Question:
If \(\tan A = 3\), find the values of \(\sin A\) and \(\cos A\).
यदि \(\tan A = 3\) है, तो \(\sin A\) और \(\cos A\) के मान ज्ञात कीजिए।
Options:
- \(\sin A = \dfrac{3}{\sqrt{10}}, \; \cos A = \dfrac{1}{\sqrt{10}}\)
- \(\sin A = \dfrac{1}{\sqrt{10}}, \; \cos A = \dfrac{3}{\sqrt{10}}\)
- \(\sin A = \dfrac{3}{5}, \; \cos A = \dfrac{4}{5}\)
- \(\sin A = \dfrac{4}{5}, \; \cos A = \dfrac{3}{5}\)
- \(\sin A = \dfrac{3}{\sqrt{10}}, \; \cos A = \dfrac{1}{\sqrt{10}}\)
- \(\sin A = \dfrac{1}{\sqrt{10}}, \; \cos A = \dfrac{3}{\sqrt{10}}\)
- \(\sin A = \dfrac{3}{5}, \; \cos A = \dfrac{4}{5}\)
- \(\sin A = \dfrac{4}{5}, \; \cos A = \dfrac{3}{5}\)
Correct Answer: a)
Explanation:
Since \(\tan A = \dfrac{\text{Perpendicular}}{\text{Base}} = \dfrac{3}{1}\), by Pythagoras theorem, \(\text{Hypotenuse} = \sqrt{3^2 + 1^2} = \sqrt{10}\).
Therefore, \(\sin A = \dfrac{3}{\sqrt{10}}\) and \(\cos A = \dfrac{1}{\sqrt{10}}\).
\(\tan A = \dfrac{\text{लम्ब}}{\text{आधार}} = \dfrac{3}{1}\) है। पाइथागोरस प्रमेय से, \(\text{कर्ण} = \sqrt{3^2 + 1^2} = \sqrt{10}\)।
अतः \(\sin A = \dfrac{3}{\sqrt{10}}\) तथा \(\cos A = \dfrac{1}{\sqrt{10}}\)। विकल्प (a) सही है।