Find the value of `sin10^(@).sin30^(@).sin50^(@).sin70^(@)`.

Options

A. 8
B. `(1)/(16)`
C. 16
D. `(1)/(2)`

Answer

To find the value of \( \sin 10^\circ \cdot \sin 30^\circ \cdot \sin 50^\circ \cdot \sin 70^\circ \), we can follow these steps:

### Step 1: Identify known values
We know that:
\[ \sin 30^\circ = \frac{1}{2} \]
So we can rewrite the expression:
\[ \sin 10^\circ \cdot \sin 30^\circ \cdot \sin 50^\circ \cdot \sin 70^\circ = \sin 10^\circ \cdot \frac{1}{2} \cdot \sin 50^\circ \cdot \sin 70^\circ \]

### Step 2: Use the identity for sine
We can use the identity:
\[ \sin \theta \cdot \sin (60^\circ - \theta) \cdot \sin (60^\circ + \theta) = \frac{1}{4} \sin 3\theta \]
In our case, let \( \theta = 10^\circ \). Then:
\[ \sin 10^\circ \cdot \sin (60^\circ - 10^\circ) \cdot \sin (60^\circ + 10^\circ) = \sin 10^\circ \cdot \sin 50^\circ \cdot \sin 70^\circ \]
This gives us:
\[ \sin 10^\circ \cdot \sin 50^\circ \cdot \sin 70^\circ = \frac{1}{4} \sin 30^\circ \]

### Step 3: Substitute the known value
Now, substituting \( \sin 30^\circ = \frac{1}{2} \):
\[ \sin 10^\circ \cdot \sin 50^\circ \cdot \sin 70^\circ = \frac{1}{4} \cdot \frac{1}{2} = \frac{1}{8} \]

### Step 4: Combine with \( \sin 30^\circ \)
Now we can substitute this back into our expression:
\[ \sin 10^\circ \cdot \sin 30^\circ \cdot \sin 50^\circ \cdot \sin 70^\circ = \frac{1}{8} \cdot \frac{1}{2} = \frac{1}{16} \]

### Final Answer
Thus, the value of \( \sin 10^\circ \cdot \sin 30^\circ \cdot \sin 50^\circ \cdot \sin 70^\circ \) is:
\[ \boxed{\frac{1}{16}} \]