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Tue 29 Mar 2005
加法定理の証明
- 1.オイラーの公式による証明
pr.
オイラーの公式eix = cos(x) + i sin(x)…(1)
で、x = α+βを代入すると、
ei(α+β) = cos(α+β) + i sin(α+β) ⇔eiα・eiβ = cos(α+β) + i sin(α+β)…(2)
((2)の左辺) = { cos(α) + i sin(α) }{ cos(β) + i sin(β)} (∵(1)) = { cos(α) cos(β) - sin(α) sin(β) } + i { sin(α) cos(β) + cos(α) sin(β) } = cos(α+β) + i sin(α+β) (∵(2)の右辺)実部と虚部が等しいので、
sin(α+β) = sin(α) cos(β) + cos(α) sin(β) cos(α+β) = cos(α) cos(β) - sin(α) sin(β)
が得られる。 Q.E.D.
※sin(α-β)、cos(α-β)は、改めてβ = -βとすることで得られる。
ただしこの証明はオイラーの公式を使うため、sin(x)、cos(x)をテイラー展開(べき級数展開)するなどの作業が必要になる。
- 2.行列を用いた証明
-
座標平面上にP(cos(α),sin(α))をとり、原点を中心に角度βだけ回転させた点をQ(cos(α+β),sin(α+β))とする。
すると、回転行列を用いて(|〜|は絶対値ではなく行列)|cos(α+β)| ＿ |cos(β) -sin(β)||cos(α)| |sin(α+β)| ￣ |sin(β) cos(β)||sin(α)| ＿ |cos(α)cos(β)-sin(α)sin(β)| ￣ |sin(α)cos(β)+cos(α)sin(β)|
となり、加法定理が示される。
ただし、回転行列はもともと加法定理を元にして導かれるものなのでこれは厳密な証明には使えない。あしからず。
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