\( \sin A=\frac{1}{2} \) এবং \( \cos B=\frac{1}{\sqrt{3}} \) হলে \( \tan A \tan B \) এর মান কত?

প্রশ্ন: \( \sin A = \frac{1}{2} \) এবং \( \cos B = \frac{1}{\sqrt{3}} \) হলে \( \tan A \times \tan B \) এর মান কত?

সমাধান:

প্রথমে, \( \sin A = \frac{1}{2} \) থেকে \( \tan A \) নির্ণয় করি।

\( \sin A = \frac{1}{2} \)

তাই, \( \cos A = \sqrt{1 - \sin^2 A} = \sqrt{1 - \left(\frac{1}{2}\right)^2} = \sqrt{1 - \frac{1}{4}} = \sqrt{\frac{3}{4}} = \frac{\sqrt{3}}{2} \)

অতএব,

\( \tan A = \frac{\sin A}{\cos A} = \frac{\frac{1}{2}}{\frac{\sqrt{3}}{2}} = \frac{1}{2} \times \frac{2}{\sqrt{3}} = \frac{1}{\sqrt{3}} \)

এখন, \( \cos B = \frac{1}{\sqrt{3}} \) থেকে \( \sin B \) নির্ণয় করি।

\( \sin B = \sqrt{1 - \cos^2 B} = \sqrt{1 - \left(\frac{1}{\sqrt{3}}\right)^2} = \sqrt{1 - \frac{1}{3}} = \sqrt{\frac{2}{3}} \)

অতএব,

\( \tan B = \frac{\sin B}{\cos B} = \frac{\sqrt{\frac{2}{3}}}{\frac{1}{\sqrt{3}}} = \sqrt{\frac{2}{3}} \times \sqrt{3} = \sqrt{\frac{2}{3} \times 3} = \sqrt{2} \)

অতএব,

\( \tan A \times \tan B = \frac{1}{\sqrt{3}} \times \sqrt{2} = \frac{\sqrt{2}}{\sqrt{3}} = \sqrt{\frac{2}{3}} \)

উত্তর: \( \boxed{\sqrt{\frac{2}{3}}} \)