Question

If \( \sin x + \sin y = a \), \( \cos x + \cos y = b \) and \( x + y = \frac{2\pi}{3} \), then the value of \( \frac{a}{b} \) is equal to

• A. \( \frac{\sqrt{3}}{3} \)
• B. \( 2\sqrt{3} \)
• C. \( \sqrt{3} \)
• D. \( 4\sqrt{3} \)
• E. \( \frac{\sqrt{3}}{6} \)

Hint:

For sums of sine and cosine, convert into product form and take ratios to simplify.

Answer 1

The Correct Option is C

Concept: \[ \sin x + \sin y = 2\sin\frac{x+y}{2}\cos\frac{x-y}{2} \] \[ \cos x + \cos y = 2\cos\frac{x+y}{2}\cos\frac{x-y}{2} \]

Step 1: Apply identities.
\[ a = 2\sin\frac{x+y}{2}\cos\frac{x-y}{2} \] \[ b = 2\cos\frac{x+y}{2}\cos\frac{x-y}{2} \]

Step 2: Take ratio.
\[ \frac{a}{b} = \frac{\sin\frac{x+y}{2}}{\cos\frac{x+y}{2}} = \tan\frac{x+y}{2} \]

Step 3: Substitute value.
\[ \frac{x+y}{2} = \frac{\pi}{3} \] \[ \Rightarrow \frac{a}{b} = \tan\frac{\pi}{3} = \sqrt{3} \]