# 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

| Angle (radians)       | Angle (degrees) | cos θ                |
|----------------------|----------------|----------------------|
| 0                    | 0°             | 1                    |
| π/6                  | 30°            | √3/2                |
| π/4                  | 45°            | √2/2                |
| π/3                  | 60°            | 1/2                  |
| π/2                  | 90°            | 0                    |
| **2π/3**            | **120°**       | **−1/2**             |
| 3π/4                | 135°           | −√2/2               |
| 5π/6                | 150°           | −√3/2               |
| π                    | 180°           | −1                   |

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.