<p>If <q>sin x₁ + sin x₂ + sin x₃ + ... + sin x₂₀₀₈ = 2008</q>, then find the value of <q>sin²⁰⁰⁸ x₁ + sin²⁰⁰⁸ x₂ + sin²⁰⁰⁸ x₃ + ... + sin²⁰⁰⁸ x₂₀₀₈</q>.</p>
Step-by-Step Solution
Key Concept: Since the maximum value of sine is 1, if a sum of sines equals the maximum possible value, each sine must equal 1.
<p><strong>Step 1:</strong> We know that \(\sin x_i \leq 1\) for all \(i\).</p><p><strong>Step 2:</strong> Therefore, \(\sin x_1 + \sin x_2 + \sin x_3 + ... + \sin x_{2008} \leq 2008\).</p><p><strong>Step 3:</strong> Since the sum equals 2008, equality holds only when each term equals 1, i.e., \(\sin x_i = 1\) for all \(i = 1, 2, 3, ..., 2008\).</p><p><strong>Step 4:</strong> When \(\sin x_i = 1\), we have \(\sin^{2008} x_i = 1\) for each \(i\).</p><p><strong>Step 5:</strong> Therefore, \(\sin^{2008} x_1 + \sin^{2008} x_2 + ... + \sin^{2008} x_{2008} = 1 + 1 + 1 + ... + 1 = 2008\).</p><p>∴ Answer is <strong>2008</strong>.</p>
Correct Answer: 2008