Solution: Using $\tan(60^\circ) = \frac{\text{height}}{\text{shadow length}}$, we substitute $\tan(60^\circ) = \sqrt{3}$ and shadow length $5\,\text{cm}$. Solving for height: $\text{height} = 5 \cdot \sqrt{3}$. Thus, $\boxed{5\sqrt{3}}$.

["Mastering Right Triangle Trigonometry: Calculating Height Using $\ an(60^\circ) = \sqrt{3}$", "In solving real-world problems involving angles and shadows, trigonometry offers powerful tools—especially the tangent function. One elegant and practical application is determining the height of an object using known shadow length and angle measurement. This article explains how to use the tangent of $60^\circ$, a key trigonometric ratio, to find height efficiently.", "### The Core Formula: $\ an(\ heta) = \frac{\ ext{height}}{\ ext{shadow length}}$", "When sunlight casts a shadow from a vertical object—such as a tree, flagpole, or building—the angle of elevation of the sun creates a right triangle. The tangent of the elevation angle relates the height of the object to the shadow’s length on the ground. Mathematically:", "$$\n\ an(\ heta) = \frac{\ ext{height}}{\ ext{shadow length}}\n$$", "In this scenario, $\ heta = 60^\circ$, and the shadow length measures $5,\ ext{cm}$. Since $\ an(60^\circ) = \sqrt{3}$, the formula becomes:", "$$\n\sqrt{3} = \frac{\ ext{height}}{5}\n$$", "### Solving for Height", "Rearranging the equation allows us to isolate and compute the height:", "$$\n\ ext{height} = 5 \cdot \sqrt{3}\n$$", "Thus, the exact height of the object is $\boxed{5\sqrt{3},\ ext{cm}}$.", "### Why This Matters in Practical Applications", "Understanding how to apply trigonometric ratios like tangent enables accurate measurements without direct physical assessment. This is particularly valuable in fields such as surveying, architecture, photography, and outdoor navigation. For example, sunlight angles and shadows help estimate tree heights, roof pitches, or solar panel efficiency—all modeled through right triangles.", "Using exact values like $5\sqrt{3}$ ensures precision in calculations and fosters deeper geometric intuition. Whether you're a student learning trigonometry or a professional solving real-world problems, mastering the tangent function is essential.", "---", "Key takeaway: When facing angle and shadow problems, apply $\ an(\ heta) = \frac{\ ext{opposite}}{\ ext{adjacent}}$, substitute known values, and solve directly—simple yet powerful."]