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\(\sin 2\theta = \cos 3\theta\) সমীকরণের সাধারণ মান নির্ণয় করার জন্য, আমরা প্রথমে সমীকরণটিকে একটি অভিন্ন ত্রিকোণমিতিক অনুপাতে রূপান্তর করব।

আমরা জানি যে, \(\sin x = \cos (\frac{\pi}{2} - x)\)।

সুতরাং, প্রদত্ত সমীকরণটিকে লেখা যায়:

\(\cos 3\theta = \cos (\frac{\pi}{2} - 2\theta)\)

যখন \(\cos A = \cos B\) হয়, তখন সাধারণ সমাধান হলো \(A = 2n\pi \pm B\), যেখানে \(n\) একটি পূর্ণসংখ্যা (\(n = 0, \pm 1, \pm 2, \dots\))।

এই সূত্রটি ব্যবহার করে আমরা পাই:

\(3\theta = 2n\pi \pm (\frac{\pi}{2} - 2\theta)\)

এখন, আমরা দুটি ক্ষেত্রে সমাধান করব:

১. '+' চিহ্ন নিয়ে:

\(3\theta = 2n\pi + \frac{\pi}{2} - 2\theta\)

\(3\theta + 2\theta = 2n\pi + \frac{\pi}{2}\)

\(5\theta = 2n\pi + \frac{\pi}{2}\)

\(\theta = \frac{1}{5}(2n\pi + \frac{\pi}{2})\)

২. '-' চিহ্ন নিয়ে:

\(3\theta = 2n\pi - (\frac{\pi}{2} - 2\theta)\)

\(3\theta = 2n\pi - \frac{\pi}{2} + 2\theta\)

\(3\theta - 2\theta = 2n\pi - \frac{\pi}{2}\)

\(\theta = 2n\pi - \frac{\pi}{2}\)

সুতরাং, \(\sin 2\theta = \cos 3\theta\) সমীকরণের সাধারণ মা

( sin 2 theta = cos 3 theta ) সমীকরণের সাধারণ মান নির্ণয় কর ।