What Is The Value Of Tan 4pi/3?
Tan 4pi/3 is $\sqrt{3}$, approximately $1.7321$ to four decimal places. Written with the angle in both units, $\tan\frac{4\pi}{3} = \tan 240^\circ = \sqrt{3}$.
The angle $\frac{4\pi}{3}$ is measured in radians. To see it as a familiar rotation, convert it:
$$\frac{4\pi}{3} \text{ rad} = \frac{4\pi}{3} \times \frac{180^\circ}{\pi} = 240^\circ$$
So every question about $\tan\frac{4\pi}{3}$ is a question about $\tan 240^\circ$. The two forms are the same angle, one written in radians and one in degrees. If the radian measure feels unfamiliar, what is a radian sets up the unit before you use it here.
How Do You Find Tan 4pi/3?
To find Tan 4pi/3, find its reference angle, decide the sign from the quadrant, then read off the reference value. Three short steps settle it.
Locate the quadrant. Since $\pi < \frac{4\pi}{3} < \frac{3\pi}{2}$ (that is, $180^\circ < 240^\circ < 270^\circ$), the angle sits in Quadrant III.
Find the reference angle. In Quadrant III the reference angle is the angle past $\pi$: $\frac{4\pi}{3} - \pi = \frac{\pi}{3}$, or $240^\circ - 180^\circ = 60^\circ$. See reference angle for the rule in every quadrant.
Attach the sign. The CAST rule (some students learn it as ASTC) says tangent is positive in Quadrant III. So $\tan\frac{4\pi}{3} = +\tan\frac{\pi}{3} = \sqrt{3}$.
The reference angle strips the problem down to a special angle you already know, and the quadrant decides only the sign. Every value in this family comes from the same two questions: which quadrant, and what is the reference angle.
Where Does 4pi/3 Sit On The Unit Circle?
On the unit circle, $\frac{4\pi}{3}$ is the point $\left(-\frac{1}{2}, -\frac{\sqrt{3}}{2}\right)$, sitting in the lower-left quarter. The tangent of any angle is the $y$-coordinate divided by the $x$-coordinate of its unit-circle point.
$$\tan\frac{4\pi}{3} = \frac{\sin\frac{4\pi}{3}}{\cos\frac{4\pi}{3}} = \frac{-\frac{\sqrt{3}}{2}}{-\frac{1}{2}} = \sqrt{3}$$
Both coordinates are negative in Quadrant III. A negative divided by a negative is positive, which is the whole reason the answer comes out $+\sqrt{3}$ and not $-\sqrt{3}$. The unit circle with tangent shows how that ratio behaves as the point travels around the circle.
How Do You Derive Tan 4pi/3 Exactly?
The cleanest exact derivation uses the period of the tangent function. Tangent repeats every $\pi$ radians, not every $2\pi$ like sine and cosine, so adding $\pi$ to an angle leaves the tangent unchanged: $\tan(\pi + \theta) = \tan\theta$.
Write $\frac{4\pi}{3}$ as $\pi$ plus the reference angle, then apply the period:
$$\tan\frac{4\pi}{3} = \tan\left(\pi + \frac{\pi}{3}\right)$$
$$= \tan\frac{\pi}{3} \qquad \text{(tangent has period } \pi\text{)}$$
$$= \sqrt{3}$$
That last value, $\tan\frac{\pi}{3} = \sqrt{3}$, is worth anchoring to a shape rather than memorising blind. In a 30-60-90 right triangle, the side opposite $60^\circ$ has length $\sqrt{3}$ and the side adjacent to it has length $1$. Tangent is opposite over adjacent, so $\tan 60^\circ = \frac{\sqrt{3}}{1} = \sqrt{3}$.
The unit circle in the section above gives the same number a second way, as $y/x$. Two independent pictures, one value. For the full family of these values see tan pi/3.
What Are The Tangents Of Angles Related To 4pi/3?
Four angles share the reference angle $\frac{\pi}{3}$ (that is $60^\circ$), one in each quadrant. Their sine and cosine only change sign; the tangent flips sign with the quadrant.
Table: The reference-angle-60° family, with sine, cosine, and tangent in both units.
