Let f be a real - valued function defined on R ( the set of real numbers) such that `f(x) = sin^(-1) ( sin x) + cos^(-1) ( cos x)`
The value of f(10) is equal to

Options

A. `6 pi - 20`
B. `7 pi - 20`
C. `20 - 7pi`
D. `20 - 6pi`

Answer

To find the value of \( f(10) \) for the function defined as:

\[ f(x) = \sin^{-1}(\sin x) + \cos^{-1}(\cos x) \]

we will follow these steps:

### Step 1: Identify the intervals for \( \sin^{-1}(\sin x) \) and \( \cos^{-1}(\cos x) \)

The function \( \sin^{-1}(\sin x) \) has specific values depending on the interval of \( x \):
- If \( x \in \left[-\frac{\pi}{2}, \frac{\pi}{2}\right] \), then \( \sin^{-1}(\sin x) = x \).
- If \( x \in \left[\frac{\pi}{2}, \frac{3\pi}{2}\right] \), then \( \sin^{-1}(\sin x) = \pi - x \).
- If \( x \in \left[\frac{3\pi}{2}, \frac{5\pi}{2}\right] \), then \( \sin^{-1}(\sin x) = x - 2\pi \).
- If \( x \in \left[\frac{5\pi}{2}, \frac{7\pi}{2}\right] \), then \( \sin^{-1}(\sin x) = 3\pi - x \).

Similarly, for \( \cos^{-1}(\cos x) \):
- If \( x \in [0, \pi] \), then \( \cos^{-1}(\cos x) = x \).
- If \( x \in [\pi, 2\pi] \), then \( \cos^{-1}(\cos x) = 2\pi - x \).
- If \( x \in [2\pi, 3\pi] \), then \( \cos^{-1}(\cos x) = x - 2\pi \).
- If \( x \in [3\pi, 4\pi] \), then \( \cos^{-1}(\cos x) = 4\pi - x \).

### Step 2: Determine the interval for \( x = 10 \)

To find \( f(10) \), we need to determine which intervals \( 10 \) falls into:
- The value \( 10 \) is approximately \( 3.18\pi \) (since \( \pi \approx 3.14 \)).
- It lies in the interval \( \left[\frac{3\pi}{2}, \frac{5\pi}{2}\right] \) which corresponds to \( \left[4.71, 7.85\right] \).

### Step 3: Calculate \( \sin^{-1}(\sin 10) \)

Since \( 10 \) is in the interval \( \left[\frac{3\pi}{2}, \frac{5\pi}{2}\right] \):
\[ \sin^{-1}(\sin 10) = 3\pi - 10 \]

### Step 4: Calculate \( \cos^{-1}(\cos 10) \)

Now, we check the interval for \( \cos^{-1}(\cos 10) \):
- \( 10 \) is in the interval \( [3\pi, 4\pi] \):
\[ \cos^{-1}(\cos 10) = 4\pi - 10 \]

### Step 5: Combine the results

Now we can combine the results:
\[ f(10) = \sin^{-1}(\sin 10) + \cos^{-1}(\cos 10) = (3\pi - 10) + (4\pi - 10) \]
\[ f(10) = 3\pi + 4\pi - 20 = 7\pi - 20 \]

Thus, the final value of \( f(10) \) is:
\[ \boxed{7\pi - 20} \]