# Mark the Correct Alternative in Each of the Following: in Any ∆Abc, the Value of 2 a C Sin ( a − B + C 2 ) is - Mathematics

MCQ

Mark the correct alternative in each of the following:

In any ∆ABC, the value of  $2ac\sin\left( \frac{A - B + C}{2} \right)$  is

#### Options

• $a^2 + b^2 - c^2$

• $c^2 + a^2 - b^2$

• $b^2 - c^2 - a^2$

• $c^2 - a^2 - b^2$

#### Solution

In ∆ABC,

$A + B + C = \pi \left( \text{ Angle sum property } \right)$
$\Rightarrow A + C = \pi - B$

$\therefore 2ac\sin\left( \frac{A - B + C}{2} \right)$
$= 2ac\sin\left( \frac{\pi - 2B}{2} \right)$
$= 2ac\sin\left( \frac{\pi}{2} - B \right)$
$= 2ac\cos B$

$= 2ac\left( \frac{c^2 + a^2 - b^2}{2ca} \right) \left( \text{ Using cosine rule } \right)$
$= c^2 + a^2 - b^2$

Hence, the correct answer is option (b).

Concept: Sine and Cosine Formulae and Their Applications
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#### APPEARS IN

RD Sharma Class 11 Mathematics Textbook
Chapter 10 Sine and cosine formulae and their applications
Q 7 | Page 27