The principal value of `sin^(-1)((sqrt(3))/(2))+cos^(-1)(cos((pi)/(6)))`, is :

Options

A. `(pi)/(6)`
B. `(pi)/(2)`
C. `(3pi)/(2)`
D. None of these

Answer

To solve the problem, we need to find the principal value of the expression:

\[ \sin^{-1}\left(\frac{\sqrt{3}}{2}\right) + \cos^{-1}\left(\cos\left(\frac{\pi}{6}\right)\right) \]

### Step 1: Evaluate \(\sin^{-1}\left(\frac{\sqrt{3}}{2}\right)\)

The value of \(\sin^{-1}(x)\) gives us the angle whose sine is \(x\). We know that:

\[ \sin\left(\frac{\pi}{3}\right) = \frac{\sqrt{3}}{2} \]

Thus,

\[ \sin^{-1}\left(\frac{\sqrt{3}}{2}\right) = \frac{\pi}{3} \]

### Step 2: Evaluate \(\cos^{-1}\left(\cos\left(\frac{\pi}{6}\right)\right)\)

The value of \(\cos^{-1}(x)\) gives us the angle whose cosine is \(x\). Since \(\cos\left(\frac{\pi}{6}\right) = \frac{\sqrt{3}}{2}\), we have:

\[ \cos^{-1}\left(\cos\left(\frac{\pi}{6}\right)\right) = \frac{\pi}{6} \]

### Step 3: Combine the results

Now we combine the results from Step 1 and Step 2:

\[ \sin^{-1}\left(\frac{\sqrt{3}}{2}\right) + \cos^{-1}\left(\cos\left(\frac{\pi}{6}\right)\right) = \frac{\pi}{3} + \frac{\pi}{6} \]

### Step 4: Find a common denominator and add

To add \(\frac{\pi}{3}\) and \(\frac{\pi}{6}\), we need a common denominator. The least common multiple of 3 and 6 is 6.

\[ \frac{\pi}{3} = \frac{2\pi}{6} \]

Now we can add:

\[ \frac{2\pi}{6} + \frac{\pi}{6} = \frac{3\pi}{6} = \frac{\pi}{2} \]

### Final Answer

Thus, the principal value of the expression is:

\[ \frac{\pi}{2} \]