Toán Lớp 11: sinx+ √3 cosx= $\frac{1}{sinx}$
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Toán Lớp 11: sinx+ √3 cosx= $\frac{1}{sinx}$
Comments ( 2 )
\sin x + \sqrt 3 \cos x = \dfrac{1}{{\sin x}}\\
\Leftrightarrow {\sin ^2}x + \sqrt 3 \sin x\cos x = 1\\
\Leftrightarrow {\sin ^2}x + \sqrt 3 \sin x\cos x – 1 = 0\\
\Leftrightarrow \left( {{{\sin }^2}x – 1} \right) + \sqrt 3 \sin x\cos x = 0\\
\Leftrightarrow \sqrt 3 \sin x\cos x – \cos x = 0\\
\Leftrightarrow \cos x\left( {\sqrt 3 \sin x – 1} \right) = 0\\
\Leftrightarrow \left[ \begin{array}{l}
\cos x = 0\\
\sin x = \dfrac{{\sqrt 3 }}{3}
\end{array} \right.\\
\Leftrightarrow \left[ \begin{array}{l}
x = \dfrac{\pi }{2} + k\pi \\
x = \arcsin \dfrac{{\sqrt 3 }}{3} + k2\pi \\
x = \pi – \arcsin \dfrac{{\sqrt 3 }}{3} + k2\pi
\end{array} \right.\left( {k \in \mathbb Z} \right)
\end{array}$