# Cos 270 Degrees = 0 — Value, Unit Circle, Radians

## TL;DR

The value of cos 270 degrees is exactly 0. In radians, 270° is \(\frac{3\pi}{2}\), so \(\cos(270°) = \cos\left(\frac{3\pi}{2}\right) = 0\). This article shows why three-quarters of a turn lands on the bottom of the unit circle where the x-coordinate vanishes, provides a standard-angle reference table in degrees and radians, and clarifies common mix-ups.

## Quick Answer:

- **Result:** \(\cos 270° = 0\)
- **Notation:** \(\cos(3\pi/2) = 0\) (radians)
- **Method shown:** unit circle — the x-coordinate at three-quarters of a turn
- **Approximate value:** 0 (exact)

## A rotation of 270°

A rotation of 270° — three-quarters of the way around the unit circle, or \(\frac{3\pi}{2}\) radians — lands at the point (0,−1). Cosine reads the x-coordinate of where you land, and at the bottom of the circle that x-coordinate is 0. The point sits directly below the center, so it has no horizontal offset at all.

## Quick Reference Table — Cosine of Standard Angles

| Angle (degrees) | Angle (radians)  | \(\cos θ\) |
| --- | --- | --- |
| 0°  | 0   | 1  |
| 90° | \(\frac{\pi}{2}\) | 0  |
| 180°| \(\pi\) | -1 |
| 210°| \(\frac{7\pi}{6}\)| -\(\frac{\sqrt{3}}{2}\) |
| 225°| \(\frac{5\pi}{4}\) | -\(\frac{\sqrt{2}}{2}\) |
| 240°| \(\frac{4\pi}{3}\) | -\(\frac{1}{2}\) |
| **270°**| **\(\frac{3\pi}{2}\)** | **0** |
| 300°| \(\frac{5\pi}{3}\) | \(\frac{1}{2}\) |
| 315°| \(\frac{7\pi}{4}\) | \(\frac{\sqrt{2}}{2}\) |
| 330°| \(\frac{11\pi}{6}\) | \(\frac{\sqrt{3}}{2}\) |
| 360°| \(2\pi\) | 1  |

Cosine crosses zero twice per turn — at 90° and again at 270°. Both are points where the circle is at its top or bottom, directly above or below the center, with no horizontal distance from the y-axis.

## Where cos 270 Degrees Appears

The value \(\cos 270° = 0\) marks a quarter-cycle turning point in any oscillation. In a cosine wave modeling a swinging pendulum or an AC voltage, the object is at midline when the phase reaches 270°. The same zero defines the vertical axis in screen and robotics geometry: a heading of 270° points straight down, signifying no left-right component, which is why navigation and robotics treat it as a vertical bearing.

## What is cos 270 Degrees?

Cosine of an angle is the x-coordinate of the point reached by rotating that angle counterclockwise from the positive x-axis. A turn of 270° covers three of the four quarter-turns. That rotation lands on (0,−1) — the lowest point of the circle. Its x-coordinate there is 0, so \(\cos 270° = 0\).

## How To Find The Value of cos 270 Degrees

### Method 1 — Unit Circle

Rotate 270° counterclockwise from (1,0). Three quarter-turns put you at the bottom, at (0,−1).

**Final answer:** \(\cos 270° = 0\).

### Method 2 — Cosine Subtraction Formula

Write 270° as 360°−90° and apply \(\cos(A−B)=\cos A\cos B+\sin A\sin B\)  :

\(\cos(360°−90°)=\cos 360°\cos 90°+\sin 360°\sin 90° = (1)(0)+(0)(1)=0\)

**Final answer:** \(\cos 270° = 0\).

### Method 3 — Co-function Shift

Use \(\cos(270°) = \cos(180°+90°) = -\cos(90°)\).

**Final answer:** \(\cos 270° = 0\).

## Common Mistakes with cos 270 Degrees

### Mistake 1: Confusing \(\cos 270°\) with \(\sin 270°\)

Where it slips in: At 270°, the unit-circle point is (0,−1). Don't confuse the coordinates.

**Correct:** \(\cos 270° = 0
\sin 270° = -1\).

### Mistake 2: Calculator in Degree vs Radian Mode

Reading \(\cos(270)\) while in radians can lead to incorrect results.

**Correct Approach:** Use degree mode for \(\cos 270°\) or enter \(\cos(3\pi/2)\) in radian mode.

### Mistake 3: Writing 0 with a Sign

Zero has no sign. Writing \(\cos 270° = -0\) is incorrect.

**Correct:** \(\cos 270° = 0\).

## Conclusion

- **Cos 270 degrees** equals 0 — the cosine of three-quarters of a turn, written \(\cos(3\pi/2)\) in radians.
- The rotation lands on (0,−1), the bottom of the unit circle, where the x-coordinate is 0.
- Three methods support this: unit circle, subtraction formula, and co-function shift.
- A common slip is confusing cosine and sine — \(\cos 270° = 0\), \(\sin 270° = -1\).

### Quick Self-Check — Try These

1. Evaluate \(\cos 270° + \sin 270°\).
2. Write \(\cos 270°\) in radians and state its value.
3. Explain in one sentence why \(\cos 270°\) and \(\cos 90°\) are equal.

## Frequently Asked Questions

**What is the value of cos 270 degrees?** 0.

**What is the exact value of cos 270 degrees?** Exactly 0.

**What is cos 270 degrees in radians?** 270° equals \(\frac{3\pi}{2}\) radians.

**Is cos 270° the same as cos 90°?** Yes, both are 0.

**Is cos 270° positive or negative?** Neither — it is exactly 0.