Evaluate the exact value of x if sin(−4π/3) = x, working carefully with signs and the position of the angle on the unit circle.

Introduction / Context:
This trigonometry question asks you to evaluate the sine of a negative angle measured in radians, specifically sin(−4π/3). To do this correctly, you need to understand the periodicity and symmetry of the sine function and be comfortable working with angles on the unit circle. Negative angles are measured clockwise, which can cause confusion if the reference angle and quadrant are not identified clearly.

Given Data / Assumptions:

• The angle is −4π/3 radians.
• We are told that sin(−4π/3) = x and asked to find x.
• Trigonometric functions are understood in terms of the unit circle.
• We assume standard properties of sine for real angles.

Concept / Approach:
The sine function is odd, which means sin(−θ) = −sin θ for any real angle θ. Therefore, we can first find sin(4π/3) and then take the negative of that value to obtain sin(−4π/3). The angle 4π/3 lies in the third quadrant, where sine is negative. By identifying its reference angle and using known values for special angles, we can determine sin(4π/3) exactly.

Step-by-Step Solution:
Step 1: Use the identity sin(−θ) = −sin θ.Step 2: Therefore sin(−4π/3) = −sin(4π/3).Step 3: Consider the angle 4π/3 radians. It is equal to 240°, which lies in the third quadrant.Step 4: The reference angle for 4π/3 is 4π/3 − π = π/3 (that is, 60°).Step 5: In the third quadrant, sine is negative and has the same magnitude as sin(π/3) at this reference angle.Step 6: We know sin(π/3) = √3/2, so sin(4π/3) = −√3/2.Step 7: Now use the odd function property: sin(−4π/3) = −sin(4π/3) = −(−√3/2) = √3/2.

Verification / Alternative check:
To check the sign, note that −4π/3 can also be represented by adding 2π: −4π/3 + 2π = 2π/3. So sin(−4π/3) = sin(2π/3). The angle 2π/3 equals 120°, which is in the second quadrant where sine is positive. Its reference angle is again π/3, so sin(2π/3) = sin(π/3) = √3/2. This alternate viewpoint confirms that sin(−4π/3) = √3/2.

Why Other Options Are Wrong:
The value −√3/2 corresponds to sin(4π/3), not sin(−4π/3), and has the opposite sign. The options −2 and √2 are outside the range of sine, which is limited to values between −1 and 1 inclusive. The value −2/√3 is also outside the valid range for sine. Only √3/2 lies within the correct range and matches the unit circle analysis for sin(−4π/3).

Common Pitfalls:
A frequent mistake is to forget that sine is an odd function and to assume sin(−θ) = sin θ. Another error is to misidentify the quadrant for 4π/3 or 2π/3, which leads to incorrect signs. Confusing degrees and radians can also cause errors. Working systematically with reference angles and quadrant signs helps ensure the correct exact value.

Final Answer:
The exact value of x in sin(−4π/3) = x is √3/2.