Tan 3pi/2: Why It Is Undefined | Unit Circle

What Is The Value Of Tan 3pi/2?

Tan 3pi/2 is undefined. There is no number, no exact surd, and no decimal, because the calculation requires dividing by zero.

Here is the reasoning in one line. The angle in radians is $\frac{3\pi}{2}$, which converts to degrees as:

$$\frac{3\pi}{2} \text{ rad} \times \frac{180^\circ}{\pi} = 270^\circ$$

On the unit circle, the angle $270^\circ$ points straight down to the coordinate $(0, -1)$. Since $\cos\theta$ is the $x$-coordinate and $\sin\theta$ is the $y$-coordinate:

$$\sin\frac{3\pi}{2} = -1, \qquad \cos\frac{3\pi}{2} = 0$$

Tangent is the ratio of sine to cosine, so:

$$\tan\frac{3\pi}{2} = \frac{\sin(3\pi/2)}{\cos(3\pi/2)} = \frac{-1}{0} = \text{undefined}$$

The bottom of the fraction is zero, and no real number can sit there. This is why $\tan 270^\circ$ and $\tan\frac{3\pi}{2}$ are reported as undefined on every calculator and in every table.

How Do You Find Tan 3pi/2?

The cleanest route uses the definition $\tan\theta = \dfrac{\sin\theta}{\cos\theta}$. You need three facts, and the answer falls out.

1. Convert or place the angle. $\frac{3\pi}{2} = 270^\circ$, which sits on the boundary between the third and fourth quadrants, pointing straight down.

2. Read sine and cosine. At $270^\circ$ the unit-circle point is $(0, -1)$, so $\cos\frac{3\pi}{2} = 0$ and $\sin\frac{3\pi}{2} = -1$.

3. Divide. $\tan\frac{3\pi}{2} = \dfrac{-1}{0}$, which is undefined.

A second route uses the reciprocal relationship. Cotangent is the reciprocal of tangent, and here it behaves neatly:

$$\cot\frac{3\pi}{2} = \frac{\cos(3\pi/2)}{\sin(3\pi/2)} = \frac{0}{-1} = 0$$

Because $\cot\frac{3\pi}{2} = 0$, its reciprocal $\tan\frac{3\pi}{2} = \frac{1}{0}$ is undefined. The two facts agree, which is a useful cross-check.

One warning on the sign rule. The ASTC (or CAST) rule tells you the sign of a ratio inside a quadrant, but $270^\circ$ is a quadrantal angle sitting exactly on an axis, not inside a quadrant. There is no positive or negative value to assign, because there is no value at all.

Where Does 3pi/2 Sit On The Unit Circle?

Start at the positive $x$-axis and rotate anticlockwise. A quarter turn ($\frac{\pi}{2}$, or $90^\circ$) reaches the top. A half turn ($\pi$, or $180^\circ$) reaches the left. Three-quarters of a turn brings you to $\frac{3\pi}{2}$, or $270^\circ$, pointing straight down to $(0, -1)$.

The $x$-coordinate at that point is exactly $0$. Tangent has a geometric meaning here: it is the length of the segment along the vertical tangent line to the circle, measured where the terminal ray crosses it. When the ray points straight down, it runs parallel to that tangent line and never meets it, so the length is not a finite number. That is the same "undefined" seen through geometry instead of algebra.

Why Is Tan 3pi/2 Undefined?

The short answer is division by zero. The fuller answer is that $\frac{3\pi}{2}$ is one of the angles where the tangent graph has a vertical asymptote, a place the curve races toward but never reaches.

• Cosine is the denominator. Tangent is $\frac{\sin\theta}{\cos\theta}$, and $\cos\frac{3\pi}{2} = 0$. Any fraction with zero on the bottom is undefined.

• This repeats on a pattern. Cosine equals zero at every angle of the form $\frac{\pi}{2} + n\pi$ (that is, $90^\circ + 180^\circ n$ for any whole number $n$). So tangent is undefined at $90^\circ, 270^\circ, 450^\circ$, and so on, not only here.

• The graph goes vertical. Near $\frac{3\pi}{2}$ the tangent curve climbs or drops without bound. It has a vertical asymptote at the dashed line $\theta = \frac{3\pi}{2}$.

• The two sides disagree. Approaching $270^\circ$ from just below (from the third quadrant), $\tan\theta \to +\infty$. Approaching from just above (from the fourth quadrant), $\tan\theta \to -\infty$. Because the two one-sided limits head to opposite infinities, no single value can be assigned.

That last point is the deep reason. "Undefined" is not a gap in our knowledge. It is a statement that the function genuinely has no value there, confirmed by the graph splitting to $+\infty$ on one side and $-\infty$ on the other.

Table 1: Tangent at the quadrantal angles, in both degrees and radians.

The table shows the rhythm clearly: tangent is $0$ where sine is $0$, and undefined where cosine is $0$. The angle $\frac{3\pi}{2}$ falls in the second group, alongside its sibling $\tan\frac{\pi}{2}$. For the full grid of values across all standard angles, see the trigonometric table.

Who Discovered The Tangent Function?

Tangent began as a practical measurement long before it became a function on a graph. Ancient astronomers and surveyors tracked the length of a shadow cast by a vertical stick, and the ratio of that shadow to the stick is exactly what we now call tangent. The Latin words umbra recta and umbra versa (the "straight shadow" and "turned shadow") survived in trigonometry tables for centuries.

Two later steps fixed the modern language:

• Abū al-Wafāʾ al-Būzjānī (940–998, Persia and Baghdad) advanced the tangent, cotangent, secant, and cosecant as measured functions and improved the accuracy of trigonometric tables.

