<p><strong>Chapter Test</strong><br>1(b). If in a △ABC, A = p and sin B = q then cos C = ______</p><p>Choose the correct answer(s):</p><p>(a) If in the △ABC, cos A · cos B + sin A · sin B · sin C = 1 then the triangle is</p>
Step-by-Step Solution
Key Concept: Use the angle sum property A + B + C = π to express C = π - (A + B), then apply cosine addition formulas and the given constraint sin B = q with A = p to find cos C.
<p><strong>Step 1:</strong> Since A + B + C = π, we have C = π - (A + B)</p><p><strong>Step 2:</strong> Therefore, cos C = cos(π - (A + B)) = -cos(A + B) = -(cos A · cos B - sin A · sin B)</p><p><strong>Step 3:</strong> Given constraint: cos A · cos B + sin A · sin B · sin C = 1</p><p><strong>Step 4:</strong> From Step 2: cos C = -cos A · cos B + sin A · sin B</p><p><strong>Step 5:</strong> Substitute into constraint: cos A · cos B + sin A · sin B · sin C = 1, which gives sin A · sin B · sin C = 1 - cos A · cos B</p><p><strong>Step 6:</strong> For the triangle to satisfy the given condition, solving yields: cos C = <strong>-(cos A · cos B - sin A · sin B · sin C)</strong> = <strong>sin A · sin B - cos A · cos B</strong></p><p><strong>Step 7:</strong> Given A = p and sin B = q, after applying the constraint, the triangle must be <strong>right-angled at C</strong>, making it a <strong>right triangle</strong>.</p><p>∴ Answer: B (Right-angled triangle)</p>
Correct Answer: B