Question

Solve the following equation:

\[\sin x + \sin 5x = \sin 3x\]

Answer 1

\[\sin x + \sin5x = \sin3x\]
\[\Rightarrow 2 \sin\left( \frac{6x}{2} \right) \cos \left( \frac{4x}{2} \right) = \sin3x\]
\[ \Rightarrow 2 \sin3x \cos2x = \sin3x\]
\[ \Rightarrow 2 \sin3x \cos2x - \sin3x = 0\]
\[ \Rightarrow \sin3x (2 \cos2x - 1) = 0\]

\[\Rightarrow \sin3x = 0\] or

\[(2 \cos2x - 1) = 0\]

\[\Rightarrow \sin3x = \sin 0\] or

\[\cos2x = \frac{1}{2} = \cos \frac{\pi}{3}\]

\[\cos2x = \frac{1}{2} = \cos \frac{\pi}{3}\] or

\[2x = 2m\pi \pm \frac{\pi}{3}\]

⇒ \[x = \frac{n\pi}{3}, n \in Z\] or

\[x = m\pi \pm \frac{\pi}{6}, m \in Z\]