Question

If \( A + B + C = \frac{\pi}{4} \), then evaluate the expression:
\[ \sin 4A + \sin 4B + \sin 4C \]

• A. \( 4\cos 2A \cos 2B \cos 2C \)
• B. \( 4\sin 2A \sin 2B \sin 2C \)
• C. \( 1 + 4\sin 2A \sin 2B \sin 2C \)
• D. \( 1 + 4\cos 2A \cos 2B \cos 2C \)

Hint:

When dealing with multiple-angle identities, break down expressions using known trigonometric transformations.

Answer 1

The Correct Option is D

Using the identity for the sum of sines, we express:
\[ \sin 4A + \sin 4B + \sin 4C \]
in a form that incorporates cosine terms. Given \( A + B + C = \frac{\pi}{4} \), we apply trigonometric transformations:
\[ \sin x + \sin y + \sin z = 1 + 4\cos 2A \cos 2B \cos 2C. \]
Thus, the expression simplifies to:
\[ \boxed{1 + 4\cos 2A \cos 2B \cos 2C}. \]