Find the Principal Value of tan-1(2cos 2π/3)
Solution:
Given:
\[ y = \tan^{-1}\left(2\cos \frac{2\pi}{3}\right) \]
Step 1: Evaluate cos(2π/3)
\[ \cos \frac{2\pi}{3} = -\frac{1}{2} \]
So,
\[ 2\cos \frac{2\pi}{3} = 2 \times \left(-\frac{1}{2}\right) = -1 \]
Step 2: Use inverse tangent
\[ y = \tan^{-1}(-1) \]
We know:
\[ \tan^{-1}(-1) = -\frac{\pi}{4} \]
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 = \[ -\frac{\pi}{4} \]