`sin^(-1)(sin10)` is `a+bpi` then `|a+b|` is

Explanation

To solve the question \( \sin^{-1}(\sin 10) \) and express it in the form \( a + b\pi \), we will follow these steps:

### Step 1: Understand the Range of \( \sin^{-1} \)
The function \( \sin^{-1}(x) \) (or arcsine) has a range of \( \left[-\frac{\pi}{2}, \frac{\pi}{2}\right] \). This means that for any angle \( x \) outside this range, we need to find an equivalent angle within this range.

**Hint:** Remember that angles can be expressed in terms of \( \pi \) and that \( \sin \) is periodic.

### Step 2: Convert 10 Radians to Degrees
To understand where \( 10 \) radians lies, we can convert it to degrees:
\[ 10 \text{ radians} \times \frac{180}{\pi} \approx 10 \times 57.3 \approx 573.6 \text{ degrees} \]
This angle is greater than \( 360 \) degrees, so we need to find its equivalent angle within the range of \( 0 \) to \( 360 \) degrees.

**Hint:** Subtract \( 360 \) degrees from the angle to find its equivalent in the first rotation.

### Step 3: Find the Equivalent Angle
Subtract \( 360 \) degrees from \( 573.6 \) degrees:
\[ 573.6 - 360 = 213.6 \text{ degrees} \]
Now, we can see that \( 213.6 \) degrees is in the third quadrant, where sine is negative.

**Hint:** Remember the quadrant rules for sine values.

### Step 4: Find the Reference Angle
The reference angle for \( 213.6 \) degrees is:
\[ 213.6 - 180 = 33.6 \text{ degrees} \]
Since sine is negative in the third quadrant, we have:
\[ \sin(213.6) = -\sin(33.6) \]

**Hint:** Use the reference angle to express the sine in terms of its positive value.

### Step 5: Express in Terms of \( \pi \)
Now we can express \( 10 \) radians in terms of \( \pi \):
\[ 10 \text{ radians} = 3\pi + \left(-\frac{\pi}{6}\right) \text{ (since } 33.6 \text{ degrees is approximately } \frac{\pi}{6} \text{ radians)} \]
Thus, we can write:
\[ \sin^{-1}(\sin 10) = \sin^{-1}(-\sin(33.6)) = -\frac{\pi}{6} \]

**Hint:** Use the periodicity and symmetry of the sine function to find the correct angle.

### Step 6: Write in the Form \( a + b\pi \)
Now we can express \( -\frac{\pi}{6} \) in the form \( a + b\pi \):
\[ -\frac{\pi}{6} = 0 - \frac{1}{6}\pi \]
Here, \( a = 0 \) and \( b = -\frac{1}{6} \).

**Hint:** Identify \( a \) and \( b \) from the expression.

### Step 7: Calculate \( |a + b| \)
Now we find \( |a + b| \):
\[ |0 - \frac{1}{6}| = \frac{1}{6} \]

### Final Answer
Thus, the required value of \( |a + b| \) is:
\[ \boxed{\frac{1}{6}} \]