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Geometry

平面直角坐标系的点绕原点旋转公式及证明

设点 \(A(x,y)\) 绕原点 \(O(0,0)\) 逆时针旋转 \(\beta\),则设在极坐标系下 \(A\) 的坐标为 \((r,\alpha)\)

这意味着 \(x=r \cos \alpha, y=r \sin \alpha\)

目标点 \(A'(x',y')\) 的极坐标即为 \((r,\alpha + \beta)\)

展开(其中 \(\sin\)\(\cos\) 的展开参考 here):

\(\sin/\cos(\alpha+\beta)\) 的展开证明


\[ \begin{aligned} \cos(α+β) &= OB \\ & = OD - BD \\ & = OD - EC \\ & = OC \cos \beta - AC \sin \beta \\ & = OA \cos \alpha \cos \beta - OA \sin \alpha \sin \beta \\ & = \cos \alpha \cos \beta - \sin \alpha \sin \beta \end{aligned} \]

\[ \begin{aligned} \sin(α+β) &= AB \\ & = AE + BE \\ & = AE + CD \\ & = AC \cos \beta + OC \sin \beta \\ & = OA \sin \alpha \cos \beta + OA \cos \alpha \sin \beta \\ & = \sin \alpha \cos \beta + \cos \alpha \sin \beta \\ \end{aligned} \]