Trigonometry & Inverse Trigonometry
Product-to-Sum and Factorization Formulas
Grade 11
Question:
<p>If <equation>a = \sin\frac{\pi}{18} \sin\frac{7\pi}{18} \sin\frac{13\pi}{18}</equation>, then <equation>a</equation> is equal to</p>
<p>(a) <equation>[x]</equation></p>
<p>(b) <equation>\frac{1}{[x]}</equation></p>
<p>(c) <equation>2[x]</equation></p>
<p>(d) <equation>[x]^2</equation></p>
Step-by-Step Solution
Key Concept: Use product-to-sum formulas and trigonometric identities to convert products of sines into sums that can be evaluated.
<p><strong>Step 1:</strong> Express in degrees: <equation>a = \sin 10° \sin 50° \sin 70°</equation></p><p><strong>Step 2:</strong> Use product-to-sum formula: <equation>a = \frac{1}{2}[2\sin 70° \sin 10°]\sin 50°</equation></p><p><strong>Step 3:</strong> Apply cosine difference: <equation>a = \frac{1}{2}[\cos 60° - \cos 80°]\sin 50°</equation></p><p><strong>Step 4:</strong> Simplify: <equation>a = \frac{1}{4}\sin 50° - \frac{1}{4}(\sin 130° - \sin 30°)</equation></p><p><strong>Step 5:</strong> Continue simplification to get <equation>a = \frac{1}{8}</equation></p><p>∴ Answer is (b) <equation>\frac{1}{[x]}</equation></p>
Correct Answer: B