Find the value of `sin1^(@).sin2^(@).sin3^(@)…sin180^(@)`.

Options

A. `-1`
B. 0
C. 1
D. `alpha`

Answer

To find the value of the expression \( \sin 1^\circ \cdot \sin 2^\circ \cdot \sin 3^\circ \cdots \sin 180^\circ \), we can follow these steps:

### Step 1: Understand the sine function values
The sine function has specific values at certain angles. Notably, \( \sin 180^\circ = 0 \).

### Step 2: Write down the product
We express the product as:
\[ \sin 1^\circ \cdot \sin 2^\circ \cdot \sin 3^\circ \cdots \sin 180^\circ \]

### Step 3: Identify the zero factor
Since \( \sin 180^\circ = 0 \), we can see that this product contains a factor that is equal to zero.

### Step 4: Conclude the product
When we multiply any number by zero, the result is zero. Therefore, the entire product becomes:
\[ \sin 1^\circ \cdot \sin 2^\circ \cdot \sin 3^\circ \cdots \sin 180^\circ = 0 \]

### Final Answer
Thus, the value of \( \sin 1^\circ \cdot \sin 2^\circ \cdot \sin 3^\circ \cdots \sin 180^\circ \) is:
\[ \boxed{0} \]

---