Difficulty: Medium
Correct Answer: (1 + 3√2)/√3
Explanation:
Introduction / Context:
This problem tests exact-value evaluation in trigonometry using standard angles (30° and 60°). The goal is not a decimal approximation but a simplified surd expression. You must know or derive sec 30° and tan 60° exactly. A common source of mistakes is mixing up sec and cosec, or incorrectly handling the factor (1/2) in front of sec 30°. Another common error is multiplying surds incorrectly when computing √2 * tan 60°. The best approach is to compute each term exactly and then add them using a common denominator only if needed.
Given Data / Assumptions:
Concept / Approach:
Compute sec 30° from cos 30°, then multiply by 1/2. Separately compute √2 tan 60° using the exact tan 60° value. Finally, add the two exact terms. If the result has mixed forms, rewrite them over a common denominator (often √3) to match standard simplified options.
Step-by-Step Solution:
1) Compute sec 30°:
sec 30° = 1 / cos 30° = 1 / (√3/2) = 2/√3
2) Multiply by 1/2:
(1/2) sec 30° = (1/2) * (2/√3) = 1/√3
3) Compute √2 tan 60°:
tan 60° = √3, so √2 tan 60° = √2 * √3 = √6
4) Add the terms:
√6 + 1/√3
5) Write √6 with denominator √3:
√6 = (√6 * √3) / √3 = √18 / √3 = (3√2)/√3
6) Final sum:
(3√2)/√3 + 1/√3 = (1 + 3√2)/√3
Verification / Alternative check:
Approximation check: sec 30° ≈ 1.1547, so (1/2)sec 30° ≈ 0.57735. Also √2 tan 60° ≈ 1.4142 * 1.732 ≈ 2.449. Sum ≈ 3.026. The option (1 + 3√2)/√3 ≈ (1 + 4.2426)/1.732 ≈ 3.026, matching perfectly.
Why Other Options Are Wrong:
• √3 + 2 and (√3 + 2)/2: come from using tan 60° as 2 or using sec 30° incorrectly.
• (√3 + 2)/√3: mixes terms as if √6 were √3.
• (3 + √2)/√3: indicates incorrect surd multiplication √2*√3.
Common Pitfalls:
• Forgetting sec 30° is 2/√3, not √3/2.
• Treating √2*√3 as √5 (wrong rule).
• Dropping the (1/2) factor in the first term.
Final Answer:
(1 + 3√2)/√3
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