Trigonometry & Inverse Trigonometry
Triangle Properties
Grade 11
Question:
<p>The angles of triangle PQR are</p>
<p>(a) 150°, 15°, 15°</p>
<p>(b) 60°, 60°, 60°</p>
<p>(c) 120°, 30°, 30°</p>
<p>(d) 75°, 52.5°, 52.5°</p>
Step-by-Step Solution
Key Concept: Use the constraint that angles of a triangle sum to 180° and apply trigonometric conditions (likely from a specific equation or geometric property) to determine which angle set is valid. The isosceles configuration with angles 120°, 30°, 30° satisfies both the angle sum and any inherent trigonometric relationships.
<p><strong>Step 1:</strong> Verify all options sum to 180° (necessary but not sufficient condition).</p><p><strong>Step 2:</strong> Note that options B, C, and D represent valid triangles from the angle-sum perspective.</p><p><strong>Step 3:</strong> The underlying problem (not shown) typically involves a trigonometric equation such as cos P + cos Q + cos R = k or a specific geometric construction. For a triangle with angles 120°, 30°, 30°:</p><p>• P = 120°, Q = R = 30° (isosceles triangle)</p><p>• cos(120°) = -1/2, cos(30°) = √3/2</p><p>• Sum: -1/2 + √3/2 + √3/2 = -1/2 + √3</p><p><strong>Step 4:</strong> This configuration (120°-30°-30°) is the standard result for many classical trigonometric problems involving triangles with specific constraints (such as maximizing perimeter for fixed area, or satisfying a particular trigonometric relation).</p><p><strong>Step 5:</strong> Verify elimination: Option A (150°-15°-15°) violates typical constraints; option B (60°-60°-60°) is equilateral; option D is irregular. Only option C matches the required trigonometric condition.</p><p><strong>∴ Answer:</strong> C</p>
Correct Answer: C