Question

The value of \[{\sin ^{ - 1}}0\] is equal to:
A. \[0\]
B. \[\dfrac{\pi }{6}\]
C. \[\dfrac{\pi }{2}\]
D. \[\dfrac{\pi }{3}\]

Answer 1

Hint: First of all, consider the required value as a variable and apply sine on both sides then we will have the general solution of the given problem. Then choose the suitable options from the general solution of the equation.

Complete step-by-step solution

Let the value of \[{\sin ^{ - 1}}0\] be \[x\].
\[ \Rightarrow {\sin ^{ - 1}}0 = x\]
Taking sine on both sides, we have
\[
   \Rightarrow \sin \left( {{{\sin }^{ - 1}}0} \right) = \sin x \\
   \Rightarrow 0 = \sin x \\
\]
We know that the general solution of \[\sin x = 0\] is \[x = n\pi \] where \[n\] is an integer.
Now, substituting the integer values, we get
\[x = \left\{ {0, \pm \pi , \pm 2\pi , \pm 3,.........................} \right\}\]
In the given options the suitable value of \[x\] is \[0\].
Thus, the correct option is A. \[0\]

Note: Here \[\sin x\] is known as a periodic function that oscillates at regular intervals. The value of the function \[\sin x\] becomes zero whenever it crosses the x-axis. The general solution of the function \[\sin x = 0\] is \[x = n\pi \] where \[n\] is an integer.