Tan 2pi/3 : Exact Value, −√3, and How to Find It

What Does Tan 2pi/3 Mean?

A radian is the angle that wraps an arc equal in length to the radius; $\dfrac{2\pi}{3}$ radians is two-thirds of the way from $0$ to $\pi$, landing at $120^\circ$. The four quadrants divide the plane, and $\dfrac{2\pi}{3}$ sits in the second quadrant, upper-left, where $x$ is negative and $y$ is positive.

The tangent of an angle is the slope of the radius to that point, or $\dfrac{\sin\theta}{\cos\theta}$. The reference angle is the acute angle between the radius and the $x$-axis; for $\dfrac{2\pi}{3}$ it is $\pi - \dfrac{2\pi}{3} = \dfrac{\pi}{3}$. The reference angle sets the size of the tangent; the quadrant sets the sign.

Where Does Tan 2pi/3 Show Up?

An angle of $\dfrac{2\pi}{3}$ points up and to the left - think of a clock hand rotated past the top into the upper-left, or a vector heading into the second quadrant. Its slope is $\tan\left(\dfrac{2\pi}{3}\right) = -\sqrt{3}$, a steep line falling from left to right, which is where the negative sign becomes physical: the direction tilts backward, not forward.

The value turns up whenever a rotation of $120^\circ$ appears - the three-fold symmetry of an equilateral triangle, the spacing of a three-phase electrical supply, or the angles between the three arms of a Mercedes-style hub. The unit circle is the reference that keeps the sign straight.

Standard-Angle Reference Table

The angle $\dfrac{2\pi}{3}$ radians is $120^\circ$ — a second-quadrant angle. This table lists the common radian angles around it so you can see where the tangent changes sign.

Notice the sign flip after $\dfrac{\pi}{2}$: once the angle crosses into the second quadrant, tangent turns negative. Tan 2π/3 carries the same magnitude as tan π/3 but the opposite sign, because $\dfrac{\pi}{3}$ is its reference angle.

How Do You Find The Exact Value Of Tan 2pi/3?

There are two clean routes, and both give $-\sqrt{3}$.

Method 1: Reference angle plus quadrant sign.

First find the reference angle: $\pi - \dfrac{2\pi}{3} = \dfrac{\pi}{3}$. The tangent of the reference angle is $\tan\left(\dfrac{\pi}{3}\right) = \sqrt{3}$.

Now apply the sign. In the second quadrant sine is positive and cosine is negative, so their ratio — the tangent — is negative:

$$\tan\left(\frac{2\pi}{3}\right) = -\tan\left(\frac{\pi}{3}\right) = -\sqrt{3}$$

Method 2: The unit-circle coordinates.

At $\dfrac{2\pi}{3}$ the radius meets the unit circle at $\left(-\dfrac{1}{2}, \dfrac{\sqrt{3}}{2}\right)$. Tangent is $\dfrac{y}{x}$:

$$\tan\left(\frac{2\pi}{3}\right) = \frac{\sin(2\pi/3)}{\cos(2\pi/3)} = \frac{\sqrt{3}/2}{-1/2} = -\sqrt{3}$$

Both routes agree, and a calculator in radian mode confirms it: $\tan(2\pi/3) = -1.7320508\ldots$, which is $-\sqrt{3}$.

Examples Of Tan 2pi/3

Example 1

Evaluate $4\tan\left(\dfrac{2\pi}{3}\right)$.

$$4\tan\left(\frac{2\pi}{3}\right) = 4 \times (-\sqrt{3}) = -4\sqrt{3} \approx -6.928$$

Example 2

Find $\tan\left(\dfrac{2\pi}{3}\right)$ using its reference angle.

Wrong attempt. A student finds the reference angle $\dfrac{\pi}{3}$ and stops at $\tan\left(\dfrac{\pi}{3}\right) = \sqrt{3}$, reporting $+\sqrt{3}$.

That misses the quadrant. The reference angle only fixes the magnitude; $\dfrac{2\pi}{3}$ sits in the second quadrant, where tangent is negative, so a positive answer contradicts the coordinates $\left(-\dfrac{1}{2}, \dfrac{\sqrt{3}}{2}\right)$.

Correct. Keep the magnitude $\sqrt{3}$ from the reference angle, then attach the second-quadrant sign: $\tan\left(\dfrac{2\pi}{3}\right) = -\sqrt{3}$.

