Question

The value of \( \cos^{-1} \left( \cos \frac{2\pi}{3} \right) \) is equal to:

Hint:

The principal value of \( \cos^{-1} x \) lies in the interval \( [0, \pi] \). When applying inverse trigonometric functions, always consider the principal value.

Answer 1

Step 1: Evaluating the cosine function.
We know that \( \cos \frac{2\pi}{3} = -\frac{1}{2} \). Step 2: Applying the inverse cosine.
The principal value of \( \cos^{-1} x \) is the angle \( \theta \) in the range \( [0, \pi] \) such that \( \cos(\theta) = x \). Therefore, we need to find the angle \( \theta \) such that: \[ \cos(\theta) = -\frac{1}{2} \] The angle \( \theta \) in the range \( [0, \pi] \) for which \( \cos(\theta) = -\frac{1}{2} \) is \( \frac{2\pi}{3} \). Step 3: Conclusion.
Thus, \( \cos^{-1} \left( \cos \frac{2\pi}{3} \right) = \frac{2\pi}{3} \).