If cos(2π/3) = x in trigonometry, then what is the exact value of x?

Introduction / Context:
This question tests familiarity with standard trigonometric values in radian measure. The angle 2π/3 radians corresponds to 120 degrees, which lies in the second quadrant. Recognizing this and recalling the reference angle allows quick evaluation of the cosine value.

Given Data / Assumptions:

• cos(2π/3) = x
• We interpret angles on the standard unit circle.
• We know basic values such as cos(π/3).

Concept / Approach:
We note that:

• 2π/3 radians equals 120 degrees.
• cos(π - θ) = -cos θ.
• π/3 radians equals 60 degrees, for which cos(π/3) = 1/2.

Since 2π/3 = π - π/3, we can apply the identity for cosine in the second quadrant.

Step-by-Step Solution:
Express 2π/3 as π - π/3 Use identity cos(π - θ) = -cos θ So cos(2π/3) = cos(π - π/3) = -cos(π/3) We know cos(π/3) = 1/2 Therefore cos(2π/3) = -1/2 Hence x = -1/2

Verification / Alternative check:
On the unit circle, 2π/3 places the terminal point in the second quadrant with x coordinate negative and magnitude equal to cos(π/3). Since the reference angle is π/3, the magnitude is 1/2, but because the point is in the second quadrant, cosine is negative. This geometric picture confirms the algebraic identity.

Why Other Options Are Wrong:
Option b, 1/2, would be correct for cos(π/3), not cos(2π/3). Option c, -√3/2, corresponds to sine of 2π/3 or cosine of other angles. Option d, √3/2, is positive and not valid for a second quadrant cosine value. Option e, 0, is the cosine of π/2, not 2π/3.

Common Pitfalls:
Learners often confuse sine and cosine values for special angles or forget the sign rules for different quadrants. Remember that in the second quadrant, cosine is negative and sine is positive. Converting radians to a familiar degree measure like 120 degrees can make reasoning more intuitive.

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