Trigonometry & Inverse Trigonometry
Trigonometric Equations
Grade 11
Question:
<p>Let <i>n</i> be a positive integer such that <i>n</i> ∈ ℕ. Then, <i>sin</i>\(\left(\frac{\pi}{2n}\right)\) + <i>cos</i>\(\left(\frac{\pi}{2n}\right)\) = \(\frac{\sqrt{2}}{2}\). Find the range of <i>n</i>.</p>
<p>(a) \(6 \leq n \leq 8\)</p>
<p>(b) \(4 < n \leq 8\)</p>
<p>(c) \(4 \leq n \leq 8\)</p>
<p>(d) \(4 < n < 8\)</p>
Step-by-Step Solution
Key Concept: Solve the trigonometric equation by recognizing the sum of sine and cosine and applying algebraic constraints.
<p><strong>Solution:</strong> Given that <i>sin</i>\(\left(\frac{\pi}{2n}\right)\) + <i>cos</i>\(\left(\frac{\pi}{2n}\right)\) = \(\frac{\sqrt{2}}{2}\). We need to find the range of positive integer <i>n</i>. This inequality gives us \(4 \leq n \leq 8\).</p>
Correct Answer: c