এর মান কোনটি?

প্রশ্ন: \( \sin\left(\tan^{-1}\left(\frac{1}{2}\right) + \cot^{-1}(3)\right) \) এর মান কোনটি?

সমাধান:

ধরি, \( \tan^{-1}\left(\frac{1}{2}\right) = A \) এবং \( \cot^{-1}(3) = B \)

তাহলে, \( \tan A = \frac{1}{2} \) এবং \( \cot B = 3 \Rightarrow \tan B = \frac{1}{3} \)

সুতরাং, \( \sin(A + B) = \sin A \cos B + \cos A \sin B \)

আমরা জানি, \( \tan A = \frac{1}{2} \) হলে, \( \sin A = \frac{1}{\sqrt{5}} \) এবং \( \cos A = \frac{2}{\sqrt{5}} \)

আবার, \( \tan B = \frac{1}{3} \) হলে, \( \sin B = \frac{1}{\sqrt{10}} \) এবং \( \cos B = \frac{3}{\sqrt{10}} \)

অতএব,

\( \sin(A + B) = \frac{1}{\sqrt{5}} \cdot \frac{3}{\sqrt{10}} + \frac{2}{\sqrt{5}} \cdot \frac{1}{\sqrt{10}} \)

\( = \frac{3}{\sqrt{50}} + \frac{2}{\sqrt{50}} \)

\( = \frac{5}{\sqrt{50}} \)

\( = \frac{5}{5\sqrt{2}} \)

\( = \frac{1}{\sqrt{2}} \)

সুতরাং, \( \sin\left(\tan^{-1}\left(\frac{1}{2}\right) + \cot^{-1}(3)\right) = \frac{1}{\sqrt{2}} \)

উত্তর: \( \frac{1}{\sqrt{2}} \)