是否存在k是不為30的倍數的有理數,且sin(k度)也是有理數?
回答 (2)
2016-10-08 1:02 am
不存在。結論是當且僅當有理數 k = 60n° + 30° 或 180n° 時(其中n是整數), k 和 sink° 也是有理數。
請參閲特級教師甘志国《有理數角度的三角函數值何時是有理數》
請參閲特級教師甘志国《有理數角度的三角函數值何時是有理數》
2016-10-07 12:31 pm
All complex numbers x, one has:-
cos nx + i sin nx= n/k=0 (ik) ( n/k) cos^n-k x sin^k x,
By adding the real and imaginary part of the complex numbers to get rid of (k),when k=30=6x5;
we've n-k+k=n,
For n=2,3,4,when k is get rid of;
sin 2x=2 sinx cos x;
sin 3x= 3sinx-4sin^3x;
sin4x=8cos^3x sinx-4cosxsinx.
May be rational nos.
cos nx + i sin nx= n/k=0 (ik) ( n/k) cos^n-k x sin^k x,
By adding the real and imaginary part of the complex numbers to get rid of (k),when k=30=6x5;
we've n-k+k=n,
For n=2,3,4,when k is get rid of;
sin 2x=2 sinx cos x;
sin 3x= 3sinx-4sin^3x;
sin4x=8cos^3x sinx-4cosxsinx.
May be rational nos.
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