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6. (5 分) 已知 $\sin 2 \alpha=\frac{2}{3}$, 则 $\cos ^{2}\left(\alpha+\frac{\pi}{4}\right)=(\quad)$

Gaokao · Math · previous-year question

  1. A.$\frac{1}{6}$correct
  2. B.$\frac{1}{3}$
  3. C.$\frac{1}{2}$
  4. D.$\frac{2}{3}$

Answer

A. $\frac{1}{6}$

Explanation

解: $\because \sin 2 \alpha=\frac{2}{3}$, $\therefore \cos ^{2}\left(\alpha+\frac{\pi}{4}\right)=\frac{1}{2}\left[1+\cos \left(2 \alpha+\frac{\pi}{2}\right)\right]=\frac{1}{2}(1-\sin 2 \alpha)=\frac{1}{2} \times\left(1-\frac{2}{3}\right)=\frac{1}{6}$. 故选: A.

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