Find the Principal Value of tan-1(−1/√3)
Solution:
Let
\[ y = \tan^{-1}\left(-\frac{1}{\sqrt{3}}\right) \]
Then,
\[ \tan y = -\frac{1}{\sqrt{3}} \]
We know:
\[ \tan\left(\frac{\pi}{6}\right) = \frac{1}{\sqrt{3}} \]
Using the identity:
\[ \tan^{-1}(-x) = -\tan^{-1}(x) \]
So,
\[ y = -\frac{\pi}{6} \]
Since the principal value range of tan-1(x) is:
\[ \left(-\frac{\pi}{2}, \frac{\pi}{2}\right) \]
This value lies within the range.
Final Answer:
Principal Value = \[ -\frac{\pi}{6} \]