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

[Trigonometry](/content/tag/trigonometry/index.html)

TL;DR  
The value of cos 120 degrees is exactly −\frac{1}{2}, or −0.5. This article explains why the value is negative (120° sits in Quadrant II), how the reference angle of 60° gives the magnitude, a standard-angle table in degrees and radians, plus worked examples and common mistakes.

BT  
Last updated on June 14, 2026  
6 min read

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

> ### **Quick Answer:**  
> - **Result:** cos⁡120°=−\frac{1}{2}  
> - **Decimal:** −0.5  
> - **In radians:** cos⁡(2π/3)=−\frac{1}{2}  
> - **Exact form:** −\frac{1}{2}  
> - **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.

| Angle (degrees) | Angle (radians) | cos⁡θ (exact)  | cos⁡θ (decimal)  |
| --- | --- | --- | --- |
| 0° | 0 | 1 | 1.000 |
| 30° | \frac{\pi}{6} | \frac{\sqrt{3}}{2} | 0.866 |
| 60° | \frac{\pi}{3} | \frac{1}{2} | 0.500 |
| 90° | \frac{\pi}{2} | 0 | 0.000 |
| 120° | \frac{2\pi}{3} | −\frac{1}{2} | −0.500 |
| 135° | \frac{3\pi}{4} | −\frac{\sqrt{2}}{2} | −0.707 |
| 150° | \frac{5\pi}{6} | −\frac{\sqrt{3}}{2} | −0.866 |
| 180° | \pi | −1 | −1.000 |

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](/content/math/geometry/unit-circle/index.html), 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 (−\frac{1}{2}, \frac{\sqrt{3}}{2}) 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 (−\frac{1}{2}, \frac{\sqrt{3}}{2}).

cos⁡120°=x-coordinate=−\frac{1}{2}.

### **Method 3: Supplementary-angle identity**  
The identity cos⁡(180°−θ)=−cos⁡θ applies directly:
cos⁡120°=cos⁡(180°−60°)=−cos⁡60°=−\frac{1}{2}.

## Examples of Cos 120 Degrees  
### Example 1  
**Evaluate 6cos⁡120°.**  
6cos⁡120°=6×(−\frac{1}{2})=−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 does not match the known value.  
Correct. Cosine does not distribute over addition. Use cos⁡(90°+θ)=−sin⁡θ:
cos⁡120°=cos⁡(90°+30°)=−sin⁡30°=−\frac{1}{2}.

### Example 3  
**Evaluate cos⁡120°+cos⁡60°.**  
−\frac{1}{2}+\frac{1}{2}=0  
### Example 4  
**Verify cos²120°+sin²120°=1, given sin⁡120°=\frac{\sqrt{3}}{2}.**  
  
(−\frac{1}{2})²+(\frac{\sqrt{3}}{2})²=\frac{1}{4} + \frac{3}{4} = 1  
### Example 5  
**Express 120° in radians and evaluate cos⁡(2π/3).**  
120°=\frac{2\pi}{3}, so cos⁡(2π/3)=−\frac{1}{2}.

## Where Things Go Sideways With Cos 120 Degrees  
### Mistake 1: Dropping the negative sign  
Reading the magnitude \frac{1}{2} off the reference angle and forgetting to apply the Quadrant II sign.  
### Mistake 2: Distributing cosine over a sum  
Rewriting 120° as 90°+30° and splitting the cosine.
### Mistake 3: Using 120° as its own reference angle  
Plugging 120° straight into a Quadrant I shortcut without reducing it.

## Key Takeaways  
- Cos 120 degrees equals −\frac{1}{2} (or −0.5).  
- The reference angle is 60°, giving the magnitude \frac{1}{2}; Quadrant II makes it negative.  
- In radians, cos⁡120°=cos⁡(2π/3).  
- The most common mistake is dropping the negative sign — always check the quadrant before writing the answer.

## Try These Before Moving On  
1. Evaluate 4cos⁡120°+2sin⁡120°.  
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}.