Question

If ${{\sin }^{2}}\theta =\dfrac{1}{4}$ then the most general value of $\theta $ is
A. \[2n\pi \pm {{(-1)}^{n}}\dfrac{\pi }{6}\]
B. \[\dfrac{n\pi }{2}\pm {{(-1)}^{n}}\dfrac{\pi }{6}\]
C. \[n\pi \pm \dfrac{\pi }{6}\]
D. \[2n\pi \pm \dfrac{\pi }{6}\]

Answer 1

Hint: To derive the general value of $\theta $ we will take the given equation and simplify it and derive the value of $\sin \theta $. After simplifying we will find the general equation for $\theta $ by defining $n$ using the trigonometric table of values for the function sine.

Formula Used: To derive the general value of $\theta $ we will take the given equation and simplify it and derive the value of $\sin \theta $. After simplifying we will find the general equation for $\theta $ by defining $n$ using the trigonometric table of values for the function sine.

Complete step by step solution:We are given ${{\sin }^{2}}\theta =\dfrac{1}{4}$ and we have to determine the general value of $\theta $.
We will take ${{\sin }^{2}}\theta =\dfrac{1}{4}$and derive the value of $\sin \theta $ by simplifying.
$\begin{align}
  & {{\sin }^{2}}\theta =\dfrac{1}{4} \\
 & \sin \theta =\sqrt{\dfrac{1}{4}} \\
 & \sin \theta =\pm \dfrac{1}{2}
\end{align}$
Now we know that $\sin \dfrac{\pi }{6}=\dfrac{1}{2},\sin \dfrac{5\pi }{6}=\dfrac{1}{2}....$and so on. Hence we will take a variable for integer \[n\] such that \[n\in Z\].



Option ‘C’ is correct

Note: A trigonometric equation can be defined as an equation which has trigonometric functions present in them like sin, cos, tan, cot, sec, cosec. The values of these functions lies in some specific interval after that it starts repeating its value. The interval of the function sine is $0\le \theta \le 2\pi $.
If the value of the angle $\theta $ has a variable like $n\pi $in its solution, then that solution is called as general solution which gives the all the solution for that trigonometric function.
The general equation we derived for angle $\theta =n\pi \pm \dfrac{\pi }{6}$will give all the solutions for the trigonometric function sine by substituting the value of $n$.
The solution which will give the precise answer without any variable in it is called as principal solution.
We should remember that $\sin x=0$ means $x=n\pi $ where $n\in Z$.
Aside from the function sine, there are also the general equations for all the other trigonometric functions according to their period interval.