Q108 / 200 · Manipal (MET) 2012 · Mathematics

The value of $\sin \left(22^{\circ} 30^{\prime}\right)$ is

  • A
    $\sqrt{\frac{\sqrt{2}+1}{2 \sqrt{2}}}$
  • B
    $\sqrt{\sqrt{2}-1}$
  • Correct answer
    C
    $\sqrt{\frac{\sqrt{2}-1}{2 \sqrt{2}}}$
  • D
    None of these

Solution

We know, $\begin{aligned} \sin ^2 A & =\left(\frac{1-\cos 2 A}{2}\right) \\ \Rightarrow \quad \sin ^2\left(22^{\circ} 30^{\prime}\right) & =\frac{1-\cos 45^{\circ}}{2} \\ & =\frac{\left(1-\frac{1}{\sqrt{2}}\right)}{2} \\ & =\frac{\sqrt{2}-1}{2 \sqrt{2}} \\ \Rightarrow \quad \sin \left(22^{\circ} 30^{\prime}\right) & =\sqrt{\frac{\sqrt{2}-1}{2 \sqrt{2}}} \end{aligned}$

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

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