Using standard trigonometric ratios, evaluate the exact value of sec 30° + tan 60° and express the result in simplest surd form.

Introduction / Context: This problem tests your knowledge of exact trigonometric ratios for special angles, particularly 30° and 60°. It asks you to use those values to evaluate an expression involving secant and tangent, and then simplify it into a compact surd form commonly used in exam answers.

Given Data / Assumptions:

• Expression: sec 30° + tan 60°.
• Angles are measured in degrees.
• We use standard exact values for sin, cos and tan at 30° and 60°.
• No calculators are required; algebraic simplification with surds is enough.

Concept / Approach: We recall that sec θ = 1 / cos θ and tan θ = sin θ / cos θ. For 30° and 60°, the cosine and sine values are well known: cos 30° = √3/2, sin 30° = 1/2, sin 60° = √3/2 and cos 60° = 1/2. Using these, we compute sec 30° and tan 60° exactly, then add them and simplify the result into a single fraction with a rational numerator and a surd denominator.

Step-by-Step Solution: cos 30° = √3/2, so sec 30° = 1 / cos 30° = 1 / (√3/2) = 2/√3. For 60°, sin 60° = √3/2 and cos 60° = 1/2. So tan 60° = sin 60° / cos 60° = (√3/2) / (1/2) = √3. Now compute sec 30° + tan 60° = 2/√3 + √3. Write √3 as 3/√3 (since 3/√3 = √3). Then 2/√3 + √3 = 2/√3 + 3/√3 = 5/√3.

Verification / Alternative check: We can approximate numerically to confirm. cos 30° ≈ 0.866 so sec 30° ≈ 1.155. tan 60° ≈ 1.732. Their sum is about 2.887. Now 5/√3 ≈ 5 / 1.732 ≈ 2.887 as well. The close numerical match confirms that the simplified expression 5/√3 is correct.

Why Other Options Are Wrong: (√6 + 1)/√3 and (√3 + 2)/√3 have different numerators and do not evaluate to approximately 2.887. (1 + √3)/2 is around 1.366, which is far from the true value. The option 2√3 is approximately 3.464, again not matching the computed sum. Only 5/√3 agrees with the exact algebraic simplification and the numerical check.

Common Pitfalls: Students sometimes confuse sec 30° with sec 60° or mistakenly use tan 30° instead of tan 60°. Another frequent error is to skip the step of rewriting √3 as 3/√3, which can make it harder to combine terms. Carefully recall the exact values for special angles and handle surds step by step to avoid mistakes.

Final Answer: Thus, sec 30° + tan 60° has the exact value 5/√3.