Question

Which one of the following quantities is not rational?

• A. $\frac{1-{\tan }^2{30}^{\circ }}{1+{\tan }^2{30}^{\circ }}$
• B. $4{\cos }^{3}30-3\cos {30}^{\circ }$
• C. $3\sin {30}^{\circ }-4{\sin }^3{30}^{\circ }$
• D. $\frac{2\cot {30}^{\circ }}{{\cot }^2{30}^{\circ }-1}$

Answer 1

Answer: (D)

$\frac{2\cot {30}^{\circ }}{{\cot }^2{30}^{\circ }-1}$

(1) $\frac{1-{\tan }^2{30}^{\circ }}{1+{\tan }^2{30}^{\circ }}$
= $\frac{1-(\frac{1}{\sqrt{3}}{)}^2}{1+(\frac{1}{\sqrt{3}}{)}^2}$
= $\frac{3-1}{3+1}$
= $\frac{1}{2}$
(2) $4{\cos }^{3}30-3\cos {30}^{\circ }$
= $4(\frac{\sqrt{3}}{2}{)}^3-3(\frac{\sqrt{3}}{2})$
= $\frac{3\sqrt{3}}{2}-\frac{3\sqrt{3}}{2}$
= $0$
(3) $3\sin {30}^{\circ }-4{\sin }^3{30}^{\circ }$
= $3\left(\frac{1}{2}\right)-4(\frac{1}{2}{)}^3$
= $\frac{3}{2}-\frac{1}{2}$
= $1$
(4) $\frac{2\cot {30}^{\circ }}{{\cot }^2{30}^{\circ }-1}$
= $\frac{2\sqrt{3}}{(\sqrt{3}{)}^2-1}$
= $\frac{2\sqrt{3}}{2}$
= $\sqrt{3}$
Thus, (4) is not rational.