If x = sin(pi/6) + sec(pi/6), what is the exact value of x?

Introduction / Context:
This question tests exact trigonometric values at special angles and the reciprocal definition of secant. Since pi/6 corresponds to 30 degrees, we can use standard known values of sin 30 degrees and cos 30 degrees to compute sec 30 degrees exactly, then add them carefully as rational surds.

Given Data / Assumptions:

• x = sin(pi/6) + sec(pi/6)
• pi/6 = 30°
• sin 30° = 1/2
• cos 30° = sqrt(3)/2
• sec 30° = 1/cos 30°

Concept / Approach:
Compute sin(pi/6) directly, compute sec(pi/6) as the reciprocal of cos(pi/6), then add them with a common denominator to match the exact option form.

Step-by-Step Solution:

Verification / Alternative check:
Approximate check: 1/2 = 0.5 and 2/sqrt(3) is about 1.1547, sum about 1.6547. The final form (sqrt(3)+4)/(2sqrt(3)) also evaluates to about 1.6547.

Why Other Options Are Wrong:

Common Pitfalls:
Using sec(pi/6) = 1/sin(pi/6) (wrong), or mixing degrees with radians and using the wrong special-angle value.

Final Answer:
(sqrt(3) + 4)/(2*sqrt(3))