Find the exact value of sin(5π/3) using unit-circle values. Give the result in simplest radical form (no decimals).

Introduction / Context:
This question tests evaluating sine at a special angle given in radians. The unit circle provides exact sine and cosine values for angles like π/6, π/3, π/2, 2π/3, 5π/3, etc. The main challenge is determining the correct quadrant to assign the correct sign. For 5π/3, the reference angle is π/3, but since 5π/3 lies in the fourth quadrant, the sine value must be negative.

Given Data / Assumptions:

• Required: sin(5π/3) • Use exact unit-circle values

Concept / Approach:
Convert or recognize: 5π/3 corresponds to 300°. In the unit circle, sine is the y-coordinate. Quadrant IV has negative y-values, so sine is negative there. The reference angle for 300° is 60° (π/3). Since sin(π/3) = √3/2, we assign a negative sign for quadrant IV: sin(5π/3) = -√3/2.

Step-by-Step Solution:
1) Identify the angle location: 5π/3 = 300° 2) Determine the quadrant: 300° lies in quadrant IV 3) Find the reference angle: 360° - 300° = 60° → reference angle is 60° (π/3) 4) Use the known value: sin(60°) = sin(π/3) = √3/2 5) Apply quadrant sign (sine is negative in quadrant IV): sin(5π/3) = -√3/2

Verification / Alternative check:
A coterminal expression approach: sin(5π/3) = sin(2π - π/3) = -sin(π/3) because sine changes sign in quadrant IV compared to the reference angle. Since sin(π/3) = √3/2, the result is -√3/2. This matches the unit-circle quadrant reasoning.

Why Other Options Are Wrong:
• √3/2 is the magnitude but misses the negative sign for quadrant IV. • ±1/2 correspond to 30°-based angles, not 60°-based angles. • 0 corresponds to angles on the x-axis (0, π, 2π), not 5π/3.

Common Pitfalls:
• Using the reference angle value but forgetting the quadrant sign. • Confusing 5π/3 (300°) with 4π/3 (240°), which lies in a different quadrant.

Final Answer:
-√3/2