sinx+sin(120°+x) =?

প্রশ্ন: \( \sin x + \sin(120^\circ + x) = ? \)

আমরা জানি, \( \sin(A+B) = \sin A \cos B + \cos A \sin B \)।

তাহলে, \( \sin(120^\circ + x) = \sin 120^\circ \cos x + \cos 120^\circ \sin x \)।

আমরা আরও জানি, \( \sin 120^\circ = \sin (180^\circ - 60^\circ) = \sin 60^\circ = \frac{\sqrt{3}}{2} \) এবং \( \cos 120^\circ = \cos (180^\circ - 60^\circ) = - \cos 60^\circ = -\frac{1}{2} \)।

সুতরাং, \( \sin(120^\circ + x) = \frac{\sqrt{3}}{2} \cos x - \frac{1}{2} \sin x \)।

এখন, \( \sin x + \sin(120^\circ + x) = \sin x + \frac{\sqrt{3}}{2} \cos x - \frac{1}{2} \sin x \)।

\(= \sin x - \frac{1}{2} \sin x + \frac{\sqrt{3}}{2} \cos x \)

\(= \frac{1}{2} \sin x + \frac{\sqrt{3}}{2} \cos x \)

\(= \cos 60^\circ \sin x + \sin 60^\circ \cos x \)

\(= \sin(x + 60^\circ) \) 🤩

অতএব, \( \sin x + \sin(120^\circ + x) = \sin(x + 60^\circ) \)। 🎉