Tan 90 Degrees : Why It Is Undefined (Not Infinity

What Does Tan 90 Degrees Mean?

Tangent is one of the trigonometric functions, defined for an angle $\theta$ as $\tan\theta = \frac{\sin\theta}{\cos\theta}$. On a circle it reads as the $y$-coordinate divided by the $x$-coordinate of the point where the angle's radius meets the circle. So $\tan 90^\circ$ asks for a ratio at the very top of the circle.

At $90^\circ$ the terminal point sits on the positive $y$-axis, at $(0, 1)$. The $x$-coordinate there is exactly $0$, so the ratio $\frac{y}{x}$ becomes $\frac{1}{0}$. Division by zero produces no number, so the tangent is undefined - a genuine gap, not a very large value.

Where Tan 90 Degrees Shows Up

A perfectly vertical line has a slope that is undefined, and $\tan 90^\circ$ is the trigonometric echo of that fact: the tangent of an angle is the slope of the line the angle's radius makes, and a straight-up line has no run to divide by. Surveyors and engineers meet this whenever a sight line points straight up, where the "rise over run" calculation loses its footing.

The same idea drives the domain and range of trigonometric functions: the tangent function simply skips $90^\circ$, $270^\circ$, and every angle a half-turn apart from them, leaving a gap in its domain at each vertical asymptote.

Standard-Angle Tangent Reference Table

Most standard angles have a clean tangent, but two of them break the pattern. Reading down the column, tangent climbs from $0$ and then has no value at all at $90^\circ$.

The two undefined rows, $90^\circ$ and $270^\circ$, are exactly the angles where cosine is zero. That is the whole story of why the tangent has no value there.

How Do You Show Tan 90 Degrees Is Undefined?

Three routes all reach the same conclusion.

Method 1: Sine over cosine.

$$\sin 90^\circ = 1, \qquad \cos 90^\circ = 0$$

$$\tan 90^\circ = \frac{\sin 90^\circ}{\cos 90^\circ} = \frac{1}{0} \quad (\text{undefined})$$

Method 2: The unit circle.

The terminal point at $90^\circ$ on the unit circle is $(0, 1)$.

$$\tan 90^\circ = \frac{y}{x} = \frac{1}{0} \quad (\text{undefined})$$

Method 3: The one-sided limits (why it is not infinity).

Approach $90^\circ$ from just below and just above, in radians for the limit:

$$\lim_{\theta \to \frac{\pi}{2}^{-}} \tan\theta = +\infty, \qquad \lim_{\theta \to \frac{\pi}{2}^{+}} \tan\theta = -\infty$$

The two sides shoot off in opposite directions, so there is no single value, finite or infinite. The graph shows this as a vertical asymptote: the curve races up to the line at $90^\circ$ from the left and comes up from far below on the right, never touching it. In radians, the same undefined behaviour appears at tan pi/2, the radian twin of this angle.

Examples of Tan 90 Degrees

Because $\tan 90^\circ$ has no value, most problems are about handling that gap correctly rather than plugging in a number. One pattern shows up again and again: learners who trust a calculator's screen over the exact cosine will report a huge finite number, because the calculator rounds $90^\circ$ before it divides.

Example 1

Evaluate $\cot 90^\circ$ using $\cot\theta = \frac{\cos\theta}{\sin\theta}$.

$$\cot 90^\circ = \frac{\cos 90^\circ}{\sin 90^\circ} = \frac{0}{1} = 0$$

Cotangent is the reciprocal partner of tangent, and while $\tan 90^\circ$ is undefined, $\cot 90^\circ$ is a clean $0$.

Example 2

A student claims $\tan 90^\circ = \infty$ and uses it as a number. Is that valid?

Wrong attempt. The student writes $\tan 90^\circ = \infty$, then computes $\frac{1}{\tan 90^\circ} = \frac{1}{\infty} = 0$ and treats the job as done, concluding tangent "has the value infinity."

That breaks on inspection. Coming from the right of $90^\circ$, the tangent heads to $-\infty$, not $+\infty$, so the two sides disagree and there is no single value to call $\infty$. Infinity is a description of unbounded growth, not a number you can divide by.

Correct. $\tan 90^\circ$ is undefined. You may say the tangent grows without bound as $\theta \to 90^\circ$ from below, but you cannot assign it the value $\infty$.

