Tan 270 Degrees — Undefined, and Why

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° = undefined

Reason: tan θ = sin θ / cos θ, and cos 270° = 0 (division by zero)

Method shown: sin/cos quotient on the unit circle

Radian equivalent: tan(3π/2)

Note: undefined is NOT the same as 0

Quick Reference Table

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

What Tangent of an Angle Means

Tangent is the ratio of sine to cosine: $\tan\theta = \frac{\sin\theta}{\cos\theta}$. 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

$$\cos 270° = 0, \qquad \sin 270° = -1$$

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 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 $\tan\frac{3\pi}{2}$ 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 $\tan 270° = \infty$ 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 $\cos\theta = 0$ tells you every undefined angle: 90°, 270°, and every 180° step from them.

To work through the unit circle and the axis angles with a live teacher, Bhanzu's trigonometry tutor and high school math tutor build up why tangent breaks at 90° and 270°, not just that it does.

Read More

• Trigonometric ratios and their definitions

• Sin Cos Tan — how the three ratios connect

• Trigonometric functions, their graphs, and asymptotes

• The trigonometric table of standard angle values

• Domain and range of trigonometric functions