The value of `sin^(-1)`(sin 10) is

Options

A. `3pi-10`
B. `10-3pi`
C. `3pi+10`
D. `4pi-10`

Explanation

To find the value of \( \sin^{-1}(\sin 10) \), we can follow these steps:

### Step 1: Understand the properties of inverse trigonometric functions
The property we need to use is:
\[ \sin^{-1}(\sin \theta) = \theta \quad \text{if } \theta \text{ is in the range } \left[-\frac{\pi}{2}, \frac{\pi}{2}\right] \]
Here, \( 10 \) (in radians) is not within this range.

### Step 2: Adjust the angle
Since \( 10 \) is not in the range of \( \left[-\frac{\pi}{2}, \frac{\pi}{2}\right] \), we need to find an equivalent angle that is within this range. We can use the property:
\[ \sin \theta = \sin(3\pi - \theta) \]
to find a suitable angle.

### Step 3: Calculate \( 3\pi - 10 \)
Now, let's calculate \( 3\pi - 10 \):
\[ 3\pi - 10 = 3 \times 3.14 - 10 = 9.42 - 10 = -0.58 \]
Now, we check if \( -0.58 \) is within the range \( \left[-\frac{\pi}{2}, \frac{\pi}{2}\right] \).

### Step 4: Check the range
The value of \( -0.58 \) is indeed within the range \( \left[-\frac{\pi}{2}, \frac{\pi}{2}\right] \) since:
\[ -\frac{\pi}{2} \approx -1.57 \quad \text{and} \quad \frac{\pi}{2} \approx 1.57 \]
Thus, we can use this angle.

### Step 5: Apply the inverse sine function
Now we can apply the inverse sine function:
\[ \sin^{-1}(\sin 10) = \sin^{-1}(\sin(3\pi - 10)) = 3\pi - 10 \]

### Final Answer
Thus, the value of \( \sin^{-1}(\sin 10) \) is:
\[ 3\pi - 10 \approx -0.58 \]