If cosec(4π/3) = x in trigonometry, where the angle 4π/3 is measured in radians, what is the exact value of x expressed as a simplified surd?

Introduction / Context:
This question checks your understanding of trigonometric ratios for standard angles, as well as your ability to work with cosecant, which is the reciprocal of sine. The angle is given in radians as 4π/3, which corresponds to 240 degrees, a standard angle in the unit circle.

Given Data / Assumptions:

• Angle θ = 4π/3 radians.
• cosec(θ) = x.
• cosec(θ) is defined as 1 / sin(θ).
• We are working with exact values on the unit circle.

Concept / Approach:
To find cosec(4π/3), we first need sin(4π/3). The angle 4π/3 is in the third quadrant, where sine is negative. Reference angles and the symmetry of the unit circle allow us to relate this to sin(π/3). Once sin(4π/3) is known, we simply take its reciprocal to obtain cosec(4π/3).

Step-by-Step Solution:
Step 1: Convert 4π/3 to degrees to understand the quadrant. 4π/3 radians corresponds to 240 degrees. Step 2: The reference angle for 240 degrees is 240 − 180 = 60 degrees, which is π/3 radians. Step 3: We know that sin(π/3) = √3 / 2. Step 4: In the third quadrant, sine is negative, so sin(4π/3) = −√3 / 2. Step 5: By definition, cosec(θ) = 1 / sin(θ). Therefore, cosec(4π/3) = 1 / (−√3 / 2). Step 6: Simplify the reciprocal: 1 / (−√3 / 2) = −2 / √3. Step 7: Thus, x = −2 / √3.

Verification / Alternative check:
You can confirm by remembering the pattern of signs in each quadrant: sine is positive in the first and second quadrants and negative in the third and fourth. Since 4π/3 lies in the third quadrant with reference angle π/3, the magnitude of sine is √3 / 2 and the sign is negative. Taking the reciprocal again gives −2 / √3, which matches our result.

Why Other Options Are Wrong:
2/√3 is incorrect because it ignores the negative sign of sine in the third quadrant.
1/√3 and −1/2 do not correspond to the reciprocal of ±√3 / 2 and therefore are not correct values of cosec(4π/3).
√3/2 is actually the sine of 60 degrees, not a cosecant value for 4π/3.

Common Pitfalls:
Many students forget to consider the quadrant and hence the sign of the trigonometric function. Another frequent error is to confuse sine and cosecant or to take the wrong reciprocal. Always identify the quadrant, find the reference angle, determine the correct sign for sine or cosine, and only then take the reciprocal if needed.

Final Answer:
The exact value of x is −2 / √3.