Tan 240 Degrees: Exact Value √3 Explained

What Is The Value Of Tan 240 Degrees?

Tan 240 degrees equals $\sqrt{3}$, or about $1.7321$ to four decimal places. In radians the angle is $\frac{4\pi}{3}$, so the same fact is written $\tan\frac{4\pi}{3} = \sqrt{3}$.

The value is exact. $\sqrt{3}$ is an irrational number, so the decimal $1.7320508\ldots$ never ends and never repeats, and the clean way to write the answer is the surd $\sqrt{3}$, not the rounded decimal.

$$\tan 240^\circ = \tan\frac{4\pi}{3} = \sqrt{3} \approx 1.7321$$

Two facts carry the whole result. The angle lands in the third quadrant, and its reference angle (the acute angle it makes with the horizontal axis) is $60^\circ$. The next sections build the value from each of those facts in turn.

How Do You Find Tan 240 Degrees?

Finding any trig value at a non-acute angle uses the same two-step routine: find the reference angle, then fix the sign from the quadrant.

Step 1: Find the reference angle. The angle $240^\circ$ is past $180^\circ$ but short of $270^\circ$, so it lives in Quadrant III. For a third-quadrant angle the reference angle is the angle minus $180^\circ$.

$$240^\circ - 180^\circ = 60^\circ$$

Step 2: Fix the sign from the quadrant. The ASTC rule (often read as "All Students Take Calculus") records which ratios are positive in each quadrant: All in Quadrant I, Sine in Quadrant II, Tangent in Quadrant III, Cