Find the value of $\sin (15^{0})$ .

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\frac{\sqrt{3} - 1}{2 \sqrt{2}}

\frac{\sqrt{3} + 1}{2 \sqrt{2}}

\frac{\sqrt{3} - 1}{\sqrt{2}}

None of these

answer is A.

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Detailed Solution

Given,

\sin 15^{\circ}

It can be written as,

\sin 15^{\circ}

= \sin (45^{\circ} - 30^{\circ})

= \sin 45^{\circ} \cos 30^{\circ} - \sin 30^{\circ} \cos 45^{\circ} [∵ \sin (θ_{1} - θ_{2}) = \sin θ_{1} \cos θ_{2} - \sin θ_{1} \cos θ_{1}]

= \frac{1}{\sqrt{2}} \cdot \frac{\sqrt{3}}{2} - \frac{1}{2} \times \frac{1}{\sqrt{2}}

(since,

\sin 45^{o} = \frac{1}{\sqrt{2}}, \sin 30^{o} = \frac{1}{2}, \cos 45 = \frac{1}{\sqrt{2}}, \cos 30 = \frac{\sqrt{3}}{2}

)

= \frac{\sqrt{3}}{2 \sqrt{2}} - \frac{1}{2 \sqrt{2}}

= \frac{\sqrt{3} - 1}{2 \sqrt{2}}

Thus, the required value

\sin 15^{\circ} = \frac{\sqrt{3} - 1}{2 \sqrt{2}}

.
Correct option is 1.

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