Cos 2pi/3 = −1/2 — Value of cos(2π/3) on Unit Circle

TL;DR

The value of cos 2pi/3 is exactly −1/2, which is −0.5. In degrees, 2π/3 is 120°, an angle in the second quadrant where cosine is negative. This article shows why cos(2π/3) = −1/2 on the unit circle, gives a standard-angle reference table in radians and degrees, and clears up the sign slip students hit most.

Quick Answer:

• Result: cos(2π/3) = −1/2
• Notation: cos 120° = −0.5 (degrees)
• Method shown: unit circle — the x-coordinate in Quadrant II
• Approximate value: −0.5 (exact)
• Exact form: −1/2

The angle 2π/3 is two-thirds of the way to a half-turn — 120°, sitting in the second quadrant on the upper-left of the unit circle. Its terminal point is (−1/2, √3/2). Cosine reads the x-coordinate, which is −1/2. The magnitude 1/2 matches cos 60°, but the sign flips negative because the point is in the left half of the circle.

Quick Reference Table — Cosine of Standard Angles

Notice the mirror: cos 60° = 1/2 and cos 120° = −1/2 share a magnitude. The angles 60° and 120° are reflections across the vertical axis, so their cosines are equal and opposite.

Where cos 2pi/3 Appears

The value cos(2π/3) = −1/2 shows up wherever three things are spaced evenly around a circle. The three phases of an alternating-current power system are set 2π/3 radians (120°) apart, and the cosine of that spacing, −1/2, is what makes the three phase voltages sum to zero at every instant.

The same 120° spacing appears in the geometry of an equilateral triangle's exterior angles and in the cube roots of unity in complex numbers, where one root sits at (−1/2, √3/2). Any "one-third of the way around" structure carries this value.

What is cos 2pi/3?

Cosine of an angle is, on the unit circle, the x-coordinate of the point reached by rotating that angle counterclockwise from the positive x-axis. The angle 2π/3 is 120°, which lands in the second quadrant.

The terminal point there is (−1/2, √3/2). Its x-coordinate is −1/2, so cos(2π/3) = −1/2. The reference angle is 60° (the gap to the negative x-axis is 180°−120°=60°), which is why the magnitude matches cos 60° = 1/2.

How To Find The Value of cos 2pi/3

Method 1 — Unit circle

Rotate 120° (2π/3 radians) counterclockwise into the second quadrant. The terminal point is (−1/2, √3/2).

Cosine is the x-coordinate of that point.

Final answer: cos(2π/3) = −1/2.

Method 2 — Reference angle

The reference angle for 120° is 180°−120°=60°, and cos 60° = 1/2.

In the second quadrant cosine is negative, so attach the minus sign:

cos 120° = −cos 60° = −1/2.

Method 3 — Supplementary-angle identity

Use cos(π−θ) = −cos θ with θ=π/3, since π−π/3 = 2π/3:

cos(2π/3) = cos(π−π/3) = −cos π/3 = −1/2.

Common Mistakes With cos 2pi/3

Mistake 1: Dropping the negative sign

Where it slips in: The reference angle 60° gives a magnitude of 1/2, and the second-quadrant sign gets forgotten.

Don't do this: Writing cos(2π/3) = 1/2 — the right size with the wrong sign.

Mistake 2: Confusing the x and y coordinates

Where it slips in: The terminal point is (−1/2, √3/2), and the two coordinates get swapped.

Don't do this: Writing cos(2π/3) = √3/2 (that is sin(2π/3)).

Mistake 3: Misreading 2π/3 as a different angle

Where it slips in: The fraction 2π/3 gets read as π/3 (60°) or 2π/6.

Conclusion

• Cos 2pi/3 equals −1/2 — the cosine of 120°, a second-quadrant angle.
• The terminal point is (−1/2, √3/2); cosine reads the x-coordinate, −1/2.
• Three routes agree: unit circle, the 60° reference angle with a second-quadrant sign, and the supplementary-angle identity.
• The most common slip is dropping the negative sign — the magnitude matches cos 60°, but the sign is negative.

Frequently Asked Questions

What is the value of cos 2pi/3? −1/2, or −0.5. The same angle in degrees is 120°.

What is the exact value of cos 2pi/3? Exactly −1/2.

What is cos 2pi/3 in degrees? 2π/3 radians equals 120°.

Why is cos 2pi/3 negative? Because 2π/3 (120°) lies in the second quadrant, where the unit-circle x-coordinate is negative.

Is cos 2pi/3 the same as cos pi/3? No. cos π/3 = 1/2 (first quadrant), while cos 2π/3 = −1/2 (second quadrant). Same magnitude, opposite sign.