Question

Solve the following equation:

\[\sin x + \sin 2x + \sin 3 = 0\]

Answer 1

\[\sin x + \sin 2x + \sin 3 = 0\]
\[\Rightarrow \sin x + \sin3x + \sin2x = 0\]
\[ \Rightarrow 2 \sin \left( \frac{4x}{2} \right) \cos \left( \frac{2x}{2} \right) + \sin2x = 0\]
\[ \Rightarrow 2 \sin2x\cos x + \sin2x = 0\]
\[ \Rightarrow \sin2x (2 \cos x + 1) = 0\]

\[\Rightarrow \sin2x = 0\] or

\[2 \cos x + 1 = 0\]

\[\Rightarrow \sin2x = \sin 0\] or

\[\cos x = - \frac{1}{2} \Rightarrow \cos x = \cos \frac{2\pi}{3}\]

⇒ \[x = \frac{n\pi}{2}, n \in Z\] or

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

\[m \in Z\]