Question

The value of \(\cos^{-1}\left(\cos\left(-\frac{\pi}{6}\right)\right)+\sin^{-1}\left(\sin\left(\frac{5\pi}{6}\right)\right)\) is:

• A. \(\frac{2\pi}{3}\)
• B. \(\frac{\pi}{3}\)
• C. \(\frac{5\pi}{3}\)
• D. Not Defined

Hint:

Always remember the principal value ranges: \[ \cos^{-1}x \in [0,\pi] \] and \[ \sin^{-1}x \in \left[-\frac{\pi}{2},\frac{\pi}{2}\right]. \] For inverse trigonometric expressions, first evaluate the trigonometric value and then choose the principal value.

Answer 1

The Correct Option is B

Concept: Inverse trigonometric functions return values only within their principal value ranges. For cosine inverse, \[ \cos^{-1}x \in [0,\pi] \] For sine inverse, \[ \sin^{-1}x \in \left[-\frac{\pi}{2},\frac{\pi}{2}\right] \] Therefore, while evaluating expressions involving inverse trigonometric functions, we must first simplify the trigonometric function and then select the corresponding principal value.

Step 1: Evaluate \(\cos^{-1}\left(\cos\left(-\frac{\pi}{6}\right)\right)\).
Since cosine is an even function, \[ \cos\left(-\frac{\pi}{6}\right) = \cos\left(\frac{\pi}{6}\right) = \frac{\sqrt{3}}{2} \] Therefore, \[ \cos^{-1}\left(\cos\left(-\frac{\pi}{6}\right)\right) = \cos^{-1}\left(\frac{\sqrt{3}}{2}\right) \] Since \[ \cos\left(\frac{\pi}{6}\right)=\frac{\sqrt{3}}{2} \] and \(\frac{\pi}{6}\) lies in the principal range \([0,\pi]\), \[ \cos^{-1}\left(\frac{\sqrt{3}}{2}\right) = \frac{\pi}{6}. \]

Step 2: Evaluate \(\sin^{-1}\left(\sin\left(\frac{5\pi}{6}\right)\right)\).
We know that \[ \sin\left(\frac{5\pi}{6}\right) = \frac{1}{2} \] Hence, \[ \sin^{-1}\left(\sin\left(\frac{5\pi}{6}\right)\right) = \sin^{-1}\left(\frac{1}{2}\right) \] Since \[ \sin\left(\frac{\pi}{6}\right)=\frac{1}{2} \] and \(\frac{\pi}{6}\) belongs to the principal range \[ \left[-\frac{\pi}{2},\frac{\pi}{2}\right], \] we obtain \[ \sin^{-1}\left(\frac{1}{2}\right) = \frac{\pi}{6}. \]

Step 3: Add the two principal values.
\[ \frac{\pi}{6} + \frac{\pi}{6} = \frac{2\pi}{6} = \frac{\pi}{3} \] Therefore, \[ \boxed{\cos^{-1}\left(\cos\left(-\frac{\pi}{6}\right)\right)+\sin^{-1}\left(\sin\left(\frac{5\pi}{6}\right)\right)=\frac{\pi}{3}} \]

Step 4: Select the correct option.
Thus the correct answer is \[ \boxed{\frac{\pi}{3}} \] which corresponds to option \((B)\).