Let x be defined by x = sin(-5π/3). Find the exact value of x using standard unit-circle values (no decimal approximation).

Introduction / Context:
This question tests unit-circle trigonometry, specifically evaluating sine at angles measured in radians, including negative angles. A negative angle means rotating clockwise from the positive x-axis. Two common strategies work well: (1) use the odd-function property sin(-θ) = -sin(θ), and (2) identify a coterminal angle between 0 and 2π (or 0° and 360°) and use known special-angle values. Accuracy is mostly about sign and quadrant.

Given Data / Assumptions:

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

Concept / Approach:
Use the identity: sin(-θ) = -sin(θ). First evaluate sin(5π/3). The angle 5π/3 corresponds to 300°, which is in the fourth quadrant, where sine is negative. The reference angle is 60° (π/3), and sin 60° = √3/2. So sin 300° = -√3/2. Then apply the negative-angle identity to get sin(-5π/3).

Step-by-Step Solution:
1) Use the odd function property: sin(-5π/3) = -sin(5π/3) 2) Convert 5π/3 to degrees (optional for recognition): 5π/3 = 300° 3) Identify quadrant and reference angle: 300° is in quadrant IV, reference angle is 60° 4) Use sin 60° = √3/2 and sine is negative in quadrant IV: sin(300°) = sin(5π/3) = -√3/2 5) Apply step 1: sin(-5π/3) = -(-√3/2) = √3/2

Verification / Alternative check:
Find a coterminal positive angle: -5π/3 + 2π = -5π/3 + 6π/3 = π/3. So sin(-5π/3) = sin(π/3) = √3/2. This confirms the same answer without using sin(-θ) = -sin(θ) explicitly.

Why Other Options Are Wrong:
• -√3/2 is sin(5π/3), but the question asks sin(-5π/3). • ±1/2 correspond to 30° reference angles, not 60°. • 0 would correspond to angles like 0, π, 2π, not π/3.

Common Pitfalls:
• Forgetting sine is an odd function and missing the sign change. • Confusing 5π/3 with 4π/3 (different quadrant and sign).

Final Answer:
√3/2