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Trigonometric Functions — Maths STD 11 Science — Question

CBSE BoardEnglish MediumSTD 11 ScienceMathsTrigonometric Functions1 Mark
MCQ
In any $\triangle\text{ABC},$ the value of $2\text{ac}\sin\Big(\frac{\text{A}-\text{B + C}}{2}\Big)$ is:
  • A
    $\text{a}^2+\text{b}^2-\text{c}^2$
  • ✓
    $\text{c}^2+\text{a}^2-\text{b}^2$
  • C
    $\text{b}^2-\text{c}^2-\text{a}^2$
  • D
    $\text{c}^2-\text{a}^2-\text{b}^2$
✓

Answer

Correct option: B.
$\text{c}^2+\text{a}^2-\text{b}^2$
In $\triangle\text{ABC},$
$\text{A + B + C}=\pi ($Angle sum property$)$
$\Rightarrow\text{A + C}=\pi-\text{B}$
$\therefore2\text{ac}\sin\Big(\frac{\text{A}-\text{B + C}}{2}\Big)$
$=2\text{ac}\sin\Big(\frac{\pi-2\text{B}}{2}\Big)$
$=2\text{ac}\sin\Big(\frac{\pi}{2}-\text{B}\Big)$
$=2\text{ac}\cos\text{B}$
$=2\text{ac}\Big(\frac{\text{c}^2+\text{a}^2-\text{b}^2}{2\text{ca}}\Big) ($Using cosine rule$)$
$=\text{c}^2+\text{a}^2-\text{b}^2$
Hence, the correct answer is option $(b).$

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