# Cos 180 Degrees = −1 — Value, Unit Circle, Radians

TL;DR

The value of cos 180 degrees is exactly −1. In radians, 180° is π, so cos(180°) = cos(π) = −1. This article shows why that single half-turn lands on the leftmost point of the unit circle, gives a standard-angle reference table in degrees and radians, and clears up the slips students hit most.

## Quick Answer:

- **Result:** cos(180°) = −1
- **Notation:** cos(π) = −1 (radians)
- **Method shown:** unit circle — the x-coordinate at the half-turn
- **Approximate value:** −1 (exact, no rounding)
- **Exact form:** −1

A half-turn around the unit circle — exactly 180°, or π radians — carries you from the starting point (1,0) to the point directly opposite, (−1,0). Cosine reads the x-coordinate of where you land, and that x-coordinate is −1. This is the most negative value cosine ever reaches, the bottom of the cosine wave.

## Quick Reference Table — Cosine of Standard Angles

This table lists cosine at the standard angles around the circle, in both degrees and radians, with 180° highlighted.

| Angle (degrees) | Angle (radians) | cos(θ) |
| --- | --- | --- |
| 0° | 0 | 1 |
| 30° | π/6 | √3/2 |
| 45° | π/4 | √2/2 |
| 60° | π/3 | 1/2 |
| 90° | π/2 | 0 |
| 120° | 2π/3 | -1/2 |
| 135° | 3π/4 | -√2/2 |
| 150° | 5π/6 | -√3/2 |
| **180°** | **π** | **−1** |
| 270° | 3π/2 | 0 |
| 360° | 2π | 1 |

Cosine is positive in quadrants I and IV, negative in II and III. At 180° the angle sits exactly on the negative x-axis, where the x-coordinate bottoms out at −1.

## Where cos 180 Degrees Appears

The value cos(180°) = −1 turns up wherever something fully reverses direction. In alternating-current circuits, a voltage and current that are 180° out of phase are exact negatives of each other at every instant, because cos(180°) = −1 scales one to the opposite of the other. It is also the heart of Euler's identity, e^(iπ) = −1, which is just cos(π) + i sin(π) with the sine term vanishing.

## What is cos 180 Degrees?

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. For 180°, that rotation is exactly half a full turn.

A half-turn lands on (−1,0) — the leftmost point of the circle. The x-coordinate there is −1, so cos(180°) = −1. The same angle written in radians is π, which is why cos(180°) and cos(π) are the same number.

## How To Find The Value of cos 180 Degrees

### Method 1 — Unit circle

Rotate 180° counterclockwise from (1,0). You land on the point directly opposite, (−1,0).

Cosine is the x-coordinate of that point.

**Final answer:** cos(180°) = −1.

### Method 2 — Cosine addition formula

Split 180° as 90° + 90° and apply cos(A+B) = cos A cos B − sin A sin B:

cos(90° + 90°) = cos(90°)cos(90°) − sin(90°)sin(90°) = (0)(0) − (1)(1) = −1.

**Final answer:** cos(180°) = −1.

### Method 3 — Supplementary-angle identity

Use cos(180°−θ) = −cos(θ) with θ=0°:

cos(180°) = cos(180°−0°) = −cos(0°) = −(1) = −1.

## Common mistakes with cos 180 degrees

### Mistake 1: Confusing cos 180° with sin 180°

At 180°, the unit-circle point is (−1,0), and the two coordinates get swapped.

**Don't do this:** Writing cos(180°) = 0 (that is sin(180°)).

**The correct way:** Cosine is the x-coordinate, so cos(180°) = −1; sine is the y-coordinate, so sin(180°) = 0.

### Mistake 2: Calculator in degree vs radian mode

Typing `cos(180)` while the calculator is in radian mode.

**Don't do this:** Reading cos(180 rad) ≈ −0.598 and reporting it as cos(180°).

**The correct way:** Set the calculator to degree mode for cos(180°), or enter cos(π) in radian mode. Both give −1.

### Mistake 3: Treating cos 180° as positive

**Where it slips in:** Recalling the magnitude 1 but dropping the sign.

**Don't do this:** Writing cos(180°) = 1.

**The correct way:** 180° is in the second-quadrant boundary on the negative x-axis, where cosine is negative, so cos(180°) = −1.

## Conclusion

- **Cos 180 degrees** equals −1 — the cosine of a half-turn, written cos(π) in radians.
- The half-turn lands on (−1,0), the leftmost point of the unit circle; cosine reads that x-coordinate.
- Three routes agree: unit circle, the addition formula on 90° + 90°, and the supplementary-angle identity.
- The most common slip is swapping cosine and sine — cos(180°) = −1, sin(180°) = 0.

## Quick self-check — try these

1. Evaluate cos(180°) + sin(180°).
2. Write cos(180°) in radians and state its value.
3. Explain in one sentence why cos(180°) is negative.