Question

\[\sin 6\theta + \sin 4\theta + \sin 2\theta = \],then \[\theta = \]
 A. \[\dfrac{{n\pi }}{4}or\left| {n\pi \pm \dfrac{\pi }{3}} \right|\]
 B. \[\dfrac{{n\pi }}{4}or\left| {n\pi \pm \dfrac{\pi }{6}} \right|\]
 C. \[\dfrac{{n\pi }}{4}or\left| {2n\pi \pm \dfrac{\pi }{6}} \right|\]
 D. none of these

Answer 1

Hint: To solve this question, A trigonometric equation is one that has one or more ratios of unknown trigonometric angles. The ratios of sine, cosine, tangent, cotangent, secant, and cosecant angles are used to express it.

Formula Used: The trigonometric formulas are used
\[\sin 6\theta + \sin 4\theta + \sin 2\theta = 0\]

Complete step by step solution: we are given the trigonometric equations
\[\sin 6\theta + \sin 4\theta + \sin 2\theta = 0\]
The values of above equations to be changed as
\[ \Rightarrow 2\sin 4\theta \cos 2\theta + \sin 4\theta = 0\]
\[ \Rightarrow \sin 4\theta (2\cos 2\theta + 1 = 0\]
The given equations to be changed as
\[ \Rightarrow \left| {2\cos 2\theta = - 1} \right|\]
\[ \Rightarrow \left| {\cos 2\theta = - \dfrac{1}{2}} \right|\]
\[ \Rightarrow \left| {2\theta = 2n\pi \pm \dfrac{{2\pi }}{3} \Rightarrow \theta = n\pi \pm \dfrac{{2\pi }}{3} \Rightarrow \theta = n\pi \pm \dfrac{\pi }{3}} \right|\]
In geometric figures, unknown angles and distances are derived from known or measured angles using trigonometric functions. The need to calculate angles and distances in areas like astronomy, mapmaking, surveying, and artillery range finding led to the development of trigonometry.

\[\sin 4\theta = 0 \Rightarrow 4\theta = n\pi \]
\[ \Rightarrow \theta = \dfrac{{n\pi }}{4}\left| {\theta = \dfrac{{n\pi }}{4}} \right|\]or
\[n\pi \pm \dfrac{\pi }{3}\]
The equations of \[\sin 6\theta + \sin 4\theta + \sin 2\theta = 0\]is that\[ \Rightarrow \theta = \dfrac{{n\pi }}{4}\left| {\theta = \dfrac{{n\pi }}{4}} \right|\]or
\[n\pi \pm \dfrac{\pi }{3}\]
Convert everything to sine and cosine terms. When possible, utilise the identities. When you run into trouble, start by simplifying the equation's left side and go on to its right side. The identity is valid as long as both sides have the same ending phrase.

Option ‘A’ is correct

Note: The three main trigonometry functions are sine, cosine, and tangent, while the other three are cotangent, secant, and cosecant. Strong trigonometry abilities allow for speedy calculations of challenging angles and dimensions. One of the most useful subfields of mathematics is trigonometry, which is widely used in engineering, architecture, and many other areas.
Hence the trigonometric equation is \[n\pi \pm \dfrac{\pi }{3}\]Students with strong trigonometry skills can calculate complicated angles and dimensions quickly. One of the most useful subfields of mathematics is trigonometry, which is widely used in engineering, architecture, and many other areas.