What is the value of 292 1/2^{} ?

Correct Answer is : 122+21/2 {2+{2}}212+2

Correct Answer is : 122+21/2 {2+{2}}212+2

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UPSC NDA Math Model Paper 10

Question : 54 of 120

Marks: +1, -0

\sin 292 \frac{1}{2}^{\circ}

Solution:

Here,

\sin 292 \frac{1}{2}^{\circ} = \sin \frac{585}{2}^{\circ}

We know,

\cos 2\theta = 1 - 2 \sin^{2} \theta

\Rightarrow \sqrt{\frac{1 - \cos 2\theta}{2}} = \sin \theta

\therefore \sin \frac{585}{2}^{\circ} = \sqrt{\frac{1 - \cos 585^{\circ}}{2}}

= \sqrt{\frac{1 - \cos (360 + 225)}{2}}

= \sqrt{\frac{1 - \cos 225^{\circ}}{2}}

\text{.....}(\cos(360+\theta) = \cos \theta)

= \sqrt{\frac{1 - \cos (180 + 45)^{\circ}}{2}}

= \sqrt{\frac{1 + \cos 45^{\circ}}{2}}

\text{.......}(\because \cos(180+\theta) = -\cos \theta)

= \sqrt{\frac{1 + \frac{1}{\sqrt{2}}}{2}} \text{......}\left(\cos 45 = \frac{1}{\sqrt{2}}\right)

= \sqrt{\frac{\sqrt{2}+1}{2\sqrt{2}} \times \frac{\sqrt{2}}{\sqrt{2}}}

= \sqrt{\frac{\sqrt{2}+2}{4}}

= \frac{1}{2} \sqrt{2+\sqrt{2}}

Hence, option (3) is correct.

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