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Compute the exact value of sin 75° + sin 15° using standard trigonometric identities.

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:

  • Find sin 75° + sin 15°
  • Use identity: sin C + sin D = 2*sin((C + D)/2)*cos((C - D)/2)
  • Known values:
    • sin 45° = 1/sqrt(2)
    • cos 30° = sqrt(3)/2


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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