.9999999=57.29577951*10000000tan89.99999999=57.29577951*100000000以上为十位计算器计算结果,即有近似,但结

Question

tan89.999 =57.29577951*1000
tan89.9999 =57.29577951*10000
tan89.99999 =57.29577951*100000
tan89.999999 =57.29577951*1000000
tan89.9999999 =57.29577951*10000000
tan89.99999999 =57.29577951*100000000
以上为十位计算器计算结果,即有近似,但结果我却不能明白,请注意：57.29577951这个数是一个弧度的量值!即57.29577951=180/π
请告诉我这是巧合呢,还是有什么内在联系.

Answer 1

注意：
89.999 = 90 - 0.001 = π/2 - (π/180)/1000
89.9999 = 90 - 0.0001 = π/2 - (π/180)/10000
tan(90 - x) = 1/tanx = cosx/sinx
展开成级数表示则为：
cosx = 1 - x^2/2!+ x^3/4!- x^6/6!+ .
sinx = x - x^3/3!+ x^5/5!- x^7/7!+ .
当 x -> 0 时,可以去近似
cosx = 1
sinx = x
所以原式变为：1/x
x = (π/180)/1000 ,10000,100000.
1/x = (180/π)*1000,10000,100000.
= 57.29577951*1000,10000,100000.
结论：这不是巧合,是必然!