MCQ
Which of the following number(s) is/are rational
- A$\sin 15^\circ $
- B$cos 15^\circ$
- ✓$\sin 15^\circ \cos 15^\circ $
- D$\sin 15^\circ \cos 75^\circ $
$\cos {15^o} = \cos ({45^o} - {30^o}) = \frac{{\sqrt 3 + 1}}{{2\sqrt 2 }}=$ irrational
$\therefore \,\,\,\sin {15^o}\cos {15^o} = \frac{1}{2}(2\sin {15^o}\cos {15^o})$
$ = \frac{1}{2}\sin {30^o} = \frac{1}{2}.\frac{1}{2} = \frac{1}{4} =$ rational
$\therefore \, \sin {15^o}\cos {75^o} = \sin {15^o}\sin {15^o} = {\sin ^2}{15^o}$
$ = {\left( {\frac{{\sqrt 3 - 1}}{{2\sqrt 2 }}} \right)^2} = \frac{{4 - 2\sqrt 3 }}{8}=$ irrational
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