Trigonometry & Inverse Trigonometry
General Solutions of Trigonometric Equations
Grade 11
Question:
<p>If \(\cot\theta = \sin 2\theta\) and \(\theta \ne n\pi\), \(n \in \mathbb{Z}\) then \(\theta\) is equal to</p>
<p>(a) 45°, 60°</p>
<p>(b) 45°, 90°</p>
<p>(c) 45° only</p>
<p>(d) 90° only</p>
Step-by-Step Solution
Key Concept: Express sin 2θ in terms of sin θ and cos θ, then use the definition cot θ = cos θ/sin θ to create an equation. The constraint θ ≠ nπ ensures sin θ ≠ 0, allowing division by sin θ.
<p><strong>Step 1:</strong> Start with cot θ = sin 2θ</p><p><strong>Step 2:</strong> Rewrite using cot θ = cos θ/sin θ and sin 2θ = 2sin θ cos θ:<br/>cos θ/sin θ = 2sin θ cos θ</p><p><strong>Step 3:</strong> Since θ ≠ nπ, we have sin θ ≠ 0. Multiply both sides by sin θ:<br/>cos θ = 2sin²θ cos θ</p><p><strong>Step 4:</strong> Rearrange: cos θ - 2sin²θ cos θ = 0<br/>cos θ(1 - 2sin²θ) = 0</p><p><strong>Step 5:</strong> Factor: cos θ · cos 2θ = 0 (since 1 - 2sin²θ = cos 2θ)</p><p><strong>Step 6:</strong> Case 1: cos θ = 0 ⟹ θ = π/2 + nπ<br/>Case 2: cos 2θ = 0 ⟹ 2θ = π/2 + nπ ⟹ θ = π/4 + nπ/2</p><p><strong>Step 7:</strong> Verify both solutions satisfy θ ≠ nπ. Both are valid.</p><p>∴ Answer: <strong>θ = π/4 + nπ/2 or θ = π/2 + nπ</strong> (or combined: <strong>θ = (2n+1)π/4</strong>)</p>
Correct Answer: B