Question

If we have a trigonometric equation as \[2\sin \theta + \tan \theta = 0\], then the general values of \[\theta \] are
1. \[2n\pi \pm \dfrac{\pi }{3}\]
2. \[n\pi ,2n\pi \pm \dfrac{{2\pi }}{3}\]
3. \[n\pi ,2n\pi \pm \dfrac{\pi }{3}\]
4. \[n\pi ,n\pi + \dfrac{{2\pi }}{3}\]

Answer 1

Hint: The given question deals with finding the general values of \[\theta \] from the given expression. We will convert all the terms into their respective sine and cosine expressions so as to ease the calculations. We will Simplify the given equation as much as possible by using various arithmetic operators and trigonometric functions. Take into consideration all the possible values of \[\theta \] and find the respective general values.

Complete step-by-step solution:
Sine function of an angle is the ratio between the opposite side length to that of the hypotenuse.
Cosine function of an angle is the ratio of the length of the adjacent side to the length of the hypotenuse.
The tangent function is the ratio of the length of the opposite side to that of the adjacent side. It should be noted that the tan can also be represented in terms of sine and cos as their ratio.
We are given \[2\sin \theta + \tan \theta = 0\]
This equation can be rewritten as \[2\sin \theta + \dfrac{{\sin \theta }}{{\cos \theta }} = 0\]
Which simplifies to
\[2\sin \theta \cos \theta + \sin \theta = 0\]
On further simplification of the expression we get ,
\[ \sin \theta \left( {2\cos \theta + 1} \right) = 0 \]
Therefore we have the following
\[\sin \theta = 0\] or \[\left( {2\cos \theta + 1} \right) = 0\]
\[\sin \theta = 0\] or \[\cos \theta = - \dfrac{1}{2}\]
Therefore we get ,
\[\sin \theta = 0\] gives us \[\theta = n\pi \]
And \[\cos \theta = \cos \left( {\dfrac{{2\pi }}{3}} \right)\]
Hence we get the general values as \[\theta = n\pi ,2n\pi \pm \dfrac{{2\pi }}{3}\]
Therefore option (2) is the correct answer.

Note: We must have a strong grip over the concepts of trigonometry , its related formulas and rules in order to solve such types of questions. In addition to this we must know the basic trigonometric identities to simplify the expressions at each step. Do the calculations very carefully as they may alter the solution.