Cos 120 Degrees - Value −1/2 Explained (2026)

The value of cos 120 degrees is $-\frac{1}{2}$, or $-0.5$.

Quick Answer:

• Result: $\cos 120° = -\dfrac{1}{2}$

• Decimal: $-0.5$

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

• Exact form: $-\dfrac{1}{2}$ (a standard angle — exact, not rounded)

• Methods shown: reference angle (Quadrant II) · unit circle x-coordinate · supplementary identity

Standard-Angle Cosine Reference Table

One hundred twenty degrees is a standard angle, so its cosine has an exact fraction. Here are the common angles spanning Quadrants I and II, in degrees and radians.

Notice the sign flip at $90°$: cosine is positive in Quadrant I and negative in Quadrant II. Cos 120° and cos 60° share the same magnitude, $\frac{1}{2}$, but opposite signs — because $60°$ is the reference angle of $120°$.

Where Cos 120 Degrees Shows Up

Angles past $90°$ appear the moment something points backward. In a three-phase electrical system, the three voltages are spaced exactly $120°$ apart, so each phase relates to the next through $\cos 120° = -\frac{1}{2}$ — the spacing that lets power grids deliver smooth, balanced current.

The same $120°$ separation defines the bonds in a trigonometric model of a graphite sheet, where carbon atoms sit at hexagon vertices, and any vector-addition problem with two forces $120°$ apart carries that $-\frac{1}{2}$ in its dot-product term. The negative cosine is read straight off the unit circle, where Quadrant II points have a negative $x$-coordinate.

What Cos 120 Degrees Means

On the unit circle — a circle of radius $1$ centred at the origin — the cosine of an angle is the $x$-coordinate of the point where the angle's radius meets the circle. The angle $120°$ rotates counterclockwise past the vertical into the upper-left region (Quadrant II), landing at $\left(-\frac{1}{2}, \frac{\sqrt{3}}{2}\right)$ whose negative $x$-coordinate gives $\cos 120° = -\frac{1}{2}$.

The right-triangle definition — adjacent over hypotenuse — only covers acute angles, so for $120°$ the unit circle is the home definition. The triangle still helps through the reference angle, the acute angle between the radius and the $x$-axis, which for $120°$ is $180° - 120° = 60°$.

How Do You Find the Value of Cos 120 Degrees?

The reference-angle method does it in two steps: find the magnitude from the acute partner, then fix the sign from the quadrant. All three routes below give $-\frac{1}{2}$.

Method 1: Reference angle

The reference angle for $120°$ is the acute angle to the negative $x$-axis:

$$180° - 120° = 60°$$

The cosine magnitude matches the reference angle: $\cos 60° = \frac{1}{2}$. Now fix the sign — $120°$ is in Quadrant II, where cosine is negative:

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

Method 2: Unit circle

Rotate the unit radius $120°$ counterclockwise. It lands at $\left(-\frac{1}{2}, \frac{\sqrt{3}}{2}\right)$.

$$\cos 120° = x\text{-coordinate} = -\frac{1}{2}$$

Method 3: Supplementary-angle identity

The identity $\cos(180° - \theta) = -\cos\theta$ applies directly:

$$\cos 120° = \cos(180° - 60°) = -\cos 60° = -\frac{1}{2}$$

Examples of Cos 120 Degrees

Example 1

Evaluate $6\cos 120°$.

$$6\cos 120° = 6 \times \left(-\frac{1}{2}\right) = -3$$

Example 2

Find $\cos 120°$ using the angle $90° + 30°$.

Wrong attempt. A student treats $\cos(90° + 30°)$ as $\cos 90° + \cos 30° = 0 + \frac{\sqrt{3}}{2} = \frac{\sqrt{3}}{2}$.

That cannot be right — it gives a positive number, but $120°$ is in Quadrant II where cosine is negative, and the magnitude $0.866$ does not match the known value $0.5$.

Correct. Cosine does not distribute over addition. Use $\cos(90° + \theta) = -\sin\theta$:

$$\cos 120° = \cos(90° + 30°) = -\sin 30° = -\frac{1}{2}$$

Example 3

Evaluate $\cos 120° + \cos 60°$.

$$-\frac{1}{2} + \frac{1}{2} = 0$$

The reference-angle partners cancel, which is the sign-symmetry of cosine across $90°$.

Example 4

Verify $\cos^2 120° + \sin^2 120° = 1$, given $\sin 120° = \frac{\sqrt{3}}{2}$.

$$\left(-\frac{1}{2}\right)^2 + \left(\frac{\sqrt{3}}{2}\right)^2 = \frac{1}{4} + \frac{3}{4} = 1$$

The Pythagorean identity holds even though cosine is negative — squaring removes the sign.

Example 5

Express $120°$ in radians and evaluate $\cos\left(\frac{2\pi}{3}\right)$.

$120° = \frac{2\pi}{3}$ radians, so $\cos\left(\frac{2\pi}{3}\right) = \cos 120° = -\frac{1}{2}$.

Where Things Go Sideways With Cos 120 Degrees

Mistake 1: Dropping the negative sign

Where it slips in: Reading the magnitude $\frac{1}{2}$ off the reference angle and forgetting to apply the Quadrant II sign.

Don't do this: Writing $\cos 120° = \frac{1}{2}$ because $\cos 60° = \frac{1}{2}$.

The correct way: Cosine is negative in Quadrant II, so $\cos 120° = -\frac{1}{2}$

Mistake 2: Distributing cosine over a sum

Where it slips in: Rewriting $120°$ as $90° + 30°$ and splitting the cosine.

Don't do this: Writing $\cos(90° + 30°) = \cos 90° + \cos 30°$.

The correct way: Cosine is not linear. Use the angle-sum identity $\cos(A+B) = \cos A\cos B - \sin A\sin B$, or the shortcut $\cos(90° + \theta) = -\sin\theta$.

Mistake 3: Using 120° as its own reference angle

Where it slips in: Plugging $120°$ straight into a Quadrant I shortcut without reducing it.

Don't do this: Treating $\cos 120°$ as if $120°$ were acute.

The correct way: Reduce to the reference angle first — $180° - 120° = 60°$ — then attach the quadrant's sign

Key Takeaways

• Cos 120 degrees equals $-\frac{1}{2}$ (or $-0.5$), an exact value because $120°$ is a standard angle.

• The reference angle is $60°$, giving the magnitude $\frac{1}{2}$; Quadrant II makes it negative.

• In radians, $\cos 120° = \cos\left(\frac{2\pi}{3}\right)$.

• The most common mistake is dropping the negative sign — always check the quadrant before writing the answer.

Try These Before Moving On

1. Evaluate $4\cos 120° + 2\sin 120°$ (use $\sin 120° = \frac{\sqrt{3}}{2}$).

2. Find the reference angle of $120°$ and use it to write $\cos 120°$ from scratch.

3. Show that $\cos 120° = \cos(360° - 240°)$ and confirm both equal $-\frac{1}{2}$.

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