What is the exact trigonometric value of sec(4π/3)?

Introduction / Context:
This question tests your ability to evaluate standard trigonometric values for special angles, in particular angles expressed in radians. Knowing the sine and cosine of standard angles allows you to find secant and other related functions quickly.

Given Data / Assumptions:

- We need to find sec(4π/3).
- 4π/3 is a standard angle on the unit circle.
- sec θ is defined as 1 / cos θ.

Concept / Approach:
The approach is to first convert 4π/3 into degrees or locate it on the unit circle, then use known values of cosine to find secant as its reciprocal. The angle 4π/3 corresponds to 240 degrees, which lies in the third quadrant where cosine is negative.

Step-by-Step Solution:
Step 1: Convert 4π/3 to degrees if you prefer: 4π/3 radians = 240 degrees.Step 2: Recognize that 240 degrees is 180 degrees + 60 degrees, which means the reference angle is 60 degrees and the angle is in the third quadrant.Step 3: The cosine of 60 degrees is 1/2. In the third quadrant, cosine is negative, so cos(240°) = −1/2.Step 4: Since sec θ = 1 / cos θ, we have sec(4π/3) = 1 / (−1/2) = −2.

Verification / Alternative check:
You can verify by using the symmetry of the unit circle. Points at 60 degrees and 240 degrees share the same absolute cosine value but with opposite signs. Since cos(60°) = 1/2 and the third quadrant has cosine negative, cos(240°) must be −1/2, making the reciprocal −2. This matches our calculation.

Why Other Options Are Wrong:
- 2 would be the secant of 60 degrees, not 240 degrees, because cosine is positive in the first quadrant and negative in the third.
- 2/√3 and −2/√3 are associated with angles whose cosine is ±√3/2, not ±1/2.
- 1 would correspond to cos θ = 1, which occurs at 0 degrees or 2π radians, not at 4π/3.

Common Pitfalls:
Many learners confuse the signs of trigonometric functions in different quadrants. Remember the ASTC rule (All, Sin, Tan, Cos) to keep track of which functions are positive in each quadrant. Also do not forget that secant is simply the reciprocal of cosine, so once you find cos θ, sec θ follows immediately.

Final Answer:
The exact value of sec(4π/3) is −2.