Angle | Radians | Quadrant | $\sin$ | $\cos$ | $\tan$ |
|---|---|---|---|---|---|
$60^\circ$ | $\frac{\pi}{3}$ | I | $\frac{\sqrt{3}}{2}$ | $\frac{1}{2}$ | $\sqrt{3}$ |
$120^\circ$ | $\frac{2\pi}{3}$ | II | $\frac{\sqrt{3}}{2}$ | $-\frac{1}{2}$ | $-\sqrt{3}$ |
$240^\circ$ | $\frac{4\pi}{3}$ | III | $-\frac{\sqrt{3}}{2}$ | $-\frac{1}{2}$ | $\sqrt{3}$ |
$300^\circ$ | $\frac{5\pi}{3}$ | IV | $-\frac{\sqrt{3}}{2}$ | $\frac{1}{2}$ | $-\sqrt{3}$ |
The two Quadrant I and III rows both give $+\sqrt{3}$, exactly the period-$\pi$ pattern from the derivation above. For the neighbours in this table, compare tan 2pi/3, sin pi/3, and cos pi/3. A complete grid of standard angles lives in the trigonometric table.
Why Is Tan 4pi/3 Positive?
The positive sign is not a rule to memorise. It falls straight out of where the angle lands and what tangent measures.
Both coordinates are negative in Quadrant III. The terminal point $\left(-\frac{1}{2}, -\frac{\sqrt{3}}{2}\right)$ has a negative $x$ and a negative $y$.
Tangent is the ratio $y/x$. Dividing one negative number by another negative number gives a positive result, so the sign turns positive before you even reach the size of the number.
The size comes from the reference angle. Stripped of sign, the magnitude is $\tan 60^\circ = \sqrt{3}$, the same steepness as the first-quadrant angle.
Put those together and $\tan\frac{4\pi}{3} = +\sqrt{3}$. Sine alone and cosine alone are both negative here, yet their ratio is positive, which is why students who track only "third quadrant is negative" get tangent wrong. Tangent asks a different question from sine and cosine: not where the point is, but how steep the ray to it is.
Who Discovered The Values Behind Tan 4pi/3?
The number we call a tangent began as a shadow, long before anyone drew a unit circle. Astronomers needed to turn the angle of the sun into a length they could measure on the ground, and that length is the tangent.
Two earlier figures built the tables this rested on:
Hipparchus of Nicaea (around 190 to 120 BCE, Greece) compiled the first known table of chords, the direct ancestor of the sine table, to predict the positions of the sun and moon.
Aryabhata (476 to 550 CE, India) recorded one of the earliest sine tables, calling the half-chord jya, in his work the Aryabhatiya around 499 CE. The word "sine" traces back through Latin to a mistranslation of jya.
Where Is Tan 4pi/3 Used In The Real World?
A tangent is a slope, so any place that measures steepness or turning uses the same value that $\tan\frac{4\pi}{3}$ carries.
Ramps and gradients: the slope of a road, a wheelchair ramp, or a car-park descent is the tangent of its angle to the horizontal, and a slope of $\sqrt{3}$ is a steep $60^\circ$ pitch.
Roofs and construction: roof pitch and staircase rise-over-run are tangents, which is why builders quote angle and slope interchangeably.
Electrical engineering: alternating current and voltage are modelled as rotating angles, and the phase between them is read using tangent-based ratios past the half-turn, exactly the region where $240^\circ$ sits.
Cameras and graphics: a camera panning past a full turn, or a game character rotating to $240^\circ$, computes its facing direction with the same signed tangent.
Astronomy and navigation: the shadow-length idea that started the tangent still fixes the sun's angle for sundials and solar tracking.
One idea, steepness of a turned ray, runs under ramps, roofs, circuits, and screens. That reach is why the special-angle tangents are worth knowing by sight.
What Are The Most Common Mistakes With Tan 4pi/3?
These four errors account for most wrong answers on this angle, confirmed against the reference-angle and quadrant confusions that surface repeatedly on student homework threads for $\tan\frac{4\pi}{3}$.
Giving the answer a negative sign.
