Evaluate the exact value of tan(-240°) using trigonometric periodicity and symmetry properties.

Introduction / Context:
This question tests your understanding of the periodicity and symmetry of the tangent function for angles measured in degrees. Instead of computing tan(-240°) directly, you can reduce the angle to a simpler equivalent angle using known properties such as tan(θ + 180°) = tan θ and tan(-θ) = -tan θ. Recognising these patterns is crucial in trigonometry and aptitude tests.

Given Data / Assumptions:

• The angle is -240°.
• We are asked to find tan(-240°).
• All trigonometric functions are considered in degrees.

Concept / Approach:
The tangent function has period 180°, which means tan(θ + 180°) = tan θ for all θ where tangent is defined. Also, tangent is an odd function, so tan(-θ) = -tan θ. We can use either or both of these properties to simplify -240° to one of the standard angles like 60° or -60°, for which exact tangent values are known. In this case, working with 180° shifts and symmetry around 0° is convenient.

Step-by-Step Solution:
Start with tan(-240°). Use periodicity: add 180° to the angle, since tan(θ + 180°) = tan θ. -240° + 180° = -60°. Therefore, tan(-240°) = tan(-60°). Now use the odd function property: tan(-θ) = -tan θ. So tan(-60°) = -tan 60°. We know tan 60° = sqrt(3). Hence, tan(-60°) = -sqrt(3). Therefore, tan(-240°) = -sqrt(3).

Verification / Alternative check:
An alternative path is to add 360° to -240° to get a coterminal angle in the standard 0° to 360° range. -240° + 360° = 120°. Now compute tan 120°. Since 120° = 180° - 60°, the reference angle is 60°, and tangent in the second quadrant (between 90° and 180°) is negative. So tan 120° = -tan 60° = -sqrt(3). This is the same value we obtained earlier, confirming the result.

Why Other Options Are Wrong:
1 / sqrt(3) and -1 / sqrt(3) correspond to tangent values at 30° and -30°, not 60° or -60°. sqrt(3) is the tangent of 60°, which would be correct for tan 60° or tan 240°, but not tan(-240°). The value 0 is tangent at angles like 0°, 180°, and 360°, which are not equivalent to -240° in terms of tangent's behaviour.

Common Pitfalls:
One common mistake is to treat tangent as having a 360° period instead of 180°, which can lead to unnecessary steps. Another error is mishandling signs when dealing with negative angles or angles in different quadrants. Always reduce angles step by step and use the known exact values for 30°, 45°, and 60° to guide you.

Final Answer:
The value of tan(-240°) is -sqrt(3).