Question

Which of the following is a rational number ?

• A. $\sin {15}^{\circ }$
• B. $\cos {15}^{\circ }$
• C. $\sin {15}^{\circ }.\cos {15}^{\circ }$
• D. $\sin {15}^{\circ }.\cos {75}^{\circ }$

Answer 1

The correct option is C $\sin {15}^{\circ }.\cos {15}^{\circ }$

$A.$ $\cos {30}^{}=1-2{\sin }^2{15}^{}$

$\Rightarrow {\sin }^2{15}^{}=\frac{2-\sqrt{3}}{4}$

$\Rightarrow \sin {15}^{}=\frac{\sqrt{3}-1}{2\sqrt{2}}$
$B.$ $\cos {30}^{}=2{\cos }^2{15}^{}-1$

$\Rightarrow {\cos }^2{15}^{}=\frac{\sqrt{3}+1}{2\sqrt{2}}$

$\Rightarrow \cos {15}^{}=\frac{\sqrt{3}+1}{2\sqrt{2}}$
$C.$ $\cos {15}^{}\sin {15}^{}=\frac{1}{2}\sin {30}^{}=\frac{1}{4}$
$D.$ $\sin {15}^{}\cos {75}^{}={\sin }^2{15}^{}=\frac{1-\cos {30}^{}}{2}=\frac{2-\sqrt{3}}{4}$

Hence, option 'C' is correct.