Difficulty: Easy
Correct Answer: (2√3 − 1)/√3
Explanation:
Introduction / Context:
This problem tests exact-value evaluation using standard angles and basic surd simplification. The key is knowing cosec 30° exactly. Since sin 30° = 1/2, cosec 30° is the reciprocal, which equals 2. After substituting this value, the expression becomes a simple difference: 2 - 1/√3. To match typical multiple-choice answers, you rewrite 2 with denominator √3 so the result becomes a single fraction. Students often lose marks here because they misremember cosec 30° as √3/2, confuse cosec with sec, or forget how to combine a whole number with a surd fraction.
Given Data / Assumptions:
Concept / Approach:
Compute cosec 30° using the reciprocal of sin 30°. Then subtract 1/√3. For a clean final form, express 2 as (2√3)/√3 so the subtraction is over a common denominator √3. This yields a single simplified surd fraction that matches standard answer formats.
Step-by-Step Solution:
1) Evaluate cosec 30°:
sin 30° = 1/2 ⇒ cosec 30° = 1/(1/2) = 2
2) Substitute into the expression:
cosec 30° − 1/√3 = 2 − 1/√3
3) Write 2 with denominator √3:
2 = (2√3)/√3
4) Subtract:
(2√3)/√3 − 1/√3 = (2√3 − 1)/√3
Verification / Alternative check:
Approximation check: 1/√3 ≈ 0.577, so 2 − 1/√3 ≈ 1.423. The expression (2√3 − 1)/√3 equals 2 − 1/√3 by construction, so it must have the same approximate value, confirming the simplification is consistent.
Why Other Options Are Wrong:
• 2/√3: misses the “− 1” part in the numerator.
• -1/√3: would imply cosec 30° = 0, which is incorrect.
• (√3 − 4)/(2√3): comes from incorrect common-denominator steps.
• (2√3 + 1)/√3: has the wrong sign (it adds instead of subtracting).
Common Pitfalls:
• Confusing cosec 30° with sec 30°.
• Forgetting to use a common denominator when subtracting.
• Treating 1/√3 as √3 (incorrect reciprocal handling).
Final Answer:
(2√3 − 1)/√3
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