Toán Lớp 11: Giúp mình với ạ
Đề : Giải các phương trình sao
a) $sin^{2}$(45$^{0}$- x) = $\frac{1}{2}$
b) $\left[\begin{array}{ccc}cos(\frac{x}{2}+\frac{\pi}{2})\end{array}\right]$ =$\frac{1} {\sqrt{2}}$
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Toán Lớp 11: Giúp mình với ạ
Đề : Giải các phương trình sao
a) $sin^{2}$(45$^{0}$- x) = $\frac{1}{2}$
b) $\left[\begin{array}{ccc}cos(\frac{x}{2}+\frac{\pi}{2})\end{array}\right]$ =$\frac{1} {\sqrt{2}}$
Comments ( 2 )
a)\quad S = \left\{\dfrac{k\pi}{2}\ \Bigg|\ k\in\Bbb Z\right\}\\
b)\quad S = \left\{- \dfrac{\pi}{4} + k2\pi;\ \dfrac{5\pi}{4} +k2\pi\ \Bigg|\ k\in\Bbb Z\right\}
\end{array}\)
a)\quad \sin^2(45^\circ – x) = \dfrac12\\
\Leftrightarrow \dfrac{1 – \cos(90^\circ – 2x)}{2} = \dfrac12\\
\Leftrightarrow \cos(90^\circ – 2x) = 0\\
\Leftrightarrow \sin2x = 0\\
\Leftrightarrow 2x = k\pi\\
\Leftrightarrow x =\dfrac{k\pi}{2}\quad (k\in\Bbb Z)\\
\text{Vậy}\ S = \left\{\dfrac{k\pi}{2}\ \Bigg|\ k\in\Bbb Z\right\}\\
b)\quad \cos\left(\dfrac{x}{2} + \dfrac{\pi}{2}\right) = \dfrac{1}{\sqrt2}\\
\Leftrightarrow -\sin x = \dfrac{1}{\sqrt2}\\
\Leftrightarrow \sin x = \sin\left(-\dfrac{\pi}{4}\right)\\
\Leftrightarrow \left[\begin{array}{l}x = – \dfrac{\pi}{4} + k2\pi\\x = \dfrac{5\pi}{4} +k2\pi\end{array}\right.\quad (k\in\Bbb Z)\\
\text{Vậy}\ S = \left\{- \dfrac{\pi}{4} + k2\pi;\ \dfrac{5\pi}{4} +k2\pi\ \Bigg|\ k\in\Bbb Z\right\}
\end{array}\)