Difficulty: Easy
Correct Answer: (√2 + √3)/√6
Explanation:
Introduction / Context:
This trigonometric simplification question tests your familiarity with standard angle values and with simplifying surd expressions. You are asked to compute 1 / √3 + sin 45°, which involves a numerical reciprocal and a trigonometric value. The final answer must be written in a neat surd form matching one of the options.
Given Data / Assumptions:
Concept / Approach:
The plan is to substitute the known exact value of sin 45° and then combine the terms as a single expression involving square roots. One efficient way to match answer options is to express both terms with a common denominator that involves √6, since several options use that pattern. Using rationalisation and multiplication by suitable forms of 1 allows us to transform the sum into the required surd form.
Step-by-Step Solution:
Verification / Alternative check:
Approximate numerically. Take √2 ≈ 1.414 and √3 ≈ 1.732. Then 1 / √3 ≈ 0.577 and sin 45° ≈ 0.707, so their sum is about 1.284. Now compute (√2 + √3) / √6 ≈ (1.414 + 1.732) / 2.449 ≈ 3.146 / 2.449 ≈ 1.284, which matches the original sum, confirming that the expression is correct.
Why Other Options Are Wrong:
The expressions √3 + 2, (1 + √6)/√3, (4 + √3)/2, and (√3 + 2)/√3 all evaluate to different numerical values that do not equal approximately 1.284. They typically come from incorrect rationalisation steps or combining surds without using a correct common denominator.
Common Pitfalls:
Common errors include using sin 45° = 1/√2 instead of √2 / 2 and then not simplifying carefully, or trying to combine the two terms without converting them to a common base. Another trap is to rationalise one term but not the other, making it harder to see the common pattern. Systematically converting both terms to a denominator involving √6 avoids these issues.
Final Answer:
The simplified surd form of 1 / √3 + sin 45° is (√2 + √3) / √6.
Discussion & Comments