Trigonometry & Inverse Trigonometry
Trigonometric Series
Grade 11
Question:
<p>Calculate <span style='display: inline-block;'>m = \sum_{k=1}^{17} \cos\left(\frac{k\pi}{9}\right) = \cos\left(\frac{\pi}{9}\right) + \cos\left(\frac{2\pi}{9}\right) + \cos\left(\frac{3\pi}{9}\right) + \ldots + \cos\left(\frac{17\pi}{9}\right)</span>, and find the value of <span style='display: inline-block;'>(m^2 + m + 2)</span>.</p>
Step-by-Step Solution
Key Concept: Use summation formulas for trigonometric series and evaluate using periodicity of cosine function.
<p><strong>Step 1:</strong> We have <span style='display: inline-block;'>m = \sum_{k=1}^{17} \cos\left(\frac{k\pi}{9}\right)</span></p><p><strong>Step 2:</strong> Using the sum-to-product formula for cosines in arithmetic progression:</p><p><span style='display: inline-block;'>m = \frac{\sin\left(\frac{17\pi}{18}\right)}{\sin\left(\frac{\pi}{18}\right)} \times \cos\left(\frac{16\pi}{9}\right)</span></p><p><strong>Step 3:</strong> Evaluating: <span style='display: inline-block;'>m = \cos\left(\frac{18\pi}{18}\right) = \cos(\pi) = -1</span></p><p><strong>Step 4:</strong> Therefore: <span style='display: inline-block;'>m^2 + m + 2 = (-1)^2 + (-1) + 2 = 1 - 1 + 2 = 2</span></p><p>∴ The value of (m² + m + 2) is 2.</p>
Correct Answer: 4