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# Find the general solution of $sinx + sin3x + sin5x = 0$.  Hint – Use the formula $\sin a + \sin b = 2\sin \left( {\dfrac{{a + b}}{2}} \right)\cos \left( {\dfrac{{a - b}}{2}} \right)$.

We have ,
$sinx + sin3x + sin5x = 0 \\ (sinx + sin5x) + sin3x = 0 \\$
We know ,
$\sin a + \sin b = 2\sin \left( {\dfrac{{a + b}}{2}} \right)\cos \left( {\dfrac{{a - b}}{2}} \right)\,\,\,\,\,\,\,\,\,\,\,\,...\left( 1 \right)$
Therefore,
$sinx + sin5x = 2\sin \left( {\dfrac{{6x}}{2}} \right)\cos \left( {\dfrac{{4x}}{2}} \right) = 2\sin \left( {3x} \right)\cos \left( {2x} \right)\,\,\,\,\,\,\,\,\,\,\,\,\,\,...(2)\,\,\,\,$ [From (1)]
$2\sin \left( {3x} \right)\cos \left( {2x} \right) + \sin 3x = 0\,\,\,\,\,\,\,$ [From (2)]
$sin3x(2cos2x + 1) = 0$
Either ${\text{ }}sin3x = 0\;$ or $2cos2x + 1 = 0$
i.e. $sin3x = 0\;\,\,\,\,{\text{or }}\;cos2x = \dfrac{{ - 1}}{2}$
$3x = n\pi ,\,\,\,n \in Z\;\,\,\,\,\,{\text{or}}\,\,\,\,\;2x = 2m\pi \pm \dfrac{{2\pi }}{3}\,\,\,\,\;{\text{where}}\;m \in Z$
Hence, $x = \dfrac{{n\pi }}{3}\;\,\,{\text{or}}\,\,\,\,\;x = m\pi \pm \dfrac{\pi }{3},{\text{ where\; }}n,m \in Z$.

Note – In these types of questions of finding general solutions, always try to simplify with the help of trigonometric formulas such that all terms on both sides are single or multiplied with each other . Then equate and then use quadrant rule in trigonometry to get the general solutions.
View Notes
Integration of Trigonometric Functions  CBSE Class 11 Maths Chapter 3 - Trigonometric Functions Formulas  General Equation of a Plane  General Equation of a Line  Trigonometric Functions  Important Properties of Inverse Trigonometric Functions  Inverse Trigonometric Functions  Graphical Representation of Inverse Trigonometric Functions  Numbers in General Form  GK Questions for Class 11  