How do you evaluate \[{{\sin }^{-1}}\left( \dfrac{\sqrt{3}}{2} \right)\] without a calculator.

Hint: In this problem, we have to solve and find the value of A. In this problem, we are going to use the inverse sine function also called the arcsine function. Since sine of an angle is equal to the opposite side by the hypotenuse, thus sine inverse of the same ration will give the measure of the angle. We can take an inverse function of sine and find its answer using a calculator. To solve these types of problems, we should know the trigonometric degree values for sine, cosine and tangent. We can recall them to evaluate this problem.

Complete step by step solution:
We know that the inverse function is used to determine the angle measure.
We know that sine inverse is denoted by \[{{\sin }^{-1}}\] or \[\arcsin \].
We are given that, \[{{\sin }^{-1}}\left( \dfrac{\sqrt{3}}{2} \right)\].
We have to solve and find the value of \[\theta \] using the inverse sine function.
We can write the given function \[\sin \theta =\dfrac{\sqrt{3}}{2}\], in terms of sine inverse function, we get
\[\Rightarrow {{\sin }^{-1}}\left( \dfrac{\sqrt{3}}{2} \right)=\theta \]
We know that the value of \[\sin {{60}^{\circ }}=\dfrac{\sqrt{3}}{2}\] from the trigonometric degree values table.
\[\Rightarrow \theta ={{60}^{\circ }}\]
Therefore, the value of the angle \[\theta \] is \[{{60}^{\circ }}\]

Note: We should always remember that, if we are given the value for a sine function, we have to find the angle for which the value is equal to the given value. We should also remember that the inverse sine function also called as arcsine function and since sine of an angle is equal to the opposite side by the hypotenuse, thus sine inverse of the same ratio will give the measure of the angle.