Question

Solve the equation $\sin \theta + sin3\theta + sin5\theta = 0$.

Answer 1

Hint : The trigonometry is the concept of mathematics which deals with the angles that are tilted with respect to the base. The terms used in the trigonometry are sin, cos, sec, tan, cot, and cosec and $\theta$ is used to denote the angle. These will help in solving the problems, so in this solution it is given the sin term to find the value of the angle $\theta$. As the terms all are sin, then try to combine them to convert into formula that was in the trigonometry concept.

Complete step-by-step answer :
The equation is given as $\sin \theta + sin3\theta + sin5\theta = 0$, then writing the equation as,
$\sin \left( {\theta + 5\theta } \right) + sin3\theta = 0$
We know that the formula for $\sin \left( {A + B} \right)$, which is given by,
$\sin \left( {A + B} \right) = 2\sin \dfrac{{A + B}}{2} \cdot \cos \dfrac{{A - B}}{2}$
Then, after applying the formula in the above relation, we get,

2\sin \left( {\dfrac{{\theta + 5\theta }}{2}} \right)\cos \left( {\dfrac{{\theta - 5\theta }}{2}} \right) + sin3\theta = 0\\\ 2\sin 3\theta \cos 2\theta + sin3\theta = 0\\\ sin3\theta (2\cos 2\theta + 1) = 0 \end{array}$$ Now, we will be separating the terms and equating with the zero, we obtain, $$\sin 3\theta = 0$$……(1) $$2\cos 2\theta + 1 = 0$$….(2) On taking equation 1, then we will get, $$\begin{array}{c} \sin 3\theta = 0\\\ 3\theta = {\sin ^{ - 1}}0\\\ 3\theta = \pi \\\ \theta = \dfrac{\pi }{3} \end{array}$$ On taking equation 2, then we will get, $$\begin{array}{c} 2\cos 2\theta + 1 = 0\\\ 2\cos 2\theta = - 1\\\ \cos 2\theta = \dfrac{{ - 1}}{2}\\\ 2\theta = {\cos ^{ - 1}}\dfrac{{ - 1}}{2} \end{array}$$ On further solving the equation, we obtain, $$\begin{array}{c} 2\theta = \dfrac{{2\pi }}{3} + 2\pi n\\\ \theta = \dfrac{\pi }{3} + \pi n \end{array}$$ Therefore, the value of $$\theta $$ is $$\dfrac{\pi }{3} + \pi n$$ and $$\dfrac{\pi }{3}$$. **Note** : In the solution, there are 3 terms of sin so it will be quite difficult to decide which terms should combine to take as common, so be sure about that and while we doing the problem it should be keep in mind that cos(-x) will be always taken as cos x, we did not care about the – symbol in the cos term but when it comes to the sin, it will be countable. After taking common terms we need to equate the terms with zero and to find the value of $$\theta $$so we need to be aware about the trigonometric table of angles for different values.