If `x=cos^(-1)(cos 4) " and " y=sin^(-1)(sin3)`, then which of the following holds?

Options

A. x-y=1
B. x+y+1=0
C. x+2y=2
D. `x+y=3pi-7`

Answer

To solve the problem, we need to find the values of \( x \) and \( y \) given the equations \( x = \cos^{-1}(\cos 4) \) and \( y = \sin^{-1}(\sin 3) \), and then determine which of the provided options holds true.

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

1. **Understanding the range of \( \cos^{-1} \)**: The function \( \cos^{-1}(a) \) is defined for \( a \) in the interval \([-1, 1]\) and its output lies in the interval \([0, \pi]\).
2. **Check the value of \( 4 \)**: Since \( 4 \) is greater than \( \pi \), we need to reduce it to an equivalent angle within the range of \( \cos^{-1} \).
3. **Using the periodic property of cosine**: We can express \( 4 \) in terms of an angle within the range:
\[ \cos(2n\pi - \theta) = \cos(\theta) \]
Here, we can take \( n = 1 \):
\[ 2\pi - 4 \]
4. **Calculate \( 2\pi - 4 \)**:
\[ 2\pi - 4 \approx 2(3.14) - 4 = 6.28 - 4 = 2.28 \]
Since \( 2.28 \) is within the range \([0, \pi]\), we have:
\[ x = 2\pi - 4 \]

### Step 2: Calculate \( y = \sin^{-1}(\sin 3) \)

1. **Understanding the range of \( \sin^{-1} \)**: The function \( \sin^{-1}(b) \) is defined for \( b \) in the interval \([-1, 1]\) and its output lies in the interval \([- \frac{\pi}{2}, \frac{\pi}{2}]\).
2. **Check the value of \( 3 \)**: Since \( 3 \) is greater than \( \frac{\pi}{2} \), we need to reduce it to an equivalent angle within the range.
3. **Using the periodic property of sine**: We can express \( 3 \) in terms of an angle within the range:
\[ \sin((2n - 1)\pi - \theta) = \sin(\theta) \]
Here, we can take \( n = 2 \):
\[ 2\pi - 3 \]
4. **Calculate \( \pi - 3 \)**:
\[ \pi - 3 \approx 3.14 - 3 = 0.14 \]
Since \( 0.14 \) is within the range \([- \frac{\pi}{2}, \frac{\pi}{2}]\), we have:
\[ y = \pi - 3 \]

### Step 3: Find \( x + y \)

1. **Combine the values of \( x \) and \( y \)**:
\[ x + y = (2\pi - 4) + (\pi - 3) \]
Simplifying this:
\[ x + y = 2\pi + \pi - 4 - 3 = 3\pi - 7 \]

### Conclusion

Thus, we have found that:
\[ x + y = 3\pi - 7 \]
The correct option is:
- **Option D: \( x + y = 3\pi - 7 \)**.