find the principle value of `cos^(-1)((sqrt3)/2)`

Explanation

To find the principal value of \( \cos^{-1} \left( \frac{\sqrt{3}}{2} \right) \), we can follow these steps:

### Step 1: Understand the function
The function \( \cos^{-1}(x) \) gives us the angle whose cosine is \( x \). We need to find the angle whose cosine is \( \frac{\sqrt{3}}{2} \).

### Step 2: Identify the angle
We know from trigonometric values that:
\[ \cos \left( \frac{\pi}{6} \right) = \frac{\sqrt{3}}{2} \]
This means that \( \frac{\pi}{6} \) (or 30 degrees) is one angle whose cosine is \( \frac{\sqrt{3}}{2} \).

### Step 3: Check the principal range
The principal value of \( \cos^{-1}(x) \) is defined in the range \( [0, \pi] \). Since \( \frac{\pi}{6} \) lies within this range, it is a valid solution.

### Step 4: Write the final answer
Thus, the principal value of \( \cos^{-1} \left( \frac{\sqrt{3}}{2} \right) \) is:
\[ \cos^{-1} \left( \frac{\sqrt{3}}{2} \right) = \frac{\pi}{6} \]

### Summary
The principal value of \( \cos^{-1} \left( \frac{\sqrt{3}}{2} \right) \) is \( \frac{\pi}{6} \).

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