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10. (5 分) 已知锐角 $\triangle A B C$ 的内角 $A, B, C$ 的对边分别为 $a, b, c, 23 \cos ^{2} A+\cos 2 A=0$ , $a=7, c=6$, 则 $b=(\quad)$

Gaokao · Math · previous-year question

  1. A.10
  2. B.9
  3. C.8
  4. D.5correct

Answer

D. 5

Explanation

解: $\because 23 \cos ^{2} \mathrm{~A}+\cos 2 \mathrm{~A}=23 \cos ^{2} \mathrm{~A}+2 \cos ^{2} \mathrm{~A}-1=0$, 即 $\cos ^{2} \mathrm{~A}=\frac{1}{25}, \mathrm{~A}$ 为锐角, $\therefore \cos A=\frac{1}{5}$, 又 $a=7, c=6$, 根据余弦定理得: $a^{2}=b^{2}+c^{2}-2 b c \bullet \cos A$, 即 $49=b^{2}+36-\frac{12}{5} b$, 解得: $b=5$ 或 $b=-\frac{13}{5}$ (舍去), 则 $b=5$. 故选: D.

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