Find the Principal Value of cos-1(sin 4π/3)
Solution:
Given:
\[ y = \cos^{-1}(\sin \tfrac{4\pi}{3}) \]
First evaluate:
\[ \sin \tfrac{4\pi}{3} = -\frac{\sqrt{3}}{2} \]
So,
\[ y = \cos^{-1}\left(-\frac{\sqrt{3}}{2}\right) \]
We know:
\[ \cos\left(\frac{\pi}{6}\right) = \frac{\sqrt{3}}{2} \]
Thus,
\[ \cos y = -\frac{\sqrt{3}}{2} = \cos\left(\frac{5\pi}{6}\right) \]
Since principal value range of cos-1(x) is:
\[ [0, \pi] \]
Therefore,
\[ y = \frac{5\pi}{6} \]
Final Answer:
Principal Value = \[ \frac{5\pi}{6} \]