Thus, the radius of the inscribed circle is \(\boxed{3}\) cm.**Question:** Compute \(\tan 45^\circ + \tan 30^\circ\) and express your answer in simplest form.
["Compute (\ an 45^\circ + \ an 30^\circ): A Step-by-Step Guide", "Understanding trigonometric functions is essential in mathematics, especially when calculating angles and their relationships. One common problem involves computing the sum of tangent values for specific angles—here we focus on (\ an 45^\circ + \ an 30^\circ).", "### Step 1: Recall the exact values of tangent at standard angles", "- (\ an 45^\circ = 1)\n This is a well-known identity derived from a right triangle where both opposite and adjacent sides are equal, resulting in a ratio of 1.", "- (\ an 30^\circ = \frac{1}{\sqrt{3}})\n In a 30-60-90 triangle, the sides are in the ratio (1 : \sqrt{3} : 2), so (\ an 30^\circ = \frac{\ ext{opposite}}{\ ext{adjacent}} = \frac{1}{\sqrt{3}}).", "### Step 2: Add the values together", "[\n\ an 45^\circ + \ an 30^\circ = 1 + \frac{1}{\sqrt{3}}\n]", "To simplify, express 1 as (\frac{\sqrt{3}}{\sqrt{3}}):", "[\n1 + \frac{1}{\sqrt{3}} = \frac{\sqrt{3}}{\sqrt{3}} + \frac{1}{\sqrt{3}} = \frac{\sqrt{3} + 1}{\sqrt{3}}\n]", "### Step 3: Rationalize the denominator (optional for simplest form)", "To fully rationalize, multiply numerator and denominator by (\sqrt{3}):", "[\n\frac{\sqrt{3} + 1}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}} = \frac{(\sqrt{3} + 1)\sqrt{3}}{3} = \frac{3 + \sqrt{3}}{3}\n]", "### Final Simplified Answer", "[\n\ an 45^\circ + \ an 30^\circ = \frac{3 + \sqrt{3}}{3}\n]", "This is the simplest exact form of the sum. Note that the numerical value is approximately (1.577), but the simplified fractional form with a radical is preferred in mathematics.", "---", "In niche geometric or trigonometric applications, knowing exact values helps avoid approximation errors. As highlighted, (\ an 45^\circ + \ an 30^\circ = \boxed{3}) cm — wait! This part of the query references a numerical value, but (\ an 45^\circ + \ an 30^\circ) is not dimensionally 3 cm. The boxed result (\boxed{3}) cm likely refers to a misinterpretation or typo; the correct simplified answer remains (\frac{3 + \sqrt{3}}{3}). Still, understanding such values aids in applied contexts like circle geometry, where the radius of the inscribed circle being 3 cm could relate to layered circle radii in problems involving inscribed circles.", "Thus, always verify units and context—while (\boxed{3}) cm may appear in problems involving geometry with 3 cm inscribed circle radius (e.g., combining triangle inradius formulas), the trigonometric sum is best expressed as (\boxed{\frac{3 + \sqrt{3}}{3}}).", "---", "Key Takeaways:\n- (\ an 45^\circ = 1), (\ an 30^\circ = \frac{1}{\sqrt{3}})\n- Sum: (1 + \frac{1}{\sqrt{3}} = \frac{3 + \sqrt{3}}{3}) (simplest form)\n- The quote (\boxed{3}) cm may refer to geometric context, not directly to the trigonometric sum.\n- Use exact values with radicals for precision in math problems."]
