Question

If $x\tan {30}^{\circ }=\frac{\sin {30}^{\circ }+\cos {60}^{\circ }}{\tan {60}^{\circ }+\sin {60}^{\circ }}$, then the value of $x$ is:

• A. $\frac{2}{3}$
• B. $\frac{2}{\sqrt{3}}$
• C. $\frac{2}{3\sqrt{3}}$
• D. $\frac{3}{2}$

Answer 1

Answer: (A)

$\frac{2}{3}$

$x\tan {30}^{\circ }=\frac{\sin {30}^{\circ }+\cos {60}^{\circ }}{\tan {60}^{\circ }+\sin {60}^{\circ }}$

$=\frac{\frac{1}{2}+\frac{1}{2}}{\sqrt{3}+\frac{\sqrt{3}}{2}}$

$=\frac{1}{\frac{2\sqrt{3}+\sqrt{3}}{2}}$

$=\frac{2}{3\sqrt{3}}$

$\Rightarrow x\times \frac{1}{\sqrt{3}}=\frac{2}{3\sqrt{3}}$

$\Rightarrow x=\frac{2\sqrt{3}}{3\sqrt{3}}=\frac{2}{3}.$