Thus, the number of distinct real roots is \(\boxed{3}\).Question: Compute $\tan 30^\circ$ in a right triangle with an opposite side of length $1$ and adjacent side of length $\sqrt{3}$.

["How to Compute $\ an 30^\circ$: A Step-by-Step Guide Using a Right Triangle", "Understanding trigonometric ratios in a right triangle is fundamental to mastering angles and their relationships. One of the most common values in trigonometry is $\ an 30^\circ$, commonly found in geometry and calculus. But how exactly do we compute $\ an 30^\circ$ using a right triangle?", "---", "### What is $\ an$ in a Right Triangle?", "The tangent of an angle in a right triangle is defined as the ratio of the length of the opposite side to the adjacent side:", "$$\n\ an \ heta = \frac{\ ext{opposite}}{\ ext{adjacent}}\n$$", "Applied to a $30^\circ$ angle, this formula helps calculate the tangent directly from triangle side lengths.", "---", "### Using the Given Triangle with Opposite Side = 1 and Adjacent Side = $\sqrt{3}$", "Imagine a right triangle where:\n- The side opposite the $30^\circ$ angle measures $1$ unit\n- The side adjacent to the $30^\circ$ angle measures $\sqrt{3}$ units", "Using the definition of tangent:", "$$\n\ an 30^\circ = \frac{\ ext{opposite}}{\ ext{adjacent}} = \frac{1}{\sqrt{3}}\n$$", "However, this fraction can be rationalized for standard representation:", "$$\n\ an 30^\circ = \frac{1}{\sqrt{3}} = \frac{\sqrt{3}}{3}\n$$", "---", "### Known Value and Verification", "It is a well-established result that:", "$$\n\ an 30^\circ = \frac{\sqrt{3}}{3}\n$$", "This matches our calculation and confirms the value. This ratio appears frequently in physics, engineering, and mathematics problems involving $30^\circ$ angles.", "---", "### Practical Tip", "Remember: When angles are $30^\circ$ and $60^\circ$, the tangent values often simplify nicely—especially using the sides in a $1 : \sqrt{3} : 2$ right triangle, where $\ an 30^\circ = \frac{1}{\sqrt{3}}$ and $\ an 60^\circ = \sqrt{3}$.", "---", "### Conclusion", "So, using a right triangle with an opposite side of $1$ and adjacent side of $\sqrt{3}$, we compute:", "$$\n\boxed{\ an 30^\circ = \frac{\sqrt{3}}{3}}\n$$", "This confirms the identity and provides a clear, visual way to understand how trigonometric functions emerge from geometric relationships."]