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Euphemia
Euphemia

Euphemia

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Euphemia
Asked: Tháng Mười 27, 20202020-10-27T00:02:35+00:00 2020-10-27T00:02:35+00:00In: Môn Toán

Phương trình 2sin(2x- pi/4)=căn 2 có bao nghiệm thuộc đoạn -pi bé hơn hoặc bằng x bé hơn hoặc bằng pi/2

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Phương trình 2sin(2x- pi/4)=căn 2 có bao nghiệm thuộc đoạn -pi bé hơn hoặc bằng x bé hơn hoặc bằng pi/2

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    1. Sapo

      Sapo

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      Sapo1
      2020-10-27T00:04:04+00:00Added an answer on Tháng Mười 27, 2020 at 12:04 sáng

      Đáp án: 3

      phuong-trinh-2sin-2-pi-4-can-2-co-bao-nghiem-thuoc-doan-pi-be-hon-hoac-bang-be-hon-hoac-bang-pi

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    2. Khoii Minh

      Khoii Minh

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      minhkhoi
      2020-10-27T00:04:21+00:00Added an answer on Tháng Mười 27, 2020 at 12:04 sáng

      Đáp án:

      $4$ nghiệm

      Giải thích các bước giải:

      $2\sin\left(2x -\dfrac{\pi}{4}\right) = \sqrt2$

      $\Leftrightarrow \sin\left(2x -\dfrac{\pi}{4}\right) = \dfrac{\sqrt2}{2}$

      $\Leftrightarrow \left[\begin{array}{l}2x – \dfrac{\pi}{4} = \dfrac{\pi}{4} + k2\pi\\2x – \dfrac{\pi}{4} = \dfrac{3\pi}{4} + k2\pi\end{array}\right.$

      $\Leftrightarrow \left[\begin{array}{l}x= \dfrac{\pi}{4} + k\pi\\x = \dfrac{\pi}{2} + k\pi\end{array}\right. \quad (k\in\Bbb Z)$

      Ta có:

      $-\pi \leq x \leq \dfrac{\pi}{2}$

      $\Leftrightarrow \left[\begin{array}{l}-\pi  \leq \dfrac{\pi}{4} + k\pi \leq \dfrac{\pi}{2}\\-\pi \leq \dfrac{\pi}{2} + k\pi\leq \dfrac{\pi}{2}\end{array}\right.$

      $\Leftrightarrow \left[\begin{array}{l} -\dfrac{5}{4} \leq  k\leq \dfrac{1}{4}\\-\dfrac{3}{2} \leq k\leq 0\end{array}\right.$

      $\Leftrightarrow \left[\begin{array}{l}k = -1;0\\k = -1;0\end{array}\right. \quad (k\in\Bbb Z)$

      $\Rightarrow \left[\begin{array}{l}x = -\dfrac{3\pi}{4}\\x= \dfrac{\pi}{4}\\x = -\dfrac{\pi}{2}\\x = \dfrac{\pi}{2}\end{array}\right.$

      $\Rightarrow$ Có $4$ nghiệm thoả mãn đề bài

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