The value of `cos^(-1)(cos10)` is

Options

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

Explanation

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

### Step 1: Understand the range of the inverse cosine function
The function \( \cos^{-1}(x) \) has a range of \( [0, \pi] \). This means that the output of \( \cos^{-1} \) will always fall within this interval.

**Hint:** Remember that the output of the inverse trigonometric functions is restricted to specific ranges.

### Step 2: Identify the angle
We have \( \cos^{-1}(\cos 10) \). The angle \( 10 \) (in radians) is not within the range \( [0, \pi] \) since \( 10 \) is greater than \( \pi \) (approximately \( 3.14 \)).

**Hint:** Check if the angle you are working with lies within the range of the inverse function.

### Step 3: Use the periodic property of cosine
We can express \( 10 \) in terms of an equivalent angle within the range of \( [0, \pi] \). We can do this by using the periodic property of the cosine function. We know that:
\[ \cos(2\pi - \theta) = \cos(\theta) \]
Thus, we can rewrite \( 10 \) as:
\[ 10 = 4\pi - 10 \]
This is valid because \( 4\pi - 10 \) will yield an angle in the range \( [0, \pi] \).

**Hint:** Use the periodic properties of trigonometric functions to find equivalent angles.

### Step 4: Substitute back into the inverse cosine function
Now we can substitute back into the inverse cosine function:
\[ \cos^{-1}(\cos 10) = \cos^{-1}(\cos(4\pi - 10)) \]

Since \( 4\pi - 10 \) is in the range \( [0, \pi] \), we can directly cancel the cosine and the inverse cosine:
\[ \cos^{-1}(\cos(4\pi - 10)) = 4\pi - 10 \]

**Hint:** When the angle is within the range of the inverse function, you can directly simplify the expression.

### Final Answer
Thus, the value of \( \cos^{-1}(\cos 10) \) is:
\[ 4\pi - 10 \]