查看“︁尼文定理”︁的源代码

'''尼文定理'''（Niven's theorem）说的是，在 0~90° 范围内，如果[[正弦函数]] sin 的自变量和因变量都要求是[[有理数]]，那么答案只有<ref>{{cite journal|first=Norman|last=Schaumberger|title=A Classroom Theorem on Trigonometric Irrationalities|journal=Two-Year College Mathematics Journal''|volume=5|pages=73–76|year=1974|jstor=3026991}}</ref>：

:<math>\sin0^\circ=0</math> 。
:<math>\sin30^\circ=\frac12</math> 。
:<math>\sin90^\circ=1</math> 。

若用[[弧度]]表示，需在0&nbsp;≤&nbsp;''x''&nbsp;≤&nbsp;{{pi}}/2的範圍內，且要求''x''/{{pi}}及sin&nbsp;''x''都是有理數。其結果是sin&nbsp;0&nbsp;=&nbsp;0, sin&nbsp;{{pi}}/6&nbsp;=&nbsp;1/2 及 sin&nbsp;{{pi}}/2&nbsp;=&nbsp;1。

此定義出現在[[伊萬·尼雲]]有關[[無理數]]的書中<ref>{{cite book|last=Niven|first=I.|author-link=伊萬·尼雲|title=Irrational Numbers|url=https://archive.org/details/irrationalnumber00nive|publisher=Wiley|page=[https://archive.org/details/irrationalnumber00nive/page/41 41]|year=1956|mr=0080123}}</ref>。

==相關條目==
*[[勾股数]]：邊長為勾股数的直角三角形，角度的正弦為有理數
*[[三角函数]]
*[[三角函數數]]

==参考资料==
{{reflist}}
*Weisstein, Eric W. "Niven's Theorem." From ''MathWorld''--A Wolfram Web Resource. http://mathworld.wolfram.com/NivensTheorem.html {{Wayback|url=http://mathworld.wolfram.com/NivensTheorem.html |date=20150211153447 }}

==延伸閱讀==
* {{cite journal
|first1=J. M. H. | last1=Olmsted
|title=Rational values of trigonometric functions
|year=1945 | volume=52 | number=9 |jstor=2304540 | pages=507–508
|journal=Am. Math. Montly}}
* {{cite journal | first1=Derik H. | last1=Lehmer
|title=A note on trigonometric algebraic numbers | journal=Am. Math. Monthly
|year=1933 | volume = 40 |number=3 | pages=165–166 |jstor=2301023}}

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[[Category:有理數]]
[[Category:三角学]]
[[Category:几何定理]]
[[Category:代数定理]]