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4. (5 分) $\triangle A B C$ 的内角 $A, B, C$ 的对边分别为 $a, b, c$, 已知 $b=2, B=\frac{\pi}{6}, C=$ $\frac{\pi}{4}$, 则 $\triangle A B C$ 的面积为 $(\quad)$

Gaokao · Math · previous-year question

  1. A.$2 \sqrt{3}+2$
  2. B.$\sqrt{3}+1$correct
  3. C.$2 \sqrt{3}-2$
  4. D.$\sqrt{3}-1$

Answer

B. $\sqrt{3}+1$

Explanation

解： $\because b=2, B=\frac{\pi}{6}, C=\frac{\pi}{4}$, $\therefore$ 由正弦定理 $\frac{b}{\sin B}=\frac{c}{\sin C}$ 得: $c=\frac{b \sin C}{\sin B}=\frac{2 \times \frac{\sqrt{2}}{2}}{\frac{1}{2}}=2 \sqrt{2}, A=\frac{7 \pi}{12}$, $\therefore \sin A=\sin \left(\frac{\pi}{2}+\frac{\pi}{12}\right)=\cos \frac{\pi}{12}=\frac{\sqrt{2}+\sqrt{6}}{4}$, 则 $S_{\triangle A B C}=\frac{1}{2} b c \sin A=\frac{1}{2} \times 2 \times 2 \sqrt{2} \times \frac{\sqrt{2}+\sqrt{6}}{4}=\sqrt{3}+1$. 故选: B.

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