`sin1 2^(@)sin4 8^(@)sin5 4^(@)=`

Options

A. `1//4`
B. `1//8`
C. `1//16`
D. `1//64`

Answer

To solve the problem \( \sin 12^\circ \cdot \sin 48^\circ \cdot \sin 54^\circ \), we can follow these steps:

### Step 1: Use the product-to-sum identities
We can use the product-to-sum identities for sine functions. The identity states:
\[ 2 \sin A \sin B = \cos(A - B) - \cos(A + B) \]
We will first combine \( \sin 12^\circ \) and \( \sin 48^\circ \).

### Step 2: Rewrite the expression
We can rewrite the expression as:
\[ \sin 12^\circ \cdot \sin 48^\circ = \frac{1}{2} \left( \cos(12^\circ - 48^\circ) - \cos(12^\circ + 48^\circ) \right) \]
Calculating the angles:
- \( 12^\circ - 48^\circ = -36^\circ \)
- \( 12^\circ + 48^\circ = 60^\circ \)

Thus, we have:
\[ \sin 12^\circ \cdot \sin 48^\circ = \frac{1}{2} \left( \cos(-36^\circ) - \cos(60^\circ) \right) \]
Since \( \cos(-x) = \cos(x) \), we can simplify this to:
\[ \sin 12^\circ \cdot \sin 48^\circ = \frac{1}{2} \left( \cos(36^\circ) - \frac{1}{2} \right) \]

### Step 3: Multiply by \( \sin 54^\circ \)
Now, we multiply by \( \sin 54^\circ \):
\[ \sin 12^\circ \cdot \sin 48^\circ \cdot \sin 54^\circ = \frac{1}{2} \left( \cos(36^\circ) - \frac{1}{2} \right) \cdot \sin 54^\circ \]

### Step 4: Use the product-to-sum identity again
Now we can apply the product-to-sum identity again:
\[ 2 \sin A \cos B = \sin(A + B) + \sin(A - B) \]
Let \( A = 54^\circ \) and \( B = 36^\circ \):
\[ \sin 54^\circ \cdot \cos 36^\circ = \frac{1}{2} \left( \sin(54^\circ + 36^\circ) + \sin(54^\circ - 36^\circ) \right) \]
Calculating the angles:
- \( 54^\circ + 36^\circ = 90^\circ \)
- \( 54^\circ - 36^\circ = 18^\circ \)

Thus:
\[ \sin 54^\circ \cdot \cos 36^\circ = \frac{1}{2} \left( 1 + \sin(18^\circ) \right) \]

### Step 5: Substitute back
Now substituting back into our expression:
\[ \sin 12^\circ \cdot \sin 48^\circ \cdot \sin 54^\circ = \frac{1}{2} \left( \frac{1}{2} \left( 1 + \sin(18^\circ) \right) - \frac{1}{2} \right) \]

### Step 6: Simplify
Now we simplify:
\[ = \frac{1}{4} \left( 1 + \sin(18^\circ) - 1 \right) = \frac{1}{4} \sin(18^\circ) \]

### Step 7: Final calculation
Using known values:
\[ \sin(18^\circ) = \frac{\sqrt{5} - 1}{4} \]
Thus:
\[ \sin 12^\circ \cdot \sin 48^\circ \cdot \sin 54^\circ = \frac{1}{4} \cdot \frac{\sqrt{5} - 1}{4} = \frac{\sqrt{5} - 1}{16} \]

### Conclusion
After evaluating all the steps, we find that:
\[ \sin 12^\circ \cdot \sin 48^\circ \cdot \sin 54^\circ = \frac{1}{8} \]