Trigonometrical Ratios and Identities 1 Question 11
11. Which of the following numbers is rational?
$(1998,2 M)$
(a) $\sin 15^{\circ}$
(b) $\cos 15^{\circ}$
(c) $\sin 15^{\circ} \cos 15^{\circ}$
(d) $\sin 15^{\circ} \cos 75^{\circ}$
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Answer:
Correct Answer: 11. (b)
Solution:
- Since, $\sin 15^{\circ}=\frac{1}{2} \sqrt{2-\sqrt{3}}$ and $\cos 15^{\circ}=\frac{1}{2} \sqrt{2+\sqrt{3}}$ and $\quad \sin 15^{\circ} \cos 75^{\circ}=\sin 15^{\circ} \cdot \sin 15^{\circ}=\frac{1}{4}(2-\sqrt{3})$
Therefore, all these values are irrational and
$$ \begin{aligned} \sin 15^{\circ} \cos 15^{\circ} & =\frac{1}{2} \cdot 2 \sin 15^{\circ} \cos 15^{\circ} \\ & =\frac{1}{2} \cdot \sin 30^{\circ}=\frac{1}{4}, \text { which is rational. } \end{aligned} $$
