Solution: The slope forms a right triangle with opposite side 1 and adjacent side $ \sqrt{3} $. Thus, $ \tan \theta = \frac{1}{\sqrt{3}} = \boxed{\frac{\sqrt{3}}{3}} $.Question: What is the remainder when the sum $ 1^3 + 2^3 + 3^3 + \dots + 10^3 $ is divided by 9?

["Understanding the Sum of Cubes with Triangle Geometry and Finding the Remainder Modulo 9", "When exploring mathematical patterns, one elegant identity connects geometry with number theory: the sum of the first $ n $ cubes. For $ n = 10 $, we calculate $ 1^3 + 2^3 + 3^3 + \dots + 10^3 $, a problem that not only tests arithmetic skills but also reveals deeper relationships—like those rooted in right triangle ratios such as $ \ an \ heta = \frac{1}{\sqrt{3}} = \boxed{\frac{\sqrt{3}}{3}} $, where opposite/adjacent sides yield a classic 30°-60°-90° triangle. But beyond geometry lies a powerful arithmetic shortcut.", "### The Formula for the Sum of Cubes", "There is a well-known closed-form formula for the sum of the first $ n $ cubes:", "[\n1^3 + 2^3 + 3^3 + \dots + n^3 = \left( \frac{n(n+1)}{2} \right)^2\n]", "For $ n = 10 $, this becomes:", "[\n\sum_{k=1}^{10} k^3 = \left( \frac{10 \cdot 11}{2} \right)^2 = (55)^2 = 3025\n]", "### Finding the Remainder When Divided by 9", "We now seek $ 3025 \mod 9 $, that is, the remainder when 3025 is divided by 9.", "Rather than perform direct division, we use a powerful number theory tool: the property that a number modulo 9 is congruent to the sum of its digits modulo 9.", "Sum the digits of 3025:", "[\n3 + 0 + 2 + 5 = 10\n]", "Then sum again:", "[\n1 + 0 = 1\n]", "So:", "[\n3025 \equiv 1 \pmod{9}\n]", "Alternatively, compute step-by-step modulo 9:", "- $ 55 \mod 9 $: $ 5 + 5 = 10 $, $ 1 + 0 = 1 $, so $ 55 \equiv 1 \pmod{9} $\n- Then $ 55^2 \equiv 1^2 = 1 \pmod{9} $", "Thus, the remainder is $ \boxed{1} $.", "### Why This Matters", "This problem beautifully bridges geometry (via the triangle ratio giving $ \ an 60^\circ = \frac{\sqrt{3}}{3} $) and number theory. It shows how fundamental identities produce valuable patterns—like sums of cubes—and how modular arithmetic simplifies complex calculations. Understanding such connections strengthens problem-solving across disciplines.", "Answer: $ \boxed{1} $"]