Example 3

For what angles in $[0^\circ, 360^\circ)$ is the tangent undefined?

Tangent is undefined wherever $\cos\theta = 0$.

$$\cos\theta = 0 \implies \theta = 90^\circ \ \text{and}\ \theta = 270^\circ$$

So the tangent has exactly two gaps in a full turn, and $90^\circ$ is one of them.

Example 4

Simplify $\dfrac{\sin 90^\circ}{\cos 90^\circ}$ and state the result.

$$\frac{\sin 90^\circ}{\cos 90^\circ} = \frac{1}{0}$$

The denominator is zero, so the expression is undefined; there is nothing further to simplify.

Example 5

Evaluate $\tan 89.9^\circ$ and $\tan 90.1^\circ$ to see the asymptote.

$$\tan 89.9^\circ \approx 572.96, \qquad \tan 90.1^\circ \approx -572.96$$

The values are enormous and of opposite sign on the two sides of $90^\circ$, which is exactly the asymptote behaviour that leaves $\tan 90^\circ$ undefined.

Where Do Students Trip Up on Tan 90 Degrees?

Mistake 1: Calling tan 90 degrees "infinity"

Where it slips in: After seeing tangent grow large near $90^\circ$, when "gets very big" gets rounded up to "equals infinity."

Don't do this: Writing $\tan 90^\circ = \infty$ and then using $\infty$ inside further arithmetic.

The correct way: State it as undefined. The learner who checks both sides of the asymptote notices the right side heads to $-\infty$, which rules out any single value; the one who only looks from the left is the one who writes $\infty$.

Mistake 2: Trusting a calculator's rounded output

Where it slips in: A calculator in degree mode returns a huge number like $1.633 \times 10^{16}$ for $\tan(90)$ because it evaluates a value a hair away from exactly $90^\circ$.

Don't do this: Copying that giant number down as "the value of tan 90 degrees."

The correct way: Exactly at $90^\circ$, $\cos 90^\circ = 0$, so the tangent is undefined. The screen's large number is a rounding artefact, not the value.

Mistake 3: Building a right triangle with a 90 degrees angle

Where it slips in: Trying to use opposite over adjacent for a $90^\circ$ angle inside a right triangle.

Don't do this: Drawing a triangle whose acute angle is $90^\circ$ and reading a ratio off it.

The correct way: A right triangle cannot have a second $90^\circ$ angle, so there is no such triangle to measure. Read $\tan 90^\circ$ from the unit circle, where the definition still holds and returns $\frac{1}{0}$.

Key Takeaways

• Tan 90 degrees is undefined because $\cos 90^\circ = 0$, and $\tan 90^\circ = \frac{\sin 90^\circ}{\cos 90^\circ} = \frac{1}{0}$.

• It is not infinity: the left limit is $+\infty$ and the right limit is $-\infty$, so no single value exists.

• On the unit circle the point is $(0, 1)$, giving $\frac{y}{x} = \frac{1}{0}$; the graph has a vertical asymptote at $90^\circ$.

• The same gap appears at $270^\circ$, and in radians at $\frac{\pi}{2}$.

To get comfortable with undefined values and asymptotes alongside a teacher, explore Bhanzu's trigonometry tutor sessions, a high school math tutor, or math classes online.

Practice These Before Moving On

1. State whether each is defined: $\tan 90^\circ$, $\cot 90^\circ$, $\sec 90^\circ$, $\csc 90^\circ$.

2. List every angle in $[0^\circ, 360^\circ)$ where the tangent is undefined.

3. Evaluate $\tan 89.99^\circ$ and $\tan 90.01^\circ$, then explain in one line why $\tan 90^\circ$ has no value.

Want a live Bhanzu trainer to unpack why tan 90 degrees is undefined? Book a free demo class.

Read More

• Sin, cos, and tan explained — the three ratios and how tangent is built from the other two.

• Tangent formula and derivation — where $\tan\theta = \frac{\sin\theta}{\cos\theta}$ comes from.

• Trigonometric ratios — the ratio definitions behind every standard-angle value.

• Periodic functions — why the tangent repeats its asymptote every half-turn.

• Tan 270 degrees — the other undefined tangent in a full turn.