The principal value of `"sin"^(-1)("sin""(5pi)/(3))` is

Options

A. `-(5pi)/(3)`
B. `(5pi)/(3)`
C. `-pi/3`
D. `(4pi)/(3)`

Answer

To find the principal value of \( \sin^{-1}(\sin(5\pi/3)) \), we will follow these steps:

### Step 1: Identify the angle
We start with the angle \( 5\pi/3 \). This angle is greater than \( 2\pi \), so we can reduce it to find an equivalent angle within the range of \( 0 \) to \( 2\pi \).

### Step 2: Reduce the angle
To find an equivalent angle, we subtract \( 2\pi \):
\[ 5\pi/3 - 2\pi = 5\pi/3 - 6\pi/3 = -\pi/3 \]
Thus, \( 5\pi/3 \) is equivalent to \( -\pi/3 \).

### Step 3: Calculate \( \sin(5\pi/3) \)
Now we can find the sine of the equivalent angle:
\[ \sin(5\pi/3) = \sin(-\pi/3) \]
Using the property of sine, we know:
\[ \sin(-\theta) = -\sin(\theta) \]
Thus,
\[ \sin(-\pi/3) = -\sin(\pi/3) \]
Now, we know that \( \sin(\pi/3) = \frac{\sqrt{3}}{2} \), so:
\[ \sin(-\pi/3) = -\frac{\sqrt{3}}{2} \]

### Step 4: Find \( \sin^{-1}(\sin(5\pi/3)) \)
Now we can find the principal value:
\[ \sin^{-1}(\sin(5\pi/3)) = \sin^{-1}(-\frac{\sqrt{3}}{2}) \]
The principal value of \( \sin^{-1}(x) \) is defined in the range \( [-\frac{\pi}{2}, \frac{\pi}{2}] \). The angle whose sine is \( -\frac{\sqrt{3}}{2} \) in this range is:
\[ -\frac{\pi}{3} \]

### Final Answer
Thus, the principal value of \( \sin^{-1}(\sin(5\pi/3)) \) is:
\[ -\frac{\pi}{3} \]

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