圆上两点所对圆心角：已知弧长相等和角的度数，求圆心角

"如图，在$⊙O$中，$\overset{\huge{\frown}}{AB}=\overset{\huge{\frown}}{AC}$，且$\angle A=30^{\circ}$，则$\angle C=$ $^\circ $．"\n\n根据题意，$\overset{\huge{\frown}}{AB}=\overset{\huge{\frown}}{AC}$，所以弧$\overset{\huge{\frown}}{AB}$和弧$\overset{\huge{\frown}}{AC}$的弧长相等．\n\n由于$\angle A=30^\circ$，所以弧$\overset{\huge{\frown}}{AB}$的圆心角为$30^\circ$，那么弧$\overset{\huge{\frown}}{AB}$的弧长为$\dfrac{30^\circ}{360^\circ}\times2\pi\cdot 1= \dfrac{\pi}{6}$．\n\n因为弧$\overset{\huge{\frown}}{AB}$和弧$\overset{\huge{\frown}}{AC}$的弧长相等，所以弧$\overset{\huge{\frown}}{AC}$的弧长也是$\dfrac{\pi}{6}$．\n\n设弧$\overset{\huge{\frown}}{AC}$对应的圆心角为$\angle C$，那么根据圆心角和弧长的关系，有$\angle C=\dfrac{\dfrac{\pi}{6}}{2\pi}\times360^\circ=30^\circ$．\n\n所以$\angle C=30^\circ$．\n\n\begin{figure}[htbp]\n\centering\n\begin{tikzpicture}[scale=0.8]\n\coordinate[label=left:$O$] (O) at (0,0);\n\fill (O) circle[radius=1.5pt];\n\draw (O) circle(2);\n\coordinate[label=above left:$A$] (A) at ({2cos(60)},{2sin(60)});\n\coordinate[label=above right:$B$] (B) at ({2cos(60)},{-2sin(60)});\n\coordinate[label=below right:$C$] (C) at ({2cos(30)},{-2sin(30)});\n\draw (A) arc(60:-60:2);\n\draw (A)--(B)--(C)--cycle;\n\draw->,>=latex--(2.5,0)nodebelow {$x$};\n\draw->,>=latex--(0,2.5)noderight {$y$};\n\draw (O)--(A)--(O)--(C);\n\draw (A)--(C);\n\end{tikzpicture}\n\caption{题目：在$$⊙O$$中，$\overset{\huge{\frown}}{AB}=\overset{\huge{\frown}}{AC}$，且$\angle A=30^{\circ}$，求$\angle C$．}\label{fig:圆上两点所对圆心角}\n\end{figure}\n