Example 3

A line through the origin makes an angle of $\dfrac{2\pi}{3}$ with the positive $x$-axis. What is its slope?

$$\text{slope} = \tan\left(\frac{2\pi}{3}\right) = -\sqrt{3} \approx -1.73$$

The negative slope means the line falls as it moves to the right, matching a direction that points up and to the left.

Example 4

Verify that $\tan\left(\dfrac{2\pi}{3}\right) = \dfrac{\sin(2\pi/3)}{\cos(2\pi/3)}$.

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

The quotient identity holds, and the negative cosine is what makes the tangent negative.

Example 5

Convert $\dfrac{2\pi}{3}$ to degrees and confirm the value.

Since $\pi$ radians $= 180^\circ$, $\dfrac{2\pi}{3} = \dfrac{2 \times 180^\circ}{3} = 120^\circ$, and $\tan 120^\circ = -\sqrt{3}$. The radian and degree forms name the same angle and the same value.

Where Students Trip Up On Tan 2pi/3

Mistake 1: Dropping the negative sign

Where it slips in: Computing the reference-angle tangent and forgetting to apply the quadrant sign.

Don't do this: Writing $\tan\left(\dfrac{2\pi}{3}\right) = \sqrt{3}$.

The correct way: The reference angle gives the size $\sqrt{3}$; the second quadrant makes it negative, so the answer is $-\sqrt{3}$. The habit that prevents this is always naming the quadrant before writing the sign, rather than after.

Mistake 2: Using the wrong reference angle

Where it slips in: Subtracting from the wrong axis, so the reference angle comes out as $\dfrac{2\pi}{3}$ itself or some other value.

Don't do this: Treating $\dfrac{2\pi}{3}$ as its own reference angle.

The correct way: In the second quadrant the reference angle is $\pi - \theta$, so $\pi - \dfrac{2\pi}{3} = \dfrac{\pi}{3}$. That is why tan 2π/3 shares its magnitude with $\tan\left(\dfrac{\pi}{3}\right)$, the value covered in tan 60 degrees.

Mistake 3: Confusing tan 2π/3 with tan π/3

Where it slips in: Reading the two as the same because they share a reference angle.

Don't do this: Writing $\tan\left(\dfrac{2\pi}{3}\right) = \tan\left(\dfrac{\pi}{3}\right)$.

The correct way: They have equal magnitude but opposite sign: $\tan\left(\dfrac{\pi}{3}\right) = +\sqrt{3}$ in the first quadrant, $\tan\left(\dfrac{2\pi}{3}\right) = -\sqrt{3}$ in the second. The reference angle links them; the quadrant separates them.

Key Takeaways

• Tan 2pi/3 equals $-\sqrt{3}$, approximately $-1.7321$ — an exact value because $\dfrac{2\pi}{3}$ has a standard reference angle.

• The reference angle is $\pi - \dfrac{2\pi}{3} = \dfrac{\pi}{3}$, which sets the magnitude $\sqrt{3}$; the second quadrant sets the negative sign.

• In degrees, $\tan\left(\dfrac{2\pi}{3}\right) = \tan 120^\circ$, and on the unit circle the point is $\left(-\dfrac{1}{2}, \dfrac{\sqrt{3}}{2}\right)$.

• The most common slip is dropping the negative sign and reporting $+\sqrt{3}$ — always name the quadrant before writing the sign.

To take radian-measure angles further with a teacher, explore Bhanzu's trigonometry tutor, our high school math tutor sessions, or structured math classes online.

Practice These Before Moving On

1. Evaluate $\tan\left(\dfrac{2\pi}{3}\right) + \tan\left(\dfrac{\pi}{3}\right)$.

2. A vector makes an angle of $\dfrac{2\pi}{3}$ with the positive $x$-axis. State its slope and say why it is negative.

3. Show that $\tan\left(\dfrac{2\pi}{3}\right) \times \tan\left(\dfrac{\pi}{3}\right) = -3$.

Want a live Bhanzu trainer to walk through more tan 2pi/3 problems? Book a free demo class — online, worldwide.

Read More

• Sin, cos, and tan — the three core trigonometric ratios

• Trigonometric ratios of specific angles

• The full trigonometric table of standard values

• What a radian is and how radian measure works