Evaluate: cosec-1(2tan 11π/6)
Solution:
Step 1: Evaluate tan(11π/6)
\[ \tan \frac{11\pi}{6} = -\frac{1}{\sqrt{3}} \]
So,
\[ 2\tan \frac{11\pi}{6} = -\frac{2}{\sqrt{3}} \]
Step 2: Convert to sine
\[ \csc y = -\frac{2}{\sqrt{3}} \Rightarrow \sin y = -\frac{\sqrt{3}}{2} \]
Step 3: Find principal value
\[ \sin y = -\frac{\sqrt{3}}{2} \Rightarrow y = -\frac{\pi}{3} \]
Principal value range of cosec⁻¹(x):
\[ \left[-\frac{\pi}{2}, 0\right) \cup \left(0, \frac{\pi}{2}\right] \]
Since \( -\frac{\pi}{3} \) lies in this range,
\[ y = -\frac{\pi}{3} \]
Final Answer:
Principal Value = \[ -\frac{\pi}{3} \]