We are asked to simplify the expression $\frac{2 \tan 60^{\circ} + \cos 30^{\circ}}{\sin 60^{\circ}}$ without using mathematical tables or calculators.

1. Problem Description

We are asked to simplify the expression $\frac{2 \tan 60^{\circ} + \cos 30^{\circ}}{\sin 60^{\circ}}$ without using mathematical tables or calculators.

2. Solution Steps

First, we need to recall the values of $\tan 60^{\circ}$, $\cos 30^{\circ}$, and $\sin 60^{\circ}$.

$\tan 60^{\circ} = \sqrt{3}$

$\cos 30^{\circ} = \frac{\sqrt{3}}{2}$

$\sin 60^{\circ} = \frac{\sqrt{3}}{2}$

Now, we substitute these values into the expression:

$\frac{2 \tan 60^{\circ} + \cos 30^{\circ}}{\sin 60^{\circ}} = \frac{2(\sqrt{3}) + \frac{\sqrt{3}}{2}}{\frac{\sqrt{3}}{2}}$

We can simplify the numerator:

$2\sqrt{3} + \frac{\sqrt{3}}{2} = \frac{4\sqrt{3}}{2} + \frac{\sqrt{3}}{2} = \frac{5\sqrt{3}}{2}$

Now, we substitute this back into the expression:

$\frac{\frac{5\sqrt{3}}{2}}{\frac{\sqrt{3}}{2}}$

To divide fractions, we multiply by the reciprocal:

$\frac{5\sqrt{3}}{2} \cdot \frac{2}{\sqrt{3}} = \frac{5\sqrt{3} \cdot 2}{2 \cdot \sqrt{3}} = \frac{10\sqrt{3}}{2\sqrt{3}}$

The $\sqrt{3}$ terms cancel out, and the $2$s cancel out as well, leaving us with:

$\frac{10}{2} = 5$

3. Final Answer

5