Which of the following numbers is/are rational? - Vidyadip

MCQ

Which of the following numbers is/are rational?

A
$\sin 15^{\circ}$
B
$\cos 15^{\circ}$
✓
$\sin 15^{\circ} \cos 15^{\circ}$
D
$\sin 15^{\circ} \cos 75^{\circ}$

✓

Answer

Correct option: C.

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

(C)
$\sin 15^{\circ}=\sin \left(45^{\circ}-30^{\circ}\right)=\frac{\sqrt{3}-1}{2 \sqrt{2}}=$ irrational
$\cos 15^{\circ}=\cos \left(45^{\circ}-30^{\circ}\right)=\frac{\sqrt{3}+1}{2 \sqrt{2}}=$ irrational
$\sin 15^{\circ} \cos 15^{\circ}=\frac{1}{2}\left(2 \sin 15^{\circ} \cos 15^{\circ}\right)$
$=\frac{1}{2} \sin 30^{\circ}=\frac{1}{2} \cdot \frac{1}{2}$
$=\frac{1}{4}=$ rational
$\sin 15^{\circ} \cos 75^{\circ}=\sin 15^{\circ} \sin 15^{\circ}$
$=\sin ^2 15^{\circ}$
$=\left(\frac{\sqrt{3}-1}{2 \sqrt{2}}\right)^2$
$=\frac{4-2 \sqrt{3}}{8}=$ irrational

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