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

Quick Answer:

• Result: $\cos\left(\dfrac{2\pi}{3}\right) = -\dfrac{1}{2}$

• Notation: $\cos 120° = -0.5$ (degrees)

• Method shown: unit circle — the $x$-coordinate in Quadrant II

• Approximate value: $-0.5$ (exact)

• Exact form: $-\dfrac{1}{2}$

The angle $\frac{2\pi}{3}$ 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 $\left(-\frac{1}{2}, \frac{\sqrt{3}}{2}\right)$. Cosine reads the $x$-coordinate, which is $-\frac{1}{2}$. The magnitude $\frac{1}{2}$ 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

This table lists cosine at the standard angles in both radians and degrees, with $\frac{2\pi}{3}$ highlighted.

Notice the mirror: $\cos 60° = \frac{1}{2}$ and $\cos 120° = -\frac{1}{2}$ 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 $\cos\left(\frac{2\pi}{3}\right) = -\frac{1}{2}$ shows up wherever three things are spaced evenly around a circle. The three phases of an alternating-current power system are set $\frac{2\pi}{3}$ radians ($120°$) apart, and the cosine of that spacing, $-\frac{1}{2}$, is exactly 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 $\left(-\frac{1}{2}, \frac{\sqrt{3}}{2}\right)$. 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 $\frac{2\pi}{3}$ is $120°$, which lands in the second quadrant.

The terminal point there is $\left(-\frac{1}{2}, \frac{\sqrt{3}}{2}\right)$. Its $x$-coordinate is $-\frac{1}{2}$, so $\cos\left(\frac{2\pi}{3}\right) = -\frac{1}{2}$. The reference angle is $60°$ (the gap to the negative $x$-axis is $180° - 120° = 60°$), which is why the magnitude matches $\cos 60° = \frac{1}{2}$.

How To Find The Value of cos 2pi/3

Method 1 — Unit circle

Rotate $120°$ ($\frac{2\pi}{3}$ radians) counterclockwise into the second quadrant. The terminal point is $\left(-\frac{1}{2}, \frac{\sqrt{3}}{2}\right)$.

Cosine is the $x$-coordinate of that point.

Final answer: $\cos\left(\frac{2\pi}{3}\right) = -\frac{1}{2}$.

Method 2 — Reference angle

The reference angle for $120°$ is $180° - 120° = 60°$, and $\cos 60° = \frac{1}{2}$.

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

$$\cos 120° = -\cos 60° = -\frac{1}{2}$$

Final answer: $\cos\left(\frac{2\pi}{3}\right) = -\frac{1}{2}$.

Method 3 — Supplementary-angle identity

Use $\cos(\pi - \theta) = -\cos\theta$ with $\theta = \frac{\pi}{3}$, since $\pi - \frac{\pi}{3} = \frac{2\pi}{3}$:

$$\cos\left(\frac{2\pi}{3}\right) = \cos\left(\pi - \frac{\pi}{3}\right) = -\cos\frac{\pi}{3} = -\frac{1}{2}$$

Common Mistakes With cos 2pi/3

Mistake 1: Dropping the negative sign

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

Don't do this: Writing $\cos\left(\frac{2\pi}{3}\right) = \frac{1}{2}$ — the right size with the wrong sign.

The correct way: At $120°$ the point is in the left half of the circle, so the $x$-coordinate is negative: $\cos\left(\frac{2\pi}{3}\right) = -\frac{1}{2}$.

Mistake 2: Confusing the x and y coordinates

Where it slips in: The terminal point is $\left(-\frac{1}{2}, \frac{\sqrt{3}}{2}\right)$, and the two coordinates get swapped.

Don't do this: Writing $\cos\left(\frac{2\pi}{3}\right) = \frac{\sqrt{3}}{2}$ (that is $\sin\left(\frac{2\pi}{3}\right)$, the $y$-coordinate).

The correct way: Cosine is the $x$-coordinate, so $\cos\left(\frac{2\pi}{3}\right) = -\frac{1}{2}$; sine is the $y$-coordinate, so $\sin\left(\frac{2\pi}{3}\right) = \frac{\sqrt{3}}{2}$.

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

Where it slips in: The fraction $\frac{2\pi}{3}$ gets read as $\frac{\pi}{3}$ ($60°$) or $\frac{2\pi}{6}$.

Don't do this: Treating $\frac{2\pi}{3}$ as $60°$ and reporting $\frac{1}{2}$.

The correct way: $\frac{2\pi}{3} = 120°$, a second-quadrant angle, so the answer is $-\frac{1}{2}$, not the first-quadrant $\frac{1}{2}$.

Conclusion

• Cos 2pi/3 equals $-\frac{1}{2}$ — the cosine of $120°$, a second-quadrant angle.

• The terminal point is $\left(-\frac{1}{2}, \frac{\sqrt{3}}{2}\right)$; cosine reads the $x$-coordinate, $-\frac{1}{2}$.

• 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.

Sharpen your cos 2pi/3 — three practice problems

1. Evaluate $\cos\left(\frac{2\pi}{3}\right) + \cos\left(\frac{\pi}{3}\right)$.

2. Convert $\frac{2\pi}{3}$ to degrees and state the cosine.

3. Explain in one sentence why $\cos\left(\frac{2\pi}{3}\right)$ is negative.

If #1 didn't give $0$, recheck the signs in each quadrant. Want a live Bhanzu trainer to walk through more unit-circle problems? Book a free demo class — online globally.