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

Trigonometry

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.

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 (−\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}.