In trigonometry, if cos 240° = x, then what is the value of x?

Introduction / Context:
This is a basic question on standard trigonometric values. It checks whether you know the cosine of key angles measured in degrees and can relate angles in different quadrants using reference angles and sign conventions for sine, cosine, and tangent.

Given Data / Assumptions:

• cos 240° = x
• We work in the usual unit circle framework.
• We know standard values for 30°, 60°, and 90°.

Concept / Approach:
The angle 240° lies in the third quadrant. The reference angle is the acute angle formed with the x axis. We use:

• cos(180° + θ) = -cos θ
• cos 60° = 1/2

Since 240° = 180° + 60°, we can evaluate cos 240° using this identity.

Step-by-Step Solution:
Write 240° as 180° + 60° Use identity: cos(180° + θ) = -cos θ So cos 240° = cos(180° + 60°) = -cos 60° We know cos 60° = 1/2 Therefore cos 240° = -1/2 Hence x = -1/2

Verification / Alternative check:
The angle 240° is in the third quadrant, where cosine values are negative because x coordinates on the unit circle are negative there. The reference angle is 60°, so the magnitude is the same as cos 60°, but with a negative sign. This reasoning agrees with the algebraic identity method and confirms that -1/2 is correct.

Why Other Options Are Wrong:
Option b, -√3/2, is the cosine of 150° or 210°, not 240°. Option c, 1/2, and option e, √3/2, are positive values that cannot be correct for a third quadrant angle. Option a, -1/√2, corresponds to cos 135° or cos 225°, not cos 240°.

Common Pitfalls:
Learners sometimes forget which quadrants give positive or negative cosine values, or they confuse the reference angles 30° and 60°. Remember that cosine is positive in the first and fourth quadrants and negative in the second and third. Breaking the angle into 180° plus or minus a standard angle helps avoid confusion.

Final Answer:
The value of x is -1/2.