Question

Evaluate the expression: \[ \frac{\sin \frac{\pi}{7} + \sin \frac{3\pi}{7}}{1 + \cos \frac{\pi}{7} + \cos \frac{2\pi}{7}} \]

• A. \( 1 \)
• B. \( 2 \)
• C. \( \frac{1}{2} \)
• D. \( \frac{\sqrt{3}}{2} \)

Hint:

When dealing with sums of sines and cosines, use sum-to-product identities to simplify the expression, especially when dealing with angles in terms of fractions of \( \pi \).

Answer 1

The Correct Option is C

To solve this, we use the sum and difference identities for trigonometric functions to simplify both the numerator and denominator. The angle values in this expression can also be simplified using known values from trigonometric tables or through advanced trigonometric identities. After simplification and using symmetry, we get: \[ \frac{\sin \frac{\pi}{7} + \sin \frac{3\pi}{7}}{1 + \cos \frac{\pi}{7} + \cos \frac{2\pi}{7}} = \frac{1}{2} \]