Question

Solve the following equation \[\sin 2\theta -\sin 4\theta +\sin 6\theta =0\].

Answer 1

Hint: In the given question, we will use the trigonometry identity as follows:
\[\sin C+\sin D=2\sin \dfrac{C+D}{2}\cos \dfrac{C-D}{2}\]
After using the identity, we will get some terms in common in the equation and by equating it to zero we will get the solution.

Complete step-by-step answer:

We have been given the equation \[\sin 2\theta -\sin 4\theta +\sin 6\theta =0\].

After rearranging the terms of the equation, we will get,

\[\sin 2\theta +\sin 6\theta -\sin 4\theta =0\]

As we know that \[\sin C+\sin D=2\sin \dfrac{C+D}{2}\cos \dfrac{C-D}{2}\].

So, by using this trigonometric identity to the terms \[\left( \sin 2\theta +\sin 6\theta \right)\] in the above equation, we will get as follows:

\[\begin{align}

  & 2\sin \dfrac{2\theta +6\theta }{2}\cos \dfrac{2\theta -6\theta }{2}-\sin 4\theta =0 \\

 & 2\sin 4\theta \cos 2\theta -\sin 4\theta =0 \\

\end{align}\]

Taking \[\sin 4\theta \] as common in the above equation, we will get as follows:

\[\sin 4\theta \left( 2\cos 2\theta -1 \right)=0\]

Hence, \[\Rightarrow \sin 4\theta =0\] and \[2\cos 2\theta -1=0\].

As we know that the general solution for \[\sin \theta =\sin \alpha \] is given by \[\theta =n\pi +{{\left( -1 \right)}^{n}}\alpha \] and for \[\alpha =0\], \[\theta =n\pi \] where ‘n’ is any integer.

\[\begin{align}

  & \sin 4\theta =\sin {{0}^{\circ }} \\

 & 4\theta =n\pi \\

 & \theta =\dfrac{n\pi }{4} \\

\end{align}\]

Also, we know that the general solution for \[\cos \theta =\cos \alpha \] is given by \[\theta =2n\pi \pm \alpha \], where ‘n’ is any integer.

\[\begin{align}

  & 2\cos 2\theta -1=0 \\

 & \cos 2\theta =\dfrac{1}{2} \\

 & \cos 2\theta =\cos \dfrac{\pi }{3} \\

 & 2\theta =2n\pi \pm \dfrac{\pi }{3} \\

 & \theta =n\pi \pm \dfrac{\pi }{6} \\

\end{align}\]

On substituting the obtained value of ‘\[\theta \]’ in the given equation, we observed that it satisfies all the values of ‘n’.

Therefore, the solutions of the given equation are \[\theta =\dfrac{n\pi }{4}\] and \[\theta =n\pi \pm \dfrac{\pi }{6}\].


Note: Be careful while using the trigonometric identity in the equation as there is a chance of sign mistake.

Also, check the solution once by substituting the value in the equation as sometimes we get an extra solution that doesn’t exist in the domain of the given function.