Trigonometry & Inverse Trigonometry
Trigonometric Identities
Grade 11
Question:
<p>If \(\cos(\theta - \phi) + \cos(\phi - \alpha) + \cos(\alpha - \theta) = \frac{3}{2}\), then which of the following is/are true?</p>
<p>(a) \(\sum \cos \alpha = 0\)</p>
<p>(b) \(\sum \sin \alpha = 0\)</p>
<p>(c) \(\sum \cos \alpha \sin \alpha = 0\)</p>
<p>(d) \(\sum (\cos \alpha + \sin \alpha) = 0\)</p>
Step-by-Step Solution
Key Concept: Expand the cosine differences using product-to-sum formulas and recognize that the sum of squares of two expressions equals zero only when both are individually zero.
<p><strong>Solution:</strong> The given expression can be written as:</p><p>\(2[\cos \theta \cos \phi + \cos \phi \cos \alpha + \cos \alpha \cos \theta]\)</p><p>\(+ 2[\sin \theta \sin \phi + \sin \phi \sin \alpha + \sin \alpha \sin \theta]\)</p><p>\(+ (\sin^2 \alpha + \cos^2 \alpha) + (\sin^2 \theta + \cos^2 \theta) + (\sin^2 \phi + \cos^2 \phi) = 0\)</p><p>\(\Rightarrow (\cos \alpha + \cos \theta + \cos \phi)^2 + (\sin \alpha + \sin \theta + \sin \phi)^2 = 0\)</p><p>\(\Rightarrow \sum \cos \alpha = 0 \text{ and } \sum \sin \alpha = 0\)</p><p>\(\Rightarrow \sum(\cos \alpha + \sin \alpha) = 0\)</p><p>Therefore, (a), (b), and (d) are correct.</p>
Correct Answer: A, B, D