First slide

Compound angles

Question

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

Easy

A

$\sin 15^{\circ}$
B

$\cos 15^{\circ}$
C

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

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

Solution

we have,
$\sin 15^{\circ} = \sin (45^{\circ} - 30^{\circ}) = \frac{\sqrt{3} - 1}{2 \sqrt{2}}$
$\Rightarrow \sin 15^{\circ} = \frac{1}{4} (\sqrt{6} - \sqrt{2})$ which is an irrational number
$\cos 15^{\circ} = \cos (45^{\circ} - 30^{\circ}) = - \frac{\sqrt{3} + 1}{2 \sqrt{2}}$
$\Rightarrow \cos 15^{\circ} = \frac{1}{4} (\sqrt{6} + \sqrt{2})$ which is an irrational number
$\sin 15^{\circ} \cos 15^{\circ} = \frac{1}{2} \sin 30^{\circ} = \frac{1}{4}$ which is a rational number
$s i n 15^{\circ} c o s 75^{\circ} = s i n^{2} 15^{\circ} = \frac{1}{2} (1 - c o s 30^{\circ}) = \frac{2 - \sqrt{3}}{4}$ which is an irrational number

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