Question

Which of the following is/are irrational numbers ?

• A. $\sin {15}^{\circ }$
• B. $\cos {15}^{\circ }$
• C. $\sin {15}^{\circ }\cos {15}^{\circ }$
• D. $\sec {15}^{\circ }+\text{cosec}{15}^{\circ }$

Answer 1

Answer: (A), (B), (D)

The correct options are
A $\sin {15}^{\circ }$
B $\cos {15}^{\circ }$
D $\sec {15}^{\circ }+\text{cosec}{15}^{\circ }$

$\sin {15}^{\circ }=\sin ({45}^{\circ }-{30}^{\circ })=\sin {45}^{\circ }\cos {30}^{\circ }-\cos {45}^{\circ }\sin {30}^{\circ }=\frac{\sqrt{3}-1}{2\sqrt{2}}$
Which is irrational number.

$\cos {15}^{\circ }=\cos ({45}^{\circ }-{30}^{\circ })=\cos {45}^{\circ }\cos {30}^{\circ }+\sin {45}^{\circ }\sin {30}^{\circ }=\frac{\sqrt{3}+1}{2\sqrt{2}}$
Which is irrational number.

$\sin {15}^{\circ }\cos {15}^{\circ }=\frac{\sqrt{3}-1}{2\sqrt{2}}\times \frac{\sqrt{3}+1}{2\sqrt{2}}=\frac{1}{4}$
Which is rational number.

$\sec {15}^{\circ }+\text{cosec}{15}^{\circ }=\frac{\sin {15}^{\circ }+\cos {15}^{\circ }}{\sin {15}^{\circ }\cos {15}^{\circ }}=\frac{\frac{\sqrt{3}-1}{2\sqrt{2}}+\frac{\sqrt{3}+1}{2\sqrt{2}}}{\frac{1}{4}}=\sqrt{2}\times 2\sqrt{3}=2\sqrt{6}$
Which is irrational.