In basic trigonometry, simplify the expression cot 30° + 2/√3 and express the result in its simplest exact form.

Introduction / Context:
This trigonometry question focuses on the exact values of standard trigonometric ratios for special angles and simple algebraic manipulation with surds. The angle 30° is a well known special angle whose sine, cosine, and tangent values are often memorised. Using these, you can quickly find cot 30° and then add a simple fractional term involving √3. Such simplifications are common in school level trigonometry and competitive exams.

Given Data / Assumptions:

• The angle of interest is 30°.
• The expression to simplify is cot 30° + 2/√3.
• We assume standard exact values: sin 30° = 1/2 and cos 30° = √3/2.
• The final answer should be expressed in exact form, not as a decimal approximation.

Concept / Approach:
Cotangent is defined as the reciprocal of tangent, or equivalently cos θ / sin θ. For the special angle 30°, we can use known sine and cosine values to obtain cot 30°. Once we express cot 30° in terms of √3, we add the second term 2/√3, combine the two fractions, and simplify the resulting expression. This shows how known trigonometric ratios and basic fraction addition work together.

Step-by-Step Solution:
Step 1: Recall that tan 30° = 1/√3.Step 2: Therefore cot 30° = 1 / tan 30° = √3.Step 3: Substitute this into the given expression: cot 30° + 2/√3 = √3 + 2/√3.Step 4: Express √3 with denominator √3: √3 = 3/√3.Step 5: Now add the fractions: 3/√3 + 2/√3 = (3 + 2)/√3 = 5/√3.Step 6: Thus the simplified exact value is 5/√3.

Verification / Alternative check:
As a quick numerical check, approximate √3 as about 1.732. Then cot 30° ≈ 1.732 and 2/√3 ≈ 2/1.732 ≈ 1.155. Their sum is approximately 2.887. Now compute 5/√3 ≈ 5/1.732 ≈ 2.887. The numerical values agree, which confirms that the simplification is correct.

Why Other Options Are Wrong:
The expression (1 + 2√3)/2 does not match the combined fraction 5/√3 and gives a different decimal value. The value 5/6 is much smaller than the correct result, since 5/6 is less than 1, whereas we know cot 30° alone is greater than 1. The option 2/√3 omits the contribution from cot 30°. The value √3 accounts only for cot 30° and ignores the extra 2/√3 term. Only 5/√3 matches the correctly simplified expression.

Common Pitfalls:
A common mistake is to forget that tan 30° is 1/√3 and wrongly assume it is √3, which reverses cot 30°. Another frequent error is to add the surd terms without using a common denominator, leading to incorrect numerators. By carefully recalling the special angle values and converting everything to a common denominator, you can avoid these errors and arrive at the exact result.

Final Answer:
The simplified exact value of cot 30° + 2/√3 is 5/√3.