If `sin x_(1)+sin x_(2) + sin x_(3)+...+sin x_(2008)=2008` then find the value of `sin^(2008) x_( 1)+sin^(2008)x_(2) + sin^(2008)x_(3)+...+sin^(2008)x_(2008)`.

Answer

To solve the problem, we start with the given equation:

\[ \sin x_1 + \sin x_2 + \sin x_3 + \ldots + \sin x_{2008} = 2008 \]

### Step 1: Analyze the Range of Sine Function
The sine function, \( \sin x \), has a range of values between -1 and 1. This means that for any \( i \), \( \sin x_i \) can take values in the interval \([-1, 1]\).

**Hint:** Remember that the maximum value of \( \sin x \) is 1.

### Step 2: Determine the Maximum Possible Sum
Since there are 2008 terms in the sum, the maximum possible value for the sum of these sine terms occurs when each sine term is at its maximum value of 1. Therefore, if all \( \sin x_i = 1 \):

\[ \sin x_1 + \sin x_2 + \ldots + \sin x_{2008} = 1 + 1 + \ldots + 1 = 2008 \]

This indicates that each \( \sin x_i \) must equal 1 for the sum to equal 2008.

**Hint:** Consider what values \( \sin x_i \) must take to achieve the given sum.

### Step 3: Conclude the Values of Sine Terms
Since we have established that each \( \sin x_i = 1 \):

\[ \sin x_1 = 1, \sin x_2 = 1, \ldots, \sin x_{2008} = 1 \]

### Step 4: Calculate the Required Expression
Now, we need to find the value of:

\[ \sin^{2008} x_1 + \sin^{2008} x_2 + \sin^{2008} x_3 + \ldots + \sin^{2008} x_{2008} \]

Substituting the values we found:

\[ \sin^{2008} x_1 + \sin^{2008} x_2 + \ldots + \sin^{2008} x_{2008} = 1^{2008} + 1^{2008} + \ldots + 1^{2008} \]

Since there are 2008 terms and \( 1^{2008} = 1 \):

\[ = 1 + 1 + \ldots + 1 \quad (\text{2008 terms}) = 2008 \]

### Final Answer
Thus, the value of \( \sin^{2008} x_1 + \sin^{2008} x_2 + \ldots + \sin^{2008} x_{2008} \) is:

\[ \boxed{2008} \]