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 \(\sin\) 和 \(\cos\) 的展开参考 here):
\(\sin/\cos(\alpha+\beta)\) 的展开证明
\[
\begin{aligned}
\cos(α+β) &= OB \\
& = OD - BD \\
& = OD - EC \\
& = OC \cos \beta - AC \sin \beta \\
& = OA \cos \alpha \cos \beta - OA \sin \alpha \sin \beta \\
& = \cos \alpha \cos \beta - \sin \alpha \sin \beta
\end{aligned}
\]
\[
\begin{aligned}
\sin(α+β) &= AB \\
& = AE + BE \\
& = AE + CD \\
& = AC \cos \beta + OC \sin \beta \\
& = OA \sin \alpha \cos \beta + OA \cos \alpha \sin \beta \\
& = \sin \alpha \cos \beta + \cos \alpha \sin \beta \\
\end{aligned}
\]