Question
Download Solution PDF\(\triangle\)ABC, B पर एक समकोण त्रिभुज है और tan A = \(\frac{3}{4}\), तो sin A + sin B + sin C का मान क्या होगा?
Answer (Detailed Solution Below)
Detailed Solution
Download Solution PDFदिया गया है:
ΔABC, B पर एक समकोण त्रिभुज है और tan A = \(\frac{3}{4}\)
प्रयुक्त अवधारणा:
1. tan θ = ऊँचाई ÷ आधार
2. sin θ = ऊँचाई ÷ कर्ण
3. पाइथागोरस प्रमेय के अनुसार "एक समकोण त्रिभुज में, कर्ण भुजा का वर्ग अन्य दो भुजाओं के वर्गों के योगफल के बराबर होता है।"
गणना:
tan A = \(\frac{3}{4}\)
⇒ \(\frac {BC}{AB}\) = \(\frac{3}{4}\)
⇒ BC : AB = 3 : 4
माना BC और AB की माप क्रमशः 3k और 4k है।
अवधारणा के अनुसार,
AC2 = \(\sqrt { (3k)^2 + (4k)^2 }\) = 5k
अब, sin A + sin B + sin C
⇒ \(\frac {BC}{AC} + sin\ 90^\circ + \frac {AB}{AC}\)
⇒ \(\frac {BC + AB}{AC} + 1\)
⇒ \(\frac {3k + 4k}{5k} + 1\)
⇒ \(\frac {7}{5} + 1\)
⇒ \(\frac {12}{5}\) = \(2\frac{2}{5}\)
∴ sin A + sin B + sin C का मान \(2\frac{2}{5}\) है।
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