Tan 2pi : Value, Unit Circle Proof, and Why It's 0

What Does Tan 2pi Mean?

Tangent is one of the three trigonometric ratios, defined as sine divided by cosine. On the unit circle, a circle of radius $1$ centred at the origin, tangent is the $y$-coordinate divided by the $x$-coordinate of the point where the angle's radius meets the circle.

An angle of $2\pi$ radians is one complete revolution. You sweep the radius all the way around and it comes to rest exactly where it started, at $(1, 0)$. So the height $y$ is $0$ and the horizontal reach $x$ is $1$, giving $\dfrac{0}{1} = 0$. The degree-first version of this same value lives at tan 0 degrees, where the two articles describe the identical value from opposite entry points.

Where Does Tan 2pi Show Up?

Any process that returns exactly to its starting state after one full cycle lives at $2\pi$: a wheel back to its start mark, a pendulum through one full swing, a rotating vector back to where it began. Because $\tan 2\pi = \tan 0 = 0$, the tangent of a "complete turn" is the same as the tangent of no turn at all.

The value also anchors how periodic motion is written. Angles in graphics and physics are often reduced modulo $2\pi$ so a full rotation counts as none, and the tangent formula inherits that wrap-around from the unit circle.

Standard-Angle Reference Table

Reading tangent around a full circle shows why $2\pi$ lands back on $0$. A radian measures an angle by arc length, and $2\pi$ radians is the arc all the way around.

Tangent is $0$ at $0$, at $\pi$, and again at $2\pi$, and undefined at $\dfrac{\pi}{2}$ and $\dfrac{3\pi}{2}$. To move between the two columns yourself, see radians to degrees.

How Do You Find The Exact Value Of Tan 2pi?

Two routes both give $0$.

Method 1: The unit circle after a full turn.

Rotate the unit radius by $2\pi$ radians, a full $360^\circ$. The tip lands at $(1, 0)$.

$$\tan 2\pi = \frac{y\text{-coordinate}}{x\text{-coordinate}} = \frac{0}{1} = 0$$

Method 2: Periodicity of tangent.

Tangent repeats every $\pi$ radians, so adding any whole number of $\pi$ leaves the value unchanged:

$$\tan(\theta + \pi) = \tan\theta$$

Starting from $\tan 0 = 0$ and adding $\pi$ twice reaches $2\pi$:

$$\tan 2\pi = \tan(0 + 2\pi) = \tan 0 = 0$$

Both methods agree, and both confirm the same value found at tan π, which is also $0$. The subtle point is that tangent's period is $\pi$, not $2\pi$: by $2\pi$ the tangent curve has completed two full periods, both ending at $0$.

Examples Of Tan 2pi

Example 1

Evaluate $4\tan 2\pi + 9$.

$$4\tan 2\pi + 9 = 4 \times 0 + 9 = 9$$

Example 2

Simplify $\dfrac{\sin 2\pi}{\cos 2\pi}$ and confirm it equals $\tan 2\pi$.

Wrong attempt. A student remembers that sine and cosine repeat every $2\pi$ and concludes tangent must repeat every $2\pi$ as well, then worries that a "full period" might make the value undefined.

That misreads tangent's period. Tangent repeats every $\pi$, so $2\pi$ is two complete tangent periods, not one, and nothing about it forces an undefined value.

Correct. Substitute the known values:

$$\frac{\sin 2\pi}{\cos 2\pi} = \frac{0}{1} = 0 = \tan 2\pi$$

The denominator is $1$, not $0$, so the ratio is defined and equals $0$.

Example 3

A wheel completes one full rotation of $2\pi$ radians. Using $\tan 2\pi$, find the slope of the radius line to the mark, relative to the horizontal, at the end of the turn.

The mark returns to the horizontal axis, so the slope is:

$$\tan 2\pi = 0$$

A slope of $0$ means the radius to the mark is level, exactly as it was before the turn.

Example 4

Verify that $\tan 2\pi = \tan 0$.

Since tangent has period $\pi$, and $2\pi = 0 + 2\pi$:

$$\tan 2\pi = \tan 0 = 0$$

Both equal $0$, which is why a value written in radians as $2\pi$ and one written in degrees as $0^\circ$ can name the same tangent.

Example 5

Evaluate $\tan 2\pi + \tan \pi + \tan 0$.

Each term is $0$:

$$\tan 2\pi + \tan \pi + \tan 0 = 0 + 0 + 0 = 0$$

Every multiple of $\pi$ gives a tangent of $0$, so the whole sum collapses.

Where Students Trip Up On Tan 2pi

Mistake 1: Assuming tangent's period is 2π

Where it slips in: Right after learning that sine and cosine have period $2\pi$, when tangent gets grouped with them.

Don't do this: Reasoning "tangent repeats every $2\pi$, so $2\pi$ is one full period." Students first meeting all three functions together tend to give tangent the same period as sine and cosine.

The correct way: Tangent's period is $\pi$, half of sine and cosine's. By $2\pi$ the tangent curve has run through two periods. The value still lands on $0$, but for the right reason.

Mistake 2: Treating tan 2π as undefined

Where it slips in: Confusing the angles where tangent is undefined ($\dfrac{\pi}{2}$, $\dfrac{3\pi}{2}$) with the full-turn angle $2\pi$.

Don't do this: Writing $\tan 2\pi =$ undefined.

The correct way: Tangent is undefined only where cosine is $0$. At $2\pi$, cosine is $1$, so $\tan 2\pi = \dfrac{0}{1} = 0$ is perfectly defined.

Mistake 3: Converting 2π radians to the wrong degree measure

Where it slips in: Mixing up $\pi = 180^\circ$ so that $2\pi$ is read as $180^\circ$ instead of $360^\circ$.

Don't do this: Writing $2\pi = 180^\circ$ and then reaching for $\tan 180^\circ$ under a wrong label.

The correct way: $\pi$ radians $= 180^\circ$, so $2\pi$ radians $= 360^\circ$. Both $\tan 180^\circ$ and $\tan 360^\circ$ happen to be $0$, but keep the conversion honest so it holds for angles where it matters.

Key Takeaways

• Tan 2π equals $0$ because $2\pi$ radians is a full turn back to $(1, 0)$, where tangent is $\dfrac{0}{1}$.

• Tangent's period is $\pi$, so $2\pi$ is two complete periods and $\tan 2\pi = \tan 0 = 0$.

• In degrees, $2\pi$ radians $= 360^\circ$, and $\tan 360^\circ = 0$.

• Tangent is undefined only where cosine is $0$; at $2\pi$ cosine is $1$, so the value is a defined $0$.

To go further with radian-measure trigonometry alongside a teacher, explore Bhanzu's trigonometry tutor, a high school math tutor, or flexible math classes online.

Practice These Before Moving On

1. Evaluate $6\tan 2\pi - 5\tan \pi$.

2. Convert $2\pi$ radians to degrees, then state $\tan$ of that angle.

3. Show that $\tan 2\pi + \cos 2\pi = 1$.

Want a live Bhanzu trainer to walk through more tan 2π problems? Book a free demo class.

Read More

• Tan 90 Degrees - why it's undefined

• Sin 2pi - exact value

• Cos 2pi - exact value

• Trigonometric table - all standard angles