Question

Which of the below options represents the value of ${{\cos }^{-1}}\left( \dfrac{\sqrt{3}}{2} \right)$
a)$\dfrac{\pi }{3}$
b) $\dfrac{\pi }{6}$
c) $\dfrac{\pi }{8}$
d) None of these

Answer 1

Hint: Let us start by assuming that the given function ${{\cos }^{-1}}\left( \dfrac{\sqrt{3}}{2} \right)=\theta $. Now, we will take the cosine of the whole equation, and use the identity $\cos \left( {{\cos }^{-1}}\theta \right)=\theta $ and then find the value of $\theta $ for which we will get $\cos \theta =\dfrac{\sqrt{3}}{2}$.

Complete step-by-step solution:
Which we will get $\cos \theta =\dfrac{\sqrt{3}}{2}$ .
We have been given an expression and let us name it as: ${{\cos }^{-1}}\left( \dfrac{\sqrt{3}}{2} \right)......(1)$
Now, as a first step, we will assume that: ${{\cos }^{-1}}\left( \dfrac{\sqrt{3}}{2} \right)=\theta ......(2)$
Now, by taking cosine on both sides of equation 2, we get:
$\cos \left\{ {{\cos }^{-1}}\left( \dfrac{\sqrt{3}}{2} \right) \right\}=\cos \theta ......(3)$
Since we know the identity that: $\cos \left( {{\cos }^{-1}}\theta \right)=\theta $, so we can write equation (3) as:
\[\begin{align}
  & \dfrac{\sqrt{3}}{2}=\cos \theta \\
 & \Rightarrow \cos \theta =\dfrac{\sqrt{3}}{2}......(4) \\
\end{align}\]
Now we can clearly see that the RHS contains a value of cosine function at an angle $\theta $ . When we consider the vale of $\cos \theta $ at standard angles, we will realsie that it is a standard angle.
We will understand that $\cos \dfrac{\pi }{6}=\dfrac{\sqrt{3}}{2}$.
Therefore, we can say that the value of $\theta $ is \[\theta =\dfrac{\pi }{6}\].
Hence, option (b) is the correct answer.

Note: Since in the options we have values of \[\left( 0,\pi \right)\] , therefore, we took \[\theta =\dfrac{\pi }{6}\]. If nothing was specified, we could have used the general form of solution which is $2n\pi \pm \theta $. So, we would have written the general solution as $2n\pi \pm \dfrac{\pi }{6}$. If the student is well versed with the trigonometric ratios at standard angles, then even without solving with steps, they can easily eliminate options and choose option (b) as the correct answer.