`sin 12^(@) sin 48^(@) sin 54^(@)` is equal to

Options

A. `(1)/(16)`
B. `(1)/(32)`
C. `(1)/(8)`
D. `(1)/(4)`

Explanation

To find the value of \( \sin 12^\circ \sin 48^\circ \sin 54^\circ \), we can follow these steps:

### Step 1: Rewrite the expression
We start with the expression:
\[ \sin 12^\circ \sin 48^\circ \sin 54^\circ \]
We can multiply and divide by 2:
\[ = \frac{1}{2} \cdot 2 \sin 12^\circ \sin 48^\circ \sin 54^\circ \]

### Step 2: Use the sine product identity
Using the identity \( 2 \sin A \sin B = \cos(A - B) - \cos(A + B) \), we can apply this to \( 2 \sin 12^\circ \sin 48^\circ \):
\[ = \frac{1}{2} \left( \cos(12^\circ - 48^\circ) - \cos(12^\circ + 48^\circ) \right) \sin 54^\circ \]

### Step 3: Simplify the cosine terms
Calculating the angles:
\[ 12^\circ - 48^\circ = -36^\circ \quad \text{and} \quad 12^\circ + 48^\circ = 60^\circ \]
Thus, we have:
\[ = \frac{1}{2} \left( \cos(-36^\circ) - \cos(60^\circ) \right) \sin 54^\circ \]
Using the property \( \cos(-x) = \cos(x) \):
\[ = \frac{1}{2} \left( \cos(36^\circ) - \cos(60^\circ) \right) \sin 54^\circ \]

### Step 4: Substitute known values
We know that:
\[ \cos(60^\circ) = \frac{1}{2} \]
Now substituting this in:
\[ = \frac{1}{2} \left( \cos(36^\circ) - \frac{1}{2} \right) \sin 54^\circ \]

### Step 5: Simplify further
Now, we can express \( \sin 54^\circ \) as \( \cos(36^\circ) \) because \( \sin(90^\circ - x) = \cos(x) \):
\[ = \frac{1}{2} \left( \cos(36^\circ) - \frac{1}{2} \right) \cos(36^\circ) \]
This simplifies to:
\[ = \frac{1}{2} \left( \cos^2(36^\circ) - \frac{1}{2} \cos(36^\circ) \right) \]

### Step 6: Substitute the value of \( \cos(36^\circ) \)
The value of \( \cos(36^\circ) \) is known to be \( \frac{\sqrt{5} + 1}{4} \):
\[ = \frac{1}{2} \left( \left(\frac{\sqrt{5} + 1}{4}\right)^2 - \frac{1}{2} \cdot \frac{\sqrt{5} + 1}{4} \right) \]

### Step 7: Calculate \( \left(\frac{\sqrt{5} + 1}{4}\right)^2 \)
Calculating:
\[ \left(\frac{\sqrt{5} + 1}{4}\right)^2 = \frac{5 + 2\sqrt{5} + 1}{16} = \frac{6 + 2\sqrt{5}}{16} = \frac{3 + \sqrt{5}}{8} \]
Now substituting back:
\[ = \frac{1}{2} \left( \frac{3 + \sqrt{5}}{8} - \frac{\sqrt{5} + 1}{8} \right) \]
This simplifies to:
\[ = \frac{1}{2} \cdot \frac{(3 + \sqrt{5}) - (\sqrt{5} + 1)}{8} = \frac{1}{2} \cdot \frac{2}{8} = \frac{1}{8} \]

### Final Answer
Thus, the value of \( \sin 12^\circ \sin 48^\circ \sin 54^\circ \) is:
\[ \boxed{\frac{1}{8}} \]