Trigonometry & Inverse Trigonometry
Trigonometric Equations
Grade 11
Question:
<p>The general solution-set of the equation <span>\(\cos x + \cos 5x = 2\)</span> is:</p>
<p>A. \(\left\{x \mid x = \dfrac{2m\pi}{6},\, m \in \mathbb{Z}\right\}\)</p>
<p>B. \(\left\{x \mid x = 2n\pi,\, n \in \mathbb{Z}\right\}\)</p>
<p>C. \(\left\{x \mid x = \dfrac{2m\pi}{5},\, m \text{ is a multiple of } 5\right\}\)</p>
<p>D. \(\phi\)</p>
Step-by-Step Solution
Key Concept: The sum cos x + cos 5x can equal 2 only when both cos x = 1 AND cos 5x = 1 simultaneously, since the maximum value of each cosine term is 1 and their sum cannot exceed 2.
<p><strong>Step 1:</strong> Recognize the constraint. Since -1 ≤ cos x ≤ 1 and -1 ≤ cos 5x ≤ 1, we have cos x + cos 5x ≤ 2.</p><p><strong>Step 2:</strong> For cos x + cos 5x = 2, we need equality throughout, meaning cos x = 1 AND cos 5x = 1 simultaneously.</p><p><strong>Step 3:</strong> cos x = 1 gives x = 2nπ, where n ∈ ℤ.</p><p><strong>Step 4:</strong> Check if cos 5x = 1 when x = 2nπ: cos(5·2nπ) = cos(10nπ) = 1 ✓</p><p><strong>Step 5:</strong> Verify no other solutions exist. If x ≠ 2nπ, then cos x < 1, making the sum less than 2.</p><p><strong>∴ Answer: x = 2nπ, n ∈ ℤ (Option B)</strong></p>
Correct Answer: B