<p>If \(\sin a\theta + \cos b\theta = 0\) then the possible values of \(\theta\) form</p>
Step-by-Step Solution
Key Concept: The equation sin(aθ) + cos(bθ) = 0 means sin(aθ) = -cos(bθ) = sin(bθ - π/2), which establishes a relationship between aθ and bθ that generates θ values in arithmetic progression when a and b are rationally related.
<p><strong>Step 1:</strong> Rewrite the equation sin(aθ) + cos(bθ) = 0 as sin(aθ) = -cos(bθ)</p><p><strong>Step 2:</strong> Use the identity -cos(bθ) = sin(bθ - π/2) to get sin(aθ) = sin(bθ - π/2)</p><p><strong>Step 3:</strong> Apply the general solution for sin(A) = sin(B):<br/>aθ = bθ - π/2 + 2nπ, or<br/>aθ = π - (bθ - π/2) + 2nπ</p><p><strong>Step 4:</strong> From the first case: (a - b)θ = -π/2 + 2nπ, giving θ = (-π/2 + 2nπ)/(a - b) [arithmetic progression with common difference 2π/(a - b)]</p><p><strong>Step 5:</strong> From the second case: (a + b)θ = 3π/2 + 2nπ, giving θ = (3π/2 + 2nπ)/(a + b) [arithmetic progression with common difference 2π/(a + b)]</p><p><strong>Step 6:</strong> The complete solution consists of values forming an arithmetic progression (or union of arithmetic progressions if both families exist)</p><p>∴ Answer: B</p>
Correct Answer: B