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Question

Question: Which of the following number is rational...

Which of the following number is rational

A

sin15o\sin 15^{o}

B

cos15o\cos 15^{o}

C

sin15o.cos15o\sin 15^{o}.\cos 15^{o}

D

sin15o.cos75o\sin 15^{o}.\cos 75^{o}

Answer

sin15o.cos15o\sin 15^{o}.\cos 15^{o}

Explanation

Solution

sin15o\sin 15^{o} = sin(45o30o)\sin(45^{o} - 30^{o}) = 3122\frac{\sqrt{3} - 1}{2\sqrt{2}} = irrational

cos15o=cos(45o30o)=3+122\cos 15^{o} = \cos(45^{o} - 30^{o}) = \frac{\sqrt{3} + 1}{2\sqrt{2}} = irrational

sin15o.cos15o=12(2sin15ocos15o)=12sin30o=14\sin 15^{o}.\cos 15^{o} = \frac{1}{2}(2\sin 15^{o}\cos 15^{o}) = \frac{1}{2}\sin 30^{o} = \frac{1}{4}= rational

sin15o.cos75o=sin15o.sin15o=sin215o\sin 15^{o}.\cos 75^{o} = \sin 15^{o}.\sin 15^{o} = \sin^{2}15^{o}

= (3122)2=4238=\left( \frac{\sqrt{3} - 1}{2\sqrt{2}} \right)^{2} = \frac{4 - 2\sqrt{3}}{8} = irrational.