Discussion
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
Answer: C
\(\frac{a}{b} = \) \(\sqrt{3}\).
Apply the sum-to-product formulas to both sums:
\[ \begin{aligned} a &= 2\sin\tfrac{x + y}{2}\cos\tfrac{x - y}{2} \\ b &= 2\cos\tfrac{x + y}{2}\cos\tfrac{x - y}{2} \\ \frac{a}{b} &= \tan\tfrac{x + y}{2} = \tan\tfrac{\pi}{3} = \sqrt{3} \end{aligned} \]
\(\frac{\sqrt{3}}{3}\) is \(\tan\frac{\pi}{6}\) or \(\cot\frac{\pi}{3}\), from inverting the ratio. Using \(\tan(x + y)\) instead of the half-angle gives \(-\sqrt{3}\), which is not offered.
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