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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: sinx + sin 3x + sin 5x = 0
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Answer

sin x + sin3x + sin 5x = 0
$ \Rightarrow $ (sin 5x + sin x) + sin 3x = 0
$ \Rightarrow 2\sin \left( {\frac{{5x + x}}{2}} \right)\cos \left( {\frac{{5x - x}}{2}} \right) + \sin 3x$ = 0
$ \Rightarrow $ 2 sin 3x cos 2x + sin 3x = 0
$ \Rightarrow $ sin 3x (2 cos 2x + 1) = 0
$ \Rightarrow $ either sin 3x = 0 or 2 cos 2x + 1= 0
$ \Rightarrow 3x = n\pi $ or $\cos 2x = - \frac{1}{2} = \cos \frac{{2\pi }}{3},n \in Z$
$ \Rightarrow x = \frac{{n\pi }}{3}$ or $2x = 2n\;\pi \pm \frac{{2\pi }}{3},n \in Z$
$ \Rightarrow x = \frac{{n\pi }}{3}$ or $x = n\pi \pm n \in Z$

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