Trigonometry & Inverse Trigonometry
Trigonometric Equations
Grade 11
Question:
<p>Let <i>a</i>, <i>b</i>, <i>c</i> be three non-zero real numbers such that the equation <i>a</i>cos<i>x</i> + 2<i>b</i>sin<i>x</i> = <i>c</i>, <i>x</i> ∈ [−π/2, π/2], has two distinct real roots <i>α</i> and <i>β</i> with <i>α</i> + <i>β</i> = π/3. Then, the value of <i>b</i>/<i>a</i> is __________________________.</p>
Step-by-Step Solution
Key Concept: Transform the trigonometric equation into standard form and use the constraint on the sum of roots to determine the phase angle.
<p><strong>Step 1:</strong> The equation <i>a</i>cos<i>x</i> + 2<i>b</i>sin<i>x</i> = <i>c</i> can be rewritten as √(<i>a</i><sup>2</sup> + 4<i>b</i><sup>2</sup>)sin(<i>x</i> + <i>φ</i>) = <i>c</i>, where tan<i>φ</i> = <i>a</i>/(2<i>b</i>).</p><p><strong>Step 2:</strong> For the equation to have two distinct roots in [−π/2, π/2] with <i>α</i> + <i>β</i> = π/3, we use the constraint on the sum of angles.</p><p><strong>Step 3:</strong> From the transformed equation, the two angles satisfy: <i>x</i> + <i>φ</i> and π − (<i>x</i> + <i>φ</i>) correspond to the same sine value.</p><p><strong>Step 4:</strong> The sum condition gives: (−<i>φ</i>) + (π − <i>φ</i>) = π/3, which yields π − 2<i>φ</i> = π/3, so <i>φ</i> = π/3.</p><p><strong>Step 5:</strong> Therefore, tan(π/3) = a/(2<i>b</i>), giving √3 = <i>a</i>/(2<i>b</i>), so <i>b</i>/<i>a</i> = 1/(2√3) = √3/6.</p><p>∴ The answer is √3/6 or equivalently 1/(2√3).</p>
Correct Answer: √3/2