# Tan 270 Degrees — Undefined, and Why

TL;DR

Tan 270 degrees is undefined, because tangent is sine over cosine and \( \cos 270° = 0 \) — and dividing by zero has no value. This article shows the reason on the unit circle, the sine-over-cosine method, and what "undefined" means as opposed to zero.

The value of \( \tan 270° \) is **undefined**: \( \tan 270° = \frac{\sin 270°}{\cos 270°} = \frac{-1}{0} \), and division by zero is not defined.

> **Quick Answer:**  
> **Result:** \( \tan 270° = 	ext{undefined} \)  
> **Reason:** \( \tan θ = \frac{\sin θ}{\cos θ} \), and \( \cos 270° = 0 \) (division by zero)  
> **Method shown:** sin/cos quotient on the unit circle  
> **Radian equivalent:** \( \tan \left( \frac{3\pi}{2} \right) \)  
> **Note:** undefined is NOT the same as 0

## Quick Reference Table

The tangent of nearby angles, showing how the value behaves around 270°.

| Angle (degrees) | Angle (radians)             | \( \cos \)  | \( \tan \)     |
|------------------|------------------------------|---------------|------------------|
| 180°             | \( \pi \)                  | −1            | 0                |
| 225°             | \( \frac{5\pi}{4} \)      | −\( \frac{\sqrt{2}}{2} \) | 1                |
| 260°             | —                            | small negative | large positive   |
| 270°             | \( \frac{3\pi}{2} \)      | 0             | undefined        |
| 280°             | —                            | small positive | large negative   |
| 315°             | \( \frac{7\pi}{4} \)      | \( \frac{\sqrt{2}}{2} \) | −1               |
| 360°             | \( 2\pi \)                | 1             | 0                |

## What Tangent of an Angle Means

Tangent is the ratio of sine to cosine: \( \tan θ = \frac{\sin θ}{\cos θ} \). On the unit circle — radius 1, centred at the origin — sine is the y-coordinate and cosine is the x-coordinate of the terminal point, so tangent is \( \frac{y}{x} \), the slope of the radius.

Whenever the x-coordinate (cosine) is zero, that slope formula divides by zero and the tangent is **undefined**. This happens on the vertical axis — at 90° and 270°. Undefined is a precise statement: there is no real number that the ratio equals. It is not zero, and it is not infinity in the sense of a value.

## Why Tan 270 Degrees Is Undefined

Why is tan 270 undefined rather than just very large? Trace it through the definition.

### **Step 1: Locate the angle on the unit circle**

At 270° the terminal side lies along the negative y-axis, so the terminal point is (0,−1).

### **Step 2: Read off sine and cosine**

\( \cos 270° = 0, \quad \sin 270° = -1 \)

### **Step 3: Form the tangent ratio**

\( \tan 270° = \frac{\sin 270°}{\cos 270°} = \frac{-1}{0} \)

### **Step 4: Apply the rule on dividing by zero**

Division by zero has no defined result. There is no number \( t \) such that \( t \times 0 = -1 \), so the ratio cannot equal anything.

**Final answer:** \( \tan 270° \) is **undefined**.

The companion value [cos 270 degrees](/content/math/trigonometry/cos-270-degrees/index.html) is exactly 0, and that zero in the denominator is the whole reason tangent breaks here. In radians, the same angle is \( \frac{3\pi}{2} \), so \( \tan \left( \frac{3\pi}{2} \right) \) is undefined for the identical reason.

## Common Mistakes With Tan 270 Degrees

### **Mistake 1: Writing tan 270° = 0**

**Where it slips in:** Confusing the case where the numerator is zero with the case where the denominator is zero.

**Don't do this:** Report \( \tan 270° = 0 \) because something on the axis is zero.

**The correct way:** Tangent is zero only when sine (the numerator) is zero — at 0° and 180°. At 270° it is cosine (the denominator) that is zero, which makes the ratio undefined, not zero.

### **Mistake 2: Calling the value "infinity"**

**Where it slips in:** Seeing the tangent graph shoot upward near 270° and naming the value ∞.

**Don't do this:** Write \( \tan 270° = ∞ \) as the answer.

**The correct way:** The tangent grows without bound as the angle approaches 270° from one side and falls without bound from the other, but at exactly 270° there is no single value — the precise word is **undefined**. The graph shows a vertical asymptote, not a point.

### **Mistake 3: Forgetting cosine controls the breakdown**

**Where it slips in:** Memorising "tan is undefined at 90° and 270°" without the reason, then guessing at other angles.

**Don't do this:** Assume tangent is undefined wherever sine is involved.

**The correct way:** Tangent is undefined exactly where cosine is zero. Checking \( \cos θ=0 \) tells you every undefined angle: 90°, 270°, and every 180° step from them.

## Frequently Asked Questions

Is tan 270 degrees zero or undefined?  
**Undefined.** Tangent is zero when sine is zero (at 0° and 180°); at 270° it is cosine that equals zero, so the ratio \( \frac{-1}{0} \) is undefined.

What is tan 270 in radians?  
**Written** \( \tan \left( \frac{3\pi}{2} \right) \), and it is undefined for the same reason — \( \cos \left( \frac{3\pi}{2} \right) = 0 \).

Why is tan 90 also undefined?  
**Yes** — for the identical reason. At 90° the point is (0,1), so \( \cos 90° = 0 \) and \( \tan 90° = \frac{1}{0} \) is undefined.

Does undefined mean the same as infinity?  
**No.** "Undefined" means no value exists. The tangent grows arbitrarily large near 270° but has no actual value at 270°, which the graph marks with a vertical asymptote.

What are sin 270 and cos 270 degrees?  
**\( \sin 270° = -1 \)** and **\( \cos 270° = 0 \)**. Those two values, plugged into \( \frac{\sin}{\cos} \), are exactly why the tangent is undefined.