Find the Principal Value of cot-1(−1/√3)
Solution:
Let
\[ y = \cot^{-1}\left(-\frac{1}{\sqrt{3}}\right) \]
Then,
\[ \cot y = -\frac{1}{\sqrt{3}} \]
We know:
\[ \cot\left(\frac{\pi}{3}\right) = \frac{1}{\sqrt{3}} \]
So,
\[ \cot y = -\frac{1}{\sqrt{3}} = \cot\left(\pi – \frac{\pi}{3}\right) = \cot\left(\frac{2\pi}{3}\right) \]
Principal value range of cot⁻¹(x):
\[ (0, \pi) \]
Since \( \frac{2\pi}{3} \in (0,\pi) \),
\[ y = \frac{2\pi}{3} \]
Final Answer:
Principal Value = \[ \frac{2\pi}{3} \]