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TRIGONOMETRIC FUNCTIONS — MATHS STD 11 Science — Question

Gujarat BoardEnglish MediumSTD 11 ScienceMATHSTRIGONOMETRIC FUNCTIONS1 Mark
Question
Find the general solution for the equation: sin 2x + cos x = 0
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Answer

sin 2x + cos x = 0
$ \Rightarrow $ 2 sin x cos x + cos x = 0
$ \Rightarrow $ cos x (2 sin x + 1) = 0
$ \Rightarrow $ Either cos x = 0 or 2 sin x + 1 = 0
$ \Rightarrow x = (2n + 1)\frac{\pi }{2}$ or $\sin x = - \frac{1}{2} = - \sin \frac{\pi }{6} = \sin \left( { - \frac{\pi }{6}} \right),n \in Z$
$ \Rightarrow x = (2n + 1)\frac{\pi }{2}$ or $x = n\pi {( - 1)^n}\left( { - \frac{\pi }{6}} \right)$
$x = (2n + 1)\frac{\pi }{2}$ or $x = n\;\pi + {( - 1)^{n + 1}}\left( {\frac{\pi }{6}} \right)$
or $x = n\;\pi {( - 1)^n}\frac{{7\pi }}{6}$$\left[ {\because \sin \left( {\pi + \frac{\pi }{6}} \right) = - \sin \frac{\pi }{6}} \right]$
$ \Rightarrow x = (2n + 1)\frac{\pi }{2}$ or $x = n\;\pi + {( - 1)^n}\frac{{7\pi }}{6},n \in Z$

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