Difficulty: Medium
Correct Answer: sqrt(3/2)
Explanation:
Introduction / Context:This problem tests the sum-to-product identity for sine. Angles like 75 degrees and 15 degrees are chosen because their average is 45 degrees and their half-difference is 30 degrees, which have well-known exact sine and cosine values. The expression becomes a clean product.
Given Data / Assumptions:
Concept / Approach:Convert the sum of sines into a product, then substitute exact standard-angle values to get an exact simplified surd form.
Step-by-Step Solution:
Step 1: Apply identity with C = 75° and D = 15°. Step 2: sin 75° + sin 15° = 2*sin((75 + 15)/2)*cos((75 - 15)/2). Step 3: (75 + 15)/2 = 90/2 = 45° and (75 - 15)/2 = 60/2 = 30°. Step 4: So expression = 2*sin 45°*cos 30°. Step 5: Substitute values: = 2*(1/sqrt(2))*(sqrt(3)/2). Step 6: Cancel 2 and 2: = sqrt(3)/sqrt(2) = sqrt(3/2).Verification / Alternative check:Numerically, sin 75° is about 0.9659 and sin 15° is about 0.2588, sum about 1.2247. sqrt(3/2) is also about 1.2247, confirming the exact result.
Why Other Options Are Wrong:
sqrt(3), 2*sqrt(3): too large compared to the numeric sum. 3/sqrt(2): equals about 2.121, far too large. sqrt(2/3): about 0.816, too small.Common Pitfalls:Using the wrong identity (difference instead of sum) or halving angles incorrectly (mixing 60 and 30).
Final Answer:sqrt(3/2)
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