Find the principal value of `cot^(-1)(-sqrt3)`

Options

A. `-pi/3`
B. `-pi/6`
C. `(2pi)/3`
D. `(5pi)/6`

Answer

To find the principal value of \( \cot^{-1}(-\sqrt{3}) \), we can follow these steps:

### Step 1: Set up the equation
Let \( y = \cot^{-1}(-\sqrt{3}) \). By the definition of the inverse cotangent function, this implies:
\[ \cot y = -\sqrt{3} \]

### Step 2: Identify the cotangent value
We know that \( \sqrt{3} \) can be expressed as:
\[ \sqrt{3} = \cot\left(\frac{\pi}{6}\right) \]
Thus, we can rewrite our equation as:
\[ \cot y = -\cot\left(\frac{\pi}{6}\right) \]

### Step 3: Use the cotangent identity
Using the identity \( \cot(\pi - \theta) = -\cot(\theta) \), we can express \( -\cot\left(\frac{\pi}{6}\right) \) as:
\[ \cot y = \cot\left(\pi - \frac{\pi}{6}\right) \]
Calculating \( \pi - \frac{\pi}{6} \):
\[ \pi - \frac{\pi}{6} = \frac{6\pi}{6} - \frac{\pi}{6} = \frac{5\pi}{6} \]
So we have:
\[ \cot y = \cot\left(\frac{5\pi}{6}\right) \]

### Step 4: Conclude the value of \( y \)
Since the cotangent function is one-to-one in the interval \( (0, \pi) \), we can conclude that:
\[ y = \frac{5\pi}{6} \]

### Final Answer
Thus, the principal value of \( \cot^{-1}(-\sqrt{3}) \) is:
\[ \frac{5\pi}{6} \]

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