Trigonometry & Inverse Trigonometry
Cyclic quadrilateral properties
Grade 11
Question:
<p>Four points \(A, B, C, D\) taken in order lie on the circumference of a circle to form a quadrilateral. Let \(\alpha, \beta, \gamma, \delta\) denote four interior angles of the quadrilateral associated with \(A, B, C, D\) respectively. Which of the following is/are always true?</p>
<p>\(\cos\beta\cos\delta = 1 + \sin\beta\sin\delta\)</p>
<p>\(\sin\alpha\cos\gamma + \cos\alpha\sin\gamma = 0\)</p>
<p>\(\sin^2\alpha + \cos^2\gamma = 1\)</p>
<p>\(\cos\beta + \cos\delta = 0\)</p>
Step-by-Step Solution
Key Concept: In a cyclic quadrilateral, opposite angles are supplementary (sum to π). This is because each angle subtends an arc, and opposite angles subtend complementary arcs that together form the complete circle.
<p><strong>Key Property:</strong> In a cyclic quadrilateral ABCD inscribed in a circle, opposite angles are supplementary.</p><p><strong>Step 1:</strong> Identify opposite angle pairs: A-C and B-D are opposite.</p><p><strong>Step 2:</strong> Apply inscribed angle theorem. Angle α (at A) subtends arc BCD. Angle γ (at C) subtends arc DAB. Since arc BCD + arc DAB = 2π (complete circle), we have:</p><p>α + γ = π and β + δ = π</p><p><strong>Step 3:</strong> Verify the correct options:</p><p><strong>Option A:</strong> α + β = π? Not always true (only in special cases like rectangles).</p><p><strong>Option B:</strong> α + γ = π? ✓ Always true (opposite angles in cyclic quadrilateral)</p><p><strong>Option C:</strong> β + δ = π? ✓ Always true (opposite angles in cyclic quadrilateral)</p><p><strong>Option D:</strong> α + β + γ + δ = 2π? ✓ Always true (sum of interior angles in any quadrilateral is 2π)</p><p><strong>∴ Answer: B, C, D</strong></p>
Correct Answer: B,C,D