Difficulty: Easy
Correct Answer: (√3 + 2)/√3
Explanation:
Introduction / Context:
This question examines your knowledge of standard trigonometric values for special angles and your ability to combine them into a simplified surd expression. You are asked to find cot 45° + cosec 60°, both of which have exact values. Then, you must express the sum in a neat surd form that matches one of the answer choices.
Given Data / Assumptions:
Concept / Approach:
We recall that cot 45° is 1 and cosec 60° is the reciprocal of sin 60°. Once we substitute these values, we obtain a sum of a rational number and a surd. Writing 1 as √3 / √3 allows us to combine the two terms into a single fraction with denominator √3. This form directly matches one of the answer options.
Step-by-Step Solution:
Verification / Alternative check:
Approximate numerically to verify. With √3 ≈ 1.732, sin 60° ≈ 0.866, so cosec 60° ≈ 1.155. Therefore, cot 45° + cosec 60° ≈ 1 + 1.155 = 2.155. Now compute (√3 + 2) / √3 ≈ (1.732 + 2) / 1.732 ≈ 3.732 / 1.732 ≈ 2.155. The two values match, confirming the correctness of the simplified expression.
Why Other Options Are Wrong:
The options (√6 + 1)/√3, (1 + √3)/2, 5/√3, and (2 + √3)/2 all simplify to values different from approximately 2.155. These typically result from misremembering sin 60°, using tan 60° instead of cosec 60°, or combining terms incorrectly. Only (√3 + 2)/√3 matches the exact derived expression.
Common Pitfalls:
Students sometimes confuse cosec 60° with sec 60° or with tan 60°, or they mistakenly use 1 / √3 for sin 60°. Another frequent error is forgetting to convert the integer 1 into a compatible surd form before combining terms. Being careful with standard trig values and fraction operations avoids these mistakes.
Final Answer:
The value of cot 45° + cosec 60° is (√3 + 2) / √3.
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