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Trigonometry – II — Maths STD 11 Science — Question

Maharashtra BoardEnglish MediumSTD 11 ScienceMathsTrigonometry – II2 Marks
MCQ
In a triangle ABC, the value of sin A + sin B + sin C is
  • A
    $4 \sin \frac{A}{2} \sin \frac{B}{2} \sin \frac{ C }{2}$
  • ✓
    $4 \cos \frac{A}{2} \cos \frac{B}{2} \cos \frac{ C }{2}$
  • C
    $4 \cos \frac{A}{2} \sin \frac{B}{2} \sin \frac{ C }{2}$
  • D
    $4 \cos \frac{A}{2} \sin \frac{B}{2} \cos \frac{ C }{2}$
✓

Answer

Correct option: B.
$4 \cos \frac{A}{2} \cos \frac{B}{2} \cos \frac{ C }{2}$
(B)
$\sin A +\sin B +\sin C$
$=2 \sin \frac{A+ B }{2} \cos \frac{A- B }{2}+2 \sin \frac{ C }{2} \cos \frac{ C }{2}$
$=2 \sin \left(\frac{\pi}{2}-\frac{ C }{2}\right) \cos \frac{ A - B }{2}$$+2 \cos \frac{ C }{2} \sin \left(\frac{\pi}{2}-\left(\frac{ A + B }{2}\right)\right)$
$=2 \cos \frac{ C }{2} \cos \frac{A- B }{2}+2 \cos \frac{ C }{2} \cos \frac{A+ B }{2}$
$=2 \cos \frac{ C }{2}\left[\cos \frac{A- B }{2}+\cos \frac{ A + B }{2}\right]$
$=2 \cos \frac{ C }{2}\left(2 \cos \frac{A}{2} \cos \frac{B}{2}\right)$
$=4 \cos \frac{A}{2} \cos \frac{B}{2} \cos \frac{ C }{2}$

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