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°.
Angle (degrees) | Angle (radians) | $\cos$ | $\tan$ value |
|---|---|---|---|
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: $\tan\theta = \frac{\sin\theta}{\cos\theta}$. 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.
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