Difficulty: Easy
Correct Answer: (3 - 2√3)/3
Explanation:
Introduction / Context:
This question examines your ability to use standard trigonometric values and manipulate surds. Evaluating expressions like tan 45° and sec 60° is basic trigonometry, while simplifying the resulting surd expression tests your algebraic skills, including rationalisation and combining terms with square roots.
Given Data / Assumptions:
Concept / Approach:
First, recall the exact values tan 45° = 1 and sec 60° = 2. Substitute these values into the expression. Then simplify the resulting combination of a rational number and a surd. Finally, rationalise or rewrite the expression if needed so that it matches one of the answer choices, paying attention to equivalent forms that might look different but represent the same value.
Step-by-Step Solution:
Verification / Alternative check:
Approximate numerically to verify. Use √3 ≈ 1.732. Then 2 / √3 ≈ 2 / 1.732 ≈ 1.155. So 1 − 2 / √3 ≈ 1 − 1.155 ≈ −0.155. Now check (3 − 2√3) / 3 ≈ (3 − 3.464) / 3 ≈ −0.464 / 3 ≈ −0.155, which matches, confirming that the expression and the selected option are equivalent.
Why Other Options Are Wrong:
The expressions (1 − 2√3)/2, (√2 − √3)/√6, (1 − 2√2)/2, and (2 − √3)/3 all evaluate to different numerical values when approximated. They do not match the exact simplified form obtained from the step-by-step derivation, so they cannot represent the original expression.
Common Pitfalls:
Learners sometimes misremember sec 60° as 1/2 instead of 2, or they forget to treat 1 as √3 / √3 when combining surds. Another common mistake is to stop at (√3 − 2) / √3 and not recognise its rationalised equivalent (3 − 2√3) / 3 that appears in the options. Carefully using standard values and algebraic simplification avoids these issues.
Final Answer:
The simplified value of tan 45° − (1 / √3) · sec 60° is (3 − 2√3) / 3.
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