<單選題>△ABC 內接於圓心為 O 之單位圓，若 \( \overset{\rightharpoonup}{OA} + \overset{\rightharpoonup}{OB} + \sqrt{3} \overset{\rightharpoonup}{OC} = \overset{\rightharpoonup}{0} \)，則 ∠BAC 之度數為何？
(1) \( 30^\circ \) (2) \( 45^\circ \) (3) \( 60^\circ \) (4) \( 75^\circ \) (5) \( 90^\circ \)。
答案
將 \( \overset{\rightharpoonup}{OB} + \sqrt{3} \overset{\rightharpoonup}{OC} = -\overset{\rightharpoonup}{OA} \) 取長度平方得 \( 1 + 3 + 2\sqrt{3} \overset{\rightharpoonup}{OB} \cdot \overset{\rightharpoonup}{OC} = 1 \)，得 \( \overset{\rightharpoonup}{OB} \cdot \overset{\rightharpoonup}{OC} = -\frac{\sqrt{3}}{2} \)，故 \( \cos \angle BOC = -\frac{\sqrt{3}}{2} \Rightarrow \angle BOC = 150^\circ \)，圓周角 \( \angle BAC = \frac{1}{2} \angle BOC = 75^\circ \)。答案：(4) 報錯
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