Trigonometry & Inverse Trigonometry
Trigonometric Equations
Grade 11
Question:
<p><span>\(2\cos x - 3\sin x = a\)</span> has real solutions for <em>x</em> if:</p>
<p>A. \(|a| \leq \sqrt{13}\)</p>
<p>B. \(|a| \leq \sqrt{13}\)</p>
<p>C. \(|a| \leq \sqrt{5}\)</p>
<p>D. \(|a| = 5\)</p>
Step-by-Step Solution
Key Concept: The linear combination A·cos(x) + B·sin(x) has range [-√(A²+B²), √(A²+B²)]. For real solutions to exist, the constant term must lie within this range.
<p><strong>Step 1:</strong> Rewrite 2cos(x) - 3sin(x) in the form R·cos(x + φ).</p><p>We have R = √(2² + 3²) = √(4 + 9) = √13</p><p><strong>Step 2:</strong> The expression 2cos(x) - 3sin(x) can be written as √13·cos(x + φ) where tan(φ) = 3/2.</p><p><strong>Step 3:</strong> The range of √13·cos(x + φ) is [-√13, √13] since -1 ≤ cos(x + φ) ≤ 1.</p><p><strong>Step 4:</strong> For the equation 2cos(x) - 3sin(x) = a to have real solutions, a must lie in the range of the left side.</p><p><strong>Step 5:</strong> Therefore, -√13 ≤ a ≤ √13, or equivalently |a| ≤ √13.</p><p>∴ Answer: A</p>
Correct Answer: A