Step 1: Understanding the Concept:
The identity \( \sin^{-1}(\sin \theta) = \theta \) is only valid if \( \theta \) lies within the restricted principal value branch of the inverse sine function.
To ensure that inverse trigonometric functions are well-defined as functions (meaning they pass the vertical line test), their outputs are restricted to specific ranges.
For the \( \sin^{-1} \) function, the principal value range is defined as the closed interval \( [-\pi/2, \pi/2] \), which corresponds to \( [-90^\circ, 90^\circ] \).
If an angle falls outside this range, the inverse sine will return an equivalent angle that yields the same sine value but stays within the boundaries of the principal branch.
Step 2: Key Formula or Approach:
1. Verify if the given angle \( \theta = 2\pi/3 \) lies in the interval \( [-\pi/2, \pi/2] \).
2. If not, use the supplementary angle identity \( \sin(x) = \sin(\pi - x) \) to find an equivalent angle in the first or fourth quadrant.
Step 3: Detailed Explanation:
The angle provided is \( \theta = \frac{2\pi}{3} \).
Let's convert this to degrees for a clearer perspective: \( \frac{2 \times 180^\circ}{3} = 120^\circ \).
The principal range is \( [-90^\circ, 90^\circ] \). Since \( 120^\circ \) is in the second quadrant and is greater than \( 90^\circ \), it is outside the principal range.
Therefore, \( \sin^{-1}(\sin(2\pi/3)) \neq 2\pi/3 \).
We must find an angle in the principal range that has the same sine value as \( 120^\circ \).
In the second quadrant, we use the identity:
\[ \sin \theta = \sin(\pi - \theta) \]
Substituting \( \theta = \frac{2\pi}{3} \):
\[ \sin \left( \frac{2\pi}{3} \right) = \sin \left( \pi - \frac{2\pi}{3} \right) = \sin \left( \frac{\pi}{3} \right) \]
Now, the expression becomes:
\[ \sin^{-1} \left( \sin \frac{\pi}{3} \right) \]
The angle \( \frac{\pi}{3} \) is \( 60^\circ \), which comfortably sits inside the principal range \( [-90^\circ, 90^\circ] \).
Thus, for this angle, the inverse function and the function effectively "cancel" out:
\[ \sin^{-1} \left( \sin \frac{\pi}{3} \right) = \frac{\pi}{3} \]
Step 4: Final Answer:
The principal value of the given expression is \( \frac{\pi}{3} \).