What is the exact value of sec(−2π/3) when the angle is measured in radians on the unit circle?

Introduction / Context:
This question tests your understanding of how the secant function behaves for angles expressed in radians, particularly negative angles. To find sec(−2π/3), you must first determine cos(−2π/3) and then take its reciprocal. Recognising the symmetry of cosine and identifying the equivalent positive angle on the unit circle are crucial steps.

Given Data / Assumptions:

• The angle is −2π/3 radians.
• sec θ is defined as 1 / cos θ.
• Standard exact values for cosine at well known angles such as π/3 and 2π/3 are used.
• The cosine function is even: cos(−θ) = cos θ.

Concept / Approach:
We first simplify cos(−2π/3). Using the even property of cosine, cos(−θ) = cos θ, so cos(−2π/3) = cos(2π/3). The angle 2π/3 corresponds to 120°, which lies in the second quadrant where cosine is negative. Its reference angle is π/3 (60°). Since cos π/3 = 1/2, it follows that cos 2π/3 = −1/2. Once we find cos(−2π/3), we invert it to obtain sec(−2π/3).

Step-by-Step Solution:
Use evenness of cosine: cos(−2π/3) = cos(2π/3).Convert 2π/3 to degrees: 2π/3 * (180/π) = 120°, which lies in the second quadrant.In the second quadrant, cosine is negative, and the reference angle is 60°.We know cos 60° = 1/2, so cos 120° = −1/2.Therefore cos(−2π/3) = −1/2.Now use the definition sec θ = 1 / cos θ: sec(−2π/3) = 1 / (−1/2) = −2.

Verification / Alternative check:
Another way is to think in terms of the unit circle coordinates. The angle 2π/3 has coordinates (cos 2π/3, sin 2π/3) = (−1/2, √3/2). By the even property, cos(−2π/3) = cos(2π/3) = −1/2. Therefore sec(−2π/3) must be −2. This agrees with our earlier reasoning and confirms that −2 is the exact value.

Why Other Options Are Wrong:

• 2 would be the secant value if cosine were 1/2, which occurs at π/3, not 2π/3.
• 2/√3 and −2/√3 correspond to cosine values of √3/2 and −√3/2, which belong to angles like π/6 and 5π/6, not 2π/3.
• 0 is impossible for secant because secant is the reciprocal of cosine, and cosine cannot be infinite; secant is undefined where cosine is zero, not zero itself.

Common Pitfalls:

• Forgetting that cosine is an even function and incorrectly negating the cosine value for negative angles.
• Mixing up sine and cosine values at 60° and 120°.
• Confusing which quadrants have positive or negative cosine values.

Final Answer:
-2