やっと、昔の記事を改修できます。
簡略化のため、1の3乗根を示すωを、
ω1 =
-1+√-3
2
2
ω2 =
-1-√-3
2
2
ω3 = 1
のようにします。
| cos(0˚)= sin(90˚)= | +1 |
| cos(5˚)= cos(355˚)= sin(95˚)= sin(85˚)= | + ω3・∛√6+√2+2√-2+√3+ω3・∛√6+√2-2√-2+√3 ∛32 |
| cos(10˚)= cos(350˚)= sin(100˚)= sin(80˚)= | + ω3・∛√3+√-1+ω3・∛√3-√-1 ∛16 |
| cos(15˚)= cos(345˚)= sin(105˚)= sin(75˚)= | + √6+√2 4 |
| cos(20˚)= cos(340˚)= sin(110˚)= sin(70˚)= | + ω3・∛1+√-3+ω3・∛1-√-3 ∛16 |
| cos(25˚)= cos(335˚)= sin(115˚)= sin(65˚)= | + ω3・∛√6-√2+2√-2-√3+ω3・∛√6-√2-2√-2-√3 ∛32 |
| cos(30˚)= cos(330˚)= sin(120˚)= sin(60˚)= | + √3 2 |
| cos(35˚)= cos(325˚)= sin(125˚)= sin(55˚)= | + ω3・∛-√6+√2+2√-2-√3+ω3・∛-√6+√2-2√-2-√3 ∛32 |
| cos(40˚)= cos(320˚)= sin(130˚)= sin(50˚)= | + ω3・∛-1+√-3+ω3・∛-1-√-3 ∛16 |
| cos(45˚)= cos(315˚)= sin(135˚)= sin(45˚)= | + √2 2 |
| cos(50˚)= cos(310˚)= sin(140˚)= sin(40˚)= | + ω3・∛-√3+√-1+ω3・∛-√3-√-1 ∛16 |
| cos(55˚)= cos(305˚)= sin(145˚)= sin(35˚)= | + ω3・∛-√6-√2+2+√-2+√3+ω3・∛-√6-√2-2+√-2+√3 ∛32 |
| cos(60˚)= cos(300˚)= sin(150˚)= sin(30˚)= | + 1 2 |
| cos(65˚)= cos(295˚)= sin(155˚)= sin(25˚)= | + ω2・∛-√6-√2+2√-2+√3+ω1・∛-√6-√2-2√-2+√3 ∛32 |
| cos(70˚)= cos(290˚)= sin(160˚)= sin(20˚)= | + ω2・∛-√3+√-1+ω1・∛-√3-√-1 ∛16 |
| cos(75˚)= cos(285˚)= sin(165˚)= sin(15˚)= | + √6-√2 4 |
| cos(80˚)= cos(280˚)= sin(170˚)= sin(10˚)= | + ω2・∛-1+√-3+ω1・∛-1-√-3 ∛16 |
| cos(85˚)= cos(275˚)= sin(175˚)= sin(5˚)= | + ω2・∛-√6+√2+2√-2-√3+ω1・∛-√6+√2-2√-2-√3 ∛32 |
| cos(90˚)= sin(0˚)= | ±0 |
| cos(95˚)= cos(265˚)= sin(185˚)= sin(355˚)= | - ω2・∛-√6+√2+2√-2-√3+ω1・∛-√6+√2-2√-2-√3 ∛32 |
| cos(100˚)= cos(260˚)= sin(190˚)= sin(350˚)= | - ω2・∛-1+√-3+ω1・∛-1-√-3 ∛16 |
| cos(105˚)= cos(255˚)= sin(195˚)= sin(345˚)= | - √6-√2 4 |
| cos(110˚)= cos(250˚)= sin(200˚)= sin(340˚)= | - ω2・∛-√3+√-1+ω1・∛-√3-√-1 ∛16 |
| cos(115˚)= cos(245˚)= sin(205˚)= sin(335˚)= | - ω2・∛-√6-√2+2√-2+√3+ω1・∛-√6-√2-2√-2+√3 ∛32 |
| cos(120˚)= cos(240˚)= sin(210˚)= sin(330˚)= | - 1 2 |
| cos(125˚)= cos(235˚)= sin(215˚)= sin(325˚)= | - ω3・∛-√6-√2+2+√-2+√3+ω3・∛-√6-√2-2+√-2+√3 ∛32 |
| cos(130˚)= cos(230˚)= sin(220˚)= sin(320˚)= | - ω3・∛-√3+√-1+ω3・∛-√3-√-1 ∛16 |
| cos(135˚)= cos(225˚)= sin(225˚)= sin(315˚)= | - √2 2 |
| cos(140˚)= cos(220˚)= sin(230˚)= sin(310˚)= | - ω3・∛-1+√-3+ω3・∛-1-√-3 ∛16 |
| cos(145˚)= cos(215˚)= sin(235˚)= sin(305˚)= | - ω3・∛-√6+√2+2√-2-√3+ω3・∛-√6+√2-2√-2-√3 ∛32 |
| cos(150˚)= cos(210˚)= sin(240˚)= sin(300˚)= | - √3 2 |
| cos(155˚)= cos(205˚)= sin(245˚)= sin(295˚)= | - ω3・∛√6-√2+2√-2-√3+ω3・∛√6-√2-2√-2-√3 ∛32 |
| cos(160˚)= cos(200˚)= sin(250˚)= sin(290˚)= | - ω3・∛1+√-3+ω3・∛1-√-3 ∛16 |
| cos(165˚)= cos(195˚)= sin(255˚)= sin(285˚)= | - √6+√2 4 |
| cos(170˚)= cos(190˚)= sin(260˚)= sin(280˚)= | - ω3・∛√3+√-1+ω3・∛√3-√-1 ∛16 |
| cos(175˚)= cos(185˚)= sin(265˚)= sin(275˚)= | - ω3・∛√6+√2+2√-2+√3+ω3・∛√6+√2-2√-2+√3 ∛32 |
| cos(180˚)= sin(270˚)= | -1 |
この表に至るまでの簡単な経緯
3倍角の公式
sin(3θ) = 3sin(θ) - 4sin3(θ)
cos(3θ) = 4cos3(θ) - 3cos(θ)
チェビシェフの多項式
T3(x) = 4x3 - 3x
どちらでも良いのですが、
例えば、
cos(3θ) = cos(60˚) = 1/2
cos(θ) = cos(20˚) = x
1/2 = 4x3 - 3x
4x3 - 3x - 1/2 = 0
8x3 - 6x - 1 = 0
計算するといっても、3次方程式なので、カルダノの解法を使う。
これによって、x = cos(20˚) が求まる。
同様の方法で、
cos(3θ) = cos(15˚) ==> cos(5˚)
cos(3θ) = cos(30˚) ==> cos(10˚)
cos(3θ) = cos(75˚) ==> cos(25˚)
…
と進めれば、5度系がすべて埋まる。
ではでは