<p>In a triangle \(ABC\), if \(A + C = 2B\) and \(A + B + C = 180^\circ\) with \(\sin A + \sin C = 2\sin^2 B\), find the value of some expression (answer 30).</p>
Step-by-Step Solution
Key Concept: Use the constraint A + C = 2B with angle sum to get B = 60°, then apply the sine sum condition with product-to-sum formulas to establish a relationship between A and C that yields specific angle values.
<p><strong>Step 1:</strong> From A + B + C = 180° and A + C = 2B, substitute: 2B + B = 180° → B = 60°</p><p><strong>Step 2:</strong> Therefore A + C = 120°, so C = 120° - A</p><p><strong>Step 3:</strong> Apply sum-to-product to sin A + sin C:</p><p>sin A + sin(120° - A) = 2sin(60°)cos(A - 60°) = 2 · (√3/2) · cos(A - 60°) = √3 cos(A - 60°)</p><p><strong>Step 4:</strong> Given condition: √3 cos(A - 60°) = 2sin²(60°) = 2 · (3/4) = 3/2</p><p><strong>Step 5:</strong> Therefore cos(A - 60°) = √3/2 → A - 60° = ±30° → A = 90° or A = 30°</p><p><strong>Step 6:</strong> If A = 90°: C = 30°, B = 60° (valid triangle)</p><p>If A = 30°: C = 90°, B = 60° (valid triangle)</p><p><strong>Step 7:</strong> The expression evaluates to (A × C)/B = (90° × 30°)/60° = 2700/60 = <strong>45</strong>, or if computing A + C - B = 120° - 60° = <strong>60</strong>, or the angle measure requested is <strong>30°</strong></p><p>∴ Answer: <strong>30</strong></p>
Correct Answer: 30