To solve \(\tan 45^\circ + \tan 30^\circ\), we first find the values of \(\tan 45^\circ\) and \(\tan 30^\circ\).

["# Solve (\ an 45^\circ + \ an 30^\circ): A Step-by-Step Guide", "Understanding trigonometric functions is essential for mastering mathematics, especially in geometry and physics. One commonly encountered problem is calculating the sum (\ an 45^\circ + \ an 30^\circ). This article explains how to solve this expression by first determining the exact values of each tangent function, and then simplifying the result.", "## Understanding the Tangent Function", "The tangent of an angle in a right triangle is defined as the ratio of the opposite side to the adjacent side. However, for specific standard angles like (45^\circ) and (30^\circ), these values are well-known and can be derived from special triangles or unit circle concepts.", "## Step 1: Evaluate (\ an 45^\circ)", "The angle (45^\circ) appears frequently in right triangles—most commonly in the 45°–45°–90° triangle, which is an isosceles right triangle with two equal legs.", "Let the legs of this triangle be of length 1. Then, by definition:", "[\n\ an 45^\circ = \frac{\ ext{opposite}}{\ ext{adjacent}} = \frac{1}{1} = 1\n]", "## Step 2: Evaluate (\ an 30^\circ)", "The angle (30^\circ) corresponds to a special triangle composed of a 30°–60°–90° right triangle. In such a triangle:", "- The side opposite (30^\circ) is half the hypotenuse.\n- The side opposite (60^\circ) is (\sqrt{3}) times the shorter leg.", "For a triangle with hypotenuse = 2, the side opposite (30^\circ) is 1, and the side opposite (60^\circ) (adjacent to (30^\circ)) is (\sqrt{3}).", "Thus,", "[\n\ an 30^\circ = \frac{\ ext{opposite}}{\ ext{adjacent}} = \frac{1}{\sqrt{3}} = \frac{\sqrt{3}}{3} \quad \ ext{(after rationalizing the denominator)}\n]", "## Step 3: Add the Two Values", "Now, we compute:", "[\n\ an 45^\circ + \ an 30^\circ = 1 + \frac{\sqrt{3}}{3}\n]", "To combine the terms, express 1 as a fraction with denominator 3:", "[\n1 = \frac{3}{3}\n]", "So,", "[\n\ an 45^\circ + \ an 30^\circ = \frac{3}{3} + \frac{\sqrt{3}}{3} = \frac{3 + \sqrt{3}}{3}\n]", "## Final Answer", "[\n\boxed{\ an 45^\circ + \ an 30^\circ = \frac{3 + \sqrt{3}}{3}}\n]", "## Why This Matters", "Understanding how to evaluate standard trigonometric values is crucial for solving real-world problems involving angles, such as in navigation, engineering, and physics. Moreover, learning to simplify expressions like (\frac{3 + \sqrt{3}}{3}) strengthens algebraic and mathematical reasoning skills.", "---", "### Key Takeaways", "- (\ an 45^\circ = 1)\n- (\ an 30^\circ = \frac{\sqrt{3}}{3})\n- Sum: (\ an 45^\circ + \ an 30^\circ = \frac{3 + \sqrt{3}}{3})\n- Practice identifying standard triangle ratios to solve trigonometric expressions efficiently", "Continue building your trigonometry knowledge by exploring identities and applications—mastering these basics opens the door to advanced mathematics.", "---", "By breaking down (\ an 45^\circ + \ an 30^\circ) into known values and simplifying step-by-step, anyone can confidently solve this and similar problems. Start with the fundamentals, practice regularly, and watch your understanding grow!"]