• Thomas Fincke (1561–1656, Denmark) introduced the term "tangent" (from the Latin for "touching") in his 1583 work Geometria Rotundi, naming the ratio after the line that touches the circle.

Where Is Tan 3pi/2 Used In The Real World?

An undefined tangent is not just a textbook curiosity. It marks the moment a slope or a rate turns vertical, and that moment shows up across engineering and technology.

• Ramps and road grades: grade is the tangent of the incline angle. As a slope approaches vertical (a quarter turn), the tangent runs to infinity, which is why a truly vertical "ramp" is impossible to drive.

• Camera and telescope tilt: panning a camera at a constant speed makes the tracked point move at a rate proportional to a tangent, and that rate blows up when the line of sight passes straight up or straight down.

• Alternating current: phase relationships in AC circuits use tangent; a phase near a quarter turn drives the tangent term toward infinity, flagging a resonance or a switching point engineers must design around.

• Computer graphics: projecting a 3D point onto a screen uses a tangent of the field-of-view angle, and the math is deliberately kept away from the undefined quarter-turn to stop the image from stretching to infinity.

• Navigation and surveying: the shadow-and-stick idea that created tangent still measures the height of a distant object from its angle of elevation, and the method fails precisely when the sightline goes vertical.

One idea, a slope going vertical, ties a Ferris wheel, a power grid, and a video game to the same $-1/0$ that makes Tan 3pi/2 undefined.

What Are The Most Common Mistakes With Tan 3pi/2?

These four slips account for most wrong answers on quadrantal-angle tangents, and each has a clean fix.

Writing $0$ or $\infty$ as "the value."

Where it slips in:

A student sees $\frac{-1}{0}$ and records the answer as $0$, or writes $\infty$ as though infinity were a number.

Don't do this:

Do not report $0$, $\infty$, or $-\infty$ as the value. Infinity is a direction the graph heads, not a value the function takes.

The correct way:

State that $\tan\frac{3\pi}{2}$ is undefined. If a limit is asked for, give the one-sided limits separately: $+\infty$ from below and $-\infty$ from above.

Mixing up degrees and radians on the calculator.

Where it slips in:

A student types $\tan(3\pi/2)$ with the calculator in degree mode, or types $\tan(270)$ in radian mode, and copies down whatever number appears.

Don't do this:

Do not trust the display before checking the angle MODE. In degree mode, "$3\pi/2$" is read as about $4.71^\circ$, a completely different angle.

The correct way:

Set the mode to match the angle. For $\frac{3\pi}{2}$ use radian mode; for $270^\circ$ use degree mode. A correct calculator returns an error or "undefined" for both.

Flipping the ratio to sine over cosine incorrectly.

Where it slips in:

A student writes $\tan\theta = \frac{\cos\theta}{\sin\theta}$ by mistake, gets $\frac{0}{-1} = 0$, and reports $0$.

Don't do this:

Do not invert the definition. That expression is cotangent, not tangent.

The correct way:

Keep tangent as $\frac{\sin\theta}{\cos\theta}$. Here that is $\frac{-1}{0}$, which is undefined, while $\frac{\cos\theta}{\sin\theta} = \frac{0}{-1} = 0$ is the separate value of $\cot\frac{3\pi}{2}$.

Misreading the unit-circle point as $(-1, 0)$.

Where it slips in:

A student confuses $270^\circ$ with $180^\circ$ and uses the point $(-1, 0)$, giving $\tan = \frac{0}{-1} = 0$.

Don't do this:

Do not swap the axis points. $180^\circ$ is $(-1, 0)$; $270^\circ$ is $(0, -1)$.

The correct way:

Place $\frac{3\pi}{2}$ pointing straight down at $(0, -1)$, so $\cos = 0$ and the tangent is undefined.

Practice Problems On Tan 3pi/2

Try each one, then check the answer beside it. Work in radians unless the problem says degrees.

1. State the value of $\tan\frac{3\pi}{2}$.
   (Answer: undefined, since $\cos\frac{3\pi}{2} = 0$.)

2. Find $\cot\frac{3\pi}{2}$.
   (Answer: $\frac{\cos(3\pi/2)}{\sin(3\pi/2)} = \frac{0}{-1} = 0$.)

3. Is $\tan\frac{7\pi}{2}$ defined?
   (Answer: no; $\frac{7\pi}{2} = 630^\circ$ is coterminal with $270^\circ$, so it is also undefined.)

4. Evaluate $\tan\frac{3\pi}{2} \cdot \sin\frac{3\pi}{2}$.
   (Answer: undefined, because any product with an undefined factor is itself undefined.)

5. Give the two one-sided limits of $\tan\theta$ as $\theta \to \frac{3\pi}{2}$.
   (Answer: $+\infty$ from below, $-\infty$ from above.)

6. At which angle between $0$ and $2\pi$, other than $\frac{3\pi}{2}$, is tangent also undefined?
   (Answer: $\frac{\pi}{2}$, where cosine is again $0$.)

Where Should You Go Next After Tan 3pi/2?

Several natural doors open from this angle, and each deepens a different part of the picture.

1. Tan Pi/2. The other quadrantal angle where tangent is undefined, and the closest sibling of this one.

2. Unit Circle With Tangent. See where every tangent value, defined and undefined, lives on the circle.

3. Sin, Cos, Tan. Revisit the three core ratios so the $\frac{\sin}{\cos}$ definition feels automatic.

4. What Is A Radian. Get comfortable with $\frac{3\pi}{2}$ as an angle measure, not just a fraction.

If your child is building these foundations, a live Bhanzu trainer teaches trigonometry from the unit circle up, starting with why an angle can have no tangent at all, in the Bhanzu trigonometry program.