Where it slips in:
A student remembers that Quadrant III coordinates are negative and writes $\tan\frac{4\pi}{3} = -\sqrt{3}$.
Don't do this:
Do not copy the sign of sine or cosine onto tangent. Both are negative here, but tangent is their ratio.
The correct way:
Divide the two negatives: $\frac{-\sqrt{3}/2}{-1/2} = +\sqrt{3}$. Tangent is positive in Quadrant III, so the answer is $+\sqrt{3}$.
Leaving the calculator in degree mode.
Where it slips in:
A student types $\tan(4\pi/3)$ with the calculator set to degrees, so it reads the input as $4\pi/3 \approx 4.19$ degrees and returns about $0.073$.
Don't do this:
Do not enter a radian angle while the MODE is set to degrees. The two settings give completely different numbers.
The correct way:
Switch the calculator to radian mode before entering $\frac{4\pi}{3}$, or convert to $240^\circ$ first and stay in degree mode. Either path gives $\sqrt{3}$.
Taking the wrong reference angle.
Where it slips in:
A student subtracts from the wrong axis, computing $\frac{3\pi}{2} - \frac{4\pi}{3} = \frac{\pi}{6}$ and reporting $\tan 30^\circ$.
Don't do this:
Do not measure a Quadrant III reference angle from $\frac{3\pi}{2}$. In Quadrant III the reference angle is measured from $\pi$.
The correct way:
Subtract $\pi$: $\frac{4\pi}{3} - \pi = \frac{\pi}{3}$. The reference angle is $60^\circ$, giving $\sqrt{3}$, not $\frac{1}{\sqrt{3}}$.
Using the wrong period.
Where it slips in:
A student assumes tangent repeats every $2\pi$ like sine and cosine, so they cannot connect $\frac{4\pi}{3}$ back to $\frac{\pi}{3}$.
Don't do this:
Do not apply a $2\pi$ period to tangent. Tangent repeats every $\pi$.
The correct way:
Use $\tan(\pi + \theta) = \tan\theta$. Because $\frac{4\pi}{3} = \pi + \frac{\pi}{3}$, the tangent equals $\tan\frac{\pi}{3} = \sqrt{3}$ directly.
Practice Problems On Tan 4pi/3
Work each one, then check against the answer beside it.
Convert $\frac{4\pi}{3}$ radians to degrees.
(Answer: $240^\circ$.)State the reference angle of $\frac{4\pi}{3}$.
(Answer: $\frac{\pi}{3}$, that is $60^\circ$.)Write $\sin\frac{4\pi}{3}$ and $\cos\frac{4\pi}{3}$.
(Answer: $\sin = -\frac{\sqrt{3}}{2}$, $\cos = -\frac{1}{2}$.)Verify $\tan\frac{4\pi}{3}$ from the unit-circle point $\left(-\frac{1}{2}, -\frac{\sqrt{3}}{2}\right)$.
(Answer: $\frac{-\sqrt{3}/2}{-1/2} = \sqrt{3}$.)Find $\tan\frac{2\pi}{3}$ and compare its sign with $\tan\frac{4\pi}{3}$.
(Answer: $\tan\frac{2\pi}{3} = -\sqrt{3}$; opposite sign, because $\frac{2\pi}{3}$ is in Quadrant II.)Evaluate $\tan\left(\frac{4\pi}{3} - \pi\right)$.
(Answer: $\tan\frac{\pi}{3} = \sqrt{3}$, the period-$\pi$ shortcut.)
Where Should You Go Next After Tan 4pi/3?
The value of $\tan\frac{4\pi}{3}$ opens onto the wider machinery of the tangent, and several natural doors follow from here.
The tangent function. How tangent behaves across a full turn, where it climbs to infinity, and why its period is $\pi$.
Sin cos tan. The three core ratios together, so the sign patterns across quadrants stop feeling separate.
Tan 2pi/3. The Quadrant II neighbour in the same reference-angle family, where the sign flips to negative.
If your child is building fluency with the unit circle and these special angles, a live Bhanzu trainer teaches them from the reference-angle idea up, so the values are reasoned rather than memorised, in the Bhanzu trigonometry program.
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