Difficulty: Medium
Correct Answer: (3*root6 + 4)/6
Explanation:
Introduction / Context:This question checks standard trigonometric values at special angles and algebraic simplification involving surds (square roots). You must substitute exact values for cot 30° and cosec 60° and then add carefully as fractions.
Given Data / Assumptions:
Concept / Approach:Recall: cot 30° = root3 sin 60° = root3/2, so cosec 60° = 1/sin 60° = 2/root3 Then substitute and simplify each term separately before adding with a common denominator.
Step-by-Step Solution:
Compute cot 30°: cot 30° = root3. First term: (1/root2)*cot 30° = (1/root2)*root3 = root3/root2. Rewrite root3/root2 as root6/2 (multiply numerator and denominator by root2): root3/root2 = (root3*root2)/(root2*root2) = root6/2. Compute cosec 60°: cosec 60° = 2/root3. Second term: (1/root3)*cosec 60° = (1/root3)*(2/root3) = 2/3. Add terms: root6/2 + 2/3. Common denominator 6: root6/2 = 3*root6/6 and 2/3 = 4/6. Sum = (3*root6 + 4)/6.Verification / Alternative check:You can confirm by approximate checking: root6 is about 2.449, so (3*2.449 + 4)/6 is about (7.347 + 4)/6 = 11.347/6 ≈ 1.891, matching root6/2 (≈1.225) plus 2/3 (≈0.667).
Why Other Options Are Wrong:
(3*root6 - 4)/6 subtracts 2/3 instead of adding. (root6 + 2)/3 changes denominators incorrectly. (3*root6 + 2)/6 comes from using 1/3 instead of 2/3. root6/2 is only the first term, missing the second term entirely.Common Pitfalls:Mixing up cosec 60° with sec 60°, or forgetting that (1/root3)*(2/root3) equals 2/3, not 2/root3.
Final Answer:(3*root6 + 4)/6
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