Q225 / 270 · NDA 2016 (Phase 1) · Mathematics

What is $\sin \left(\frac{\theta}{2}\right)$ equal to?

  • Correct answer
    A
    $\frac{|\hat{a}-\hat{b}|}{2}$
  • B
    $\frac{|\hat{\mathrm{a}}+\hat{\mathrm{b}}|}{2}$
  • C
    $\frac{|\hat{a}-\hat{b}|}{4}$
  • D
    $\frac{|\hat{a}+\hat{b}|}{4}$

Solution

$\begin{aligned}|\hat{a}-\hat{b}|^2 & =(\hat{a}-\hat{b}) \cdot(\hat{a}-\hat{b}) \\ & =\hat{a} \cdot \hat{a}-\hat{b} \cdot \hat{a}-\hat{a} \cdot \hat{b}+\hat{b} \cdot \hat{b} \\ & =|\hat{a}|^2-2|\hat{a}||\hat{b}| \cos \theta+|\hat{b}|^2 \\ & =2-2 \cos \theta \\ & =2(1-\cos \theta) \\ |\hat{a}-\hat{b}|^2 & =2 \cdot 2 \sin ^2 \frac{\theta}{2} \\ \sin \frac{\theta}{2}= & \frac{|\hat{a}-\hat{b}|}{2}\end{aligned}$

Hence, the correct answer is option: 'A'.

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