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

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

1. A

$\mathrm{sin}{15}^{\circ }$

2. B

$\mathrm{cos}{15}^{\circ }$

3. C

$\mathrm{sin}{15}^{\circ }\mathrm{cos}{15}^{\circ }$

4. D

$\mathrm{sin}{15}^{\circ }\mathrm{cos}{75}^{\circ }$

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### Solution:

we have,

$\mathrm{sin}{15}^{\circ }=\mathrm{sin}\left({45}^{\circ }-{30}^{\circ }\right)=\frac{\sqrt{3}-1}{2\sqrt{2}}$

which is an irrational number

$\mathrm{cos}{15}^{\circ }=\mathrm{cos}\left({45}^{\circ }-{30}^{\circ }\right)=-\frac{\sqrt{3}+1}{2\sqrt{2}}$

which is an irrational number

$\mathrm{sin}{15}^{\circ }\mathrm{cos}{15}^{\circ }=\frac{1}{2}\mathrm{sin}{30}^{\circ }=\frac{1}{4}$which is a rational number

$sin{15}^{\circ }cos{75}^{\circ }=si{n}^{2}{15}^{\circ }=\frac{1}{2}\left(1-cos{30}^{\circ }\right)=\frac{2-\sqrt{3}}{4}$ which is an irrational number

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