If `a=sin, pi/18 sin (5pi)/18 si, (7pi)/18, and x` is the solution of the equation `y=2[x]+2 and y=3[x-2], where [x]` denotes the interal part of x then a= (A) [x] (B) `1/[x]` (C) 2[x] (D) `[x]^2`

Options

A. `[x]`
B. `(1)/([x])`
C. `2[x]`
D. `[x]^(2)`

Answer

To solve the problem, we need to find the value of \( a \) given by the product of certain sine functions, and then determine the relationship between \( a \) and \( [x] \), where \( [x] \) denotes the greatest integer less than or equal to \( x \).

### Step-by-Step Solution:

1. **Calculate \( a \)**:
We are given:
\[ a = \sin\left(\frac{\pi}{18}\right) \cdot \sin\left(\frac{5\pi}{18}\right) \cdot \sin\left(\frac{7\pi}{18}\right) \]

2. **Use the sine product-to-sum identities**:
We can simplify the product of sine functions. We will use the identity:
\[ \sin A \cdot \sin B = \frac{1}{2} [\cos(A-B) - \cos(A+B)] \]
Let's first calculate \( \sin\left(\frac{5\pi}{18}\right) \) using the complementary angle:
\[ \sin\left(\frac{5\pi}{18}\right) = \cos\left(\frac{\pi}{2} - \frac{5\pi}{18}\right) = \cos\left(\frac{9\pi}{18} - \frac{5\pi}{18}\right) = \cos\left(\frac{4\pi}{18}\right) = \cos\left(\frac{2\pi}{9}\right) \]
Now we can rewrite \( a \):
\[ a = \sin\left(\frac{\pi}{18}\right) \cdot \cos\left(\frac{2\pi}{9}\right) \cdot \sin\left(\frac{7\pi}{18}\right) \]

3. **Multiply by \( 2 \cos\left(\frac{\pi}{18}\right) \)**:
We can use the double angle identity:
\[ 2 \sin A \cos A = \sin(2A) \]
This gives us:
\[ 2 \sin\left(\frac{\pi}{18}\right) \cos\left(\frac{\pi}{18}\right) = \sin\left(\frac{2\pi}{18}\right) = \sin\left(\frac{\pi}{9}\right) \]

4. **Continue simplifying**:
We can continue this process to express \( a \) in terms of sine functions. After several steps of simplification, we find that:
\[ a = \frac{1}{8} \]

5. **Solve for \( [x] \)**:
We are given the equations:
\[ y = 2[x] + 2 \quad \text{and} \quad y = 3[x - 2] \]
Setting them equal to each other:
\[ 2[x] + 2 = 3[x] - 6 \]
Rearranging gives:
\[ 3[x] - 2[x] = 8 \implies [x] = 8 \]

6. **Relate \( a \) and \( [x] \)**:
Since we found \( a = \frac{1}{8} \) and \( [x] = 8 \), we can express \( a \) in terms of \( [x] \):
\[ a = \frac{1}{[x]} \]

### Conclusion:
Thus, the correct option for \( a \) is:
\[ \text{(B) } \frac{1}{[x]} \]