Which of the following number(s) is/are ratio | vedclass

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(C) We evaluate each option:$A) \sin 15^\circ = \sin (45^\circ - 30^\circ) = \sin 45^\circ \cos 30^\circ - \cos 45^\circ \sin 30^\circ = \frac{1}{\sqr

Vedclass · Accepted Answer

$\sin 15^\circ \cos 15^\circ$

Vedclass · Answer

(C) We evaluate each option:$A) \sin 15^\circ = \sin (45^\circ - 30^\circ) = \sin 45^\circ \cos 30^\circ - \cos 45^\circ \sin 30^\circ = \frac{1}{\sqrt{2}} \cdot \frac{\sqrt{3}}{2} - \frac{1}{\sqrt{2}} \cdot \frac{1}{2} = \frac{\sqrt{3} - 1}{2\sqrt{2}}$,which is irrational.$B) \cos 15^\circ = \cos (45^\circ - 30^\circ) = \cos 45^\circ \cos 30^\circ + \sin 45^\circ \sin 30^\circ = \frac{1}{\sqrt{2}} \cdot \frac{\sqrt{3}}{2} + \frac{1}{\sqrt{2}} \cdot \frac{1}{2} = \frac{\sqrt{3} + 1}{2\sqrt{2}}$,which is irrational.$C) \sin 15^\circ \cos 15^\circ = \frac{1}{2} (2 \sin 15^\circ \cos 15^\circ) = \frac{1}{2} \sin (2 	imes 15^\circ) = \frac{1}{2} \sin 30^\circ = \frac{1}{2} \cdot \frac{1}{2} = \frac{1}{4}$,which is rational.$D) \sin 15^\circ \cos 75^\circ = \sin 15^\circ \sin (90^\circ - 75^\circ) = \sin 15^\circ \sin 15^\circ = \sin^2 15^\circ = \left( \frac{\sqrt{3} - 1}{2\sqrt{2}} ight)^2 = \frac{3 + 1 - 2\sqrt{3}}{8} = \frac{4 - 2\sqrt{3}}{8} = \frac{2 - \sqrt{3}}{4}$,which is irrational.Thus,the correct option is $C$.

Which of the following number$(s)$ is/are rational?

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