Compute the exact value of the trigonometric expression sin 30° + cos 30°. Do not use decimal approximations; express the answer in simplest exact form.

Introduction / Context: This question checks your knowledge of standard trigonometric values for special angles. The angles 30°, 45°, and 60° are foundational because their sine and cosine values are commonly used in simplification problems. Here, we directly use the known exact values of sin 30° and cos 30°, then add them carefully and keep the result in exact radical form.

Given Data / Assumptions:

• Required: sin 30° + cos 30° • Use exact special-angle values (no rounding)

Concept / Approach: Recall: sin 30° = 1/2 cos 30° = √3/2 These come from the 30°-60°-90° right triangle ratios. Then: sin 30° + cos 30° = 1/2 + √3/2 = (1 + √3)/2.

Step-by-Step Solution: 1) Write exact values: sin 30° = 1/2 cos 30° = √3/2 2) Add them using common denominator 2: sin 30° + cos 30° = 1/2 + √3/2 3) Combine numerators: = (1 + √3)/2

Verification / Alternative check: Approximate check only for sanity: √3 ≈ 1.732, so (1 + √3)/2 ≈ (2.732)/2 ≈ 1.366. Since sin 30° = 0.5 and cos 30° ≈ 0.866, their sum is about 1.366, matching the exact expression’s value.

Why Other Options Are Wrong: • (√3 - 1)/2 would correspond to cos 30° - sin 30° instead of the sum. • √3/2 is only cos 30°, missing sin 30°. • 2 is too large for a sum of two values each ≤ 1. • (√3 + 2)/√3 is a different rearrangement and does not equal 1/2 + √3/2.

Common Pitfalls: • Mixing up sin 30° and cos 30°. • Writing √3/2 + 1/2 but not combining into a single fraction.

Final Answer: (1 + √3) / 2