If `y=cos^(-1)(cos10), " then " y=`

Options

A. 10
B. `4pi-10`
C. `2pi+10`
D. `2pi-10`

Answer

To solve the problem \( y = \cos^{-1}(\cos 10) \), we need to find the value of \( y \) within the defined range of the inverse cosine function. The range of \( \cos^{-1}(x) \) is from \( 0 \) to \( \pi \).

### Step-by-step Solution:

1. **Understanding the Function**:
The function \( y = \cos^{-1}(\cos x) \) returns the principal value of the angle \( x \) in the range \( [0, \pi] \).

2. **Finding the Equivalent Angle**:
Since \( 10 \) radians is outside the range \( [0, \pi] \), we need to find an equivalent angle that lies within this range.
- The angle \( 10 \) radians can be expressed in terms of \( 2\pi \) (approximately \( 6.2832 \) radians).
- We can subtract \( 2\pi \) from \( 10 \) to find an equivalent angle:
\[ 10 - 2\pi \approx 10 - 6.2832 \approx 3.7168 \text{ radians} \]
- Since \( 3.7168 \) is still greater than \( \pi \), we subtract \( 2\pi \) again:
\[ 3.7168 - 2\pi \approx 3.7168 - 6.2832 \approx -2.5664 \text{ radians} \]
- This angle is negative, so we add \( 2\pi \) again to find a positive equivalent:
\[ -2.5664 + 2\pi \approx -2.5664 + 6.2832 \approx 3.7168 \text{ radians} \]

3. **Finding the Correct Angle**:
- The angle \( 10 \) radians can be expressed as:
\[ 10 = 2\pi + (10 - 2\pi) = 2\pi + 3.7168 \]
- Thus, \( \cos(10) = \cos(3.7168) \).

4. **Using the Inverse Cosine**:
- Now, we can use the property of the inverse cosine:
\[ y = \cos^{-1}(\cos(10)) = \cos^{-1}(\cos(3.7168)) \]
- Since \( 3.7168 \) is not in the range \( [0, \pi] \), we need to find the equivalent angle in that range. The equivalent angle is:
\[ y = 2\pi - 3.7168 \]

5. **Calculating the Final Value**:
- Calculate \( 2\pi - 3.7168 \):
\[ y = 6.2832 - 3.7168 \approx 2.5664 \text{ radians} \]

### Conclusion:
Thus, the final value of \( y \) is:
\[ y \approx 2.5664 \text{ radians} \]