In basic trigonometry with angles measured in degrees, evaluate the expression sin 30° + 2 * cos 30° and express your answer in exact surd form (without decimals).

Introduction / Context:
This question asks you to evaluate a simple trigonometric expression involving standard angles and express the result using square roots, known as surd form. The angles 30° and 60° have well known sine and cosine values, which are frequently used in trigonometry and geometry problems. Being comfortable with these values and how to combine them algebraically is essential for solving many exam questions quickly.

Given Data / Assumptions:

• The expression to evaluate is sin 30° + 2 * cos 30°.
• Angles are given in degrees, not radians.
• We should use exact values for the trigonometric functions, not decimal approximations.
• The final answer should be written as a simplified surd expression.

Concept / Approach:
For standard angles like 30°, 45°, and 60°, sine and cosine values are derived from special right triangles. For 30°, the sine and cosine values are sin 30° = 1/2 and cos 30° = √3 / 2. Substituting these values into the given expression and simplifying leads directly to the exact answer. No advanced identities are required; only substitution and basic fraction and surd manipulation are needed.

Step-by-Step Solution:
1) Recall the standard values: sin 30° = 1/2 and cos 30° = √3 / 2. 2) Substitute into the expression sin 30° + 2 * cos 30°. 3) This gives 1/2 + 2 * (√3 / 2). 4) Simplify the second term: 2 * (√3 / 2) = √3. 5) Therefore the expression becomes 1/2 + √3. 6) To match the answer format, express this as a single fraction: 1/2 + √3 = (1/2) + (2√3 / 2) = (1 + 2√3) / 2. 7) This is the simplified exact surd form.

Verification / Alternative check:
You can verify numerically by using approximate decimal values, though the final answer must remain in surd form. Taking √3 ≈ 1.732, we get sin 30° + 2 * cos 30° ≈ 0.5 + 2 * 0.866 = 0.5 + 1.732 = 2.232. Evaluating (1 + 2√3) / 2 numerically gives (1 + 2 * 1.732) / 2 = (1 + 3.464) / 2 = 4.464 / 2 ≈ 2.232, which matches the earlier approximation and supports the correctness of the surd expression.

Why Other Options Are Wrong:
Option e (√3) ignores the sin 30° term. Options b, c, and d have more complicated structures and yield different approximate values when evaluated numerically. For example, substituting √3 ≈ 1.732 into option b gives a value that does not match 2.232. Only option a, (1 + 2√3) / 2, simplifies exactly to 1/2 + √3 and matches both the algebraic derivation and numerical check.

Common Pitfalls:
Some learners mistakenly use incorrect standard values, such as taking cos 30° as 1/2 instead of √3 / 2, which leads to a completely different result. Others may convert the angle to radians unnecessarily or try to use a calculator in decimal mode, losing the surd form. Remembering the basic special triangle values and practising simple substitutions helps avoid these errors and makes such problems straightforward.

Final Answer:
The exact surd value of sin 30° + 2 * cos 30° is (1 + 2√3)/2.