Tan 270 Degrees — Undefined, and Why
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 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.