In trigonometric simplification, find the exact value of sec(330°) using standard trigonometric values and quadrant knowledge.

Introduction / Context:
This trigonometry question asks you to evaluate sec(330°), which is the reciprocal of cos(330°). To solve it, you need to use your knowledge of standard angles on the unit circle and the signs of trigonometric functions in different quadrants. Such questions commonly appear in aptitude tests and school exams to check conceptual clarity about angles and trigonometric ratios.

Given Data / Assumptions:

• The angle is 330°, measured from the positive x-axis in standard position.
• We must find sec(330°) exactly, not approximately.
• Cosine and secant are reciprocal functions: sec(θ) = 1 / cos(θ).
• We assume standard trigonometric values for special angles like 30°, 45°, and 60° are known.

Concept / Approach:
First, express 330° as a related reference angle. Since 330° = 360° − 30°, the reference angle is 30°. In the fourth quadrant (between 270° and 360°), cos(θ) is positive and sin(θ) is negative. Therefore, cos(330°) has the same magnitude as cos(30°) but is positive. After finding cos(330°), we take its reciprocal to obtain sec(330°) and then match the result with the given options.

Step-by-Step Solution:
1) Recognize that 330° = 360° − 30°, so the reference angle is 30° in the fourth quadrant. 2) In the fourth quadrant, the cosine of an angle is positive. 3) We know that cos(30°) = √3 / 2. 4) Therefore, cos(330°) = cos(360° − 30°) = cos(30°) = √3 / 2. 5) By definition, sec(θ) = 1 / cos(θ). Thus sec(330°) = 1 / (√3 / 2) = 2 / √3. 6) This matches option c exactly.

Verification / Alternative check:
We can verify sign and magnitude by considering coordinates on the unit circle. The point corresponding to 330° has coordinates (cos 330°, sin 330°). Because it is 30° below the positive x-axis, the x-coordinate is cos 330° = √3 / 2 and the y-coordinate is sin 330° = −1/2. The secant is the reciprocal of the x-coordinate, giving sec 330° = 1 / (√3 / 2) = 2 / √3. This matches the earlier calculation and confirms the correctness of the result.

Why Other Options Are Wrong:
Option a (2) would correspond to sec(60°) or sec(300°), not sec(330°). Option b (−2) suggests a negative cosine value, which is not correct for an angle in the fourth quadrant where cosine is positive. Option d (−2/√3) has the wrong sign. Option e (√3/2) is actually a cosine value (cos 30°), not a secant value. Only 2/√3 correctly represents sec(330°).

Common Pitfalls:
A typical error is to confuse 330° with 300° or another nearby angle, leading to the wrong standard value. Another mistake is misremembering which trigonometric functions are positive in each quadrant; for the fourth quadrant, cosine and secant are positive, while sine and tangent are negative. Carefully identifying the reference angle and quadrant before applying standard values helps avoid such errors.

Final Answer:
The exact value of sec(330°) is 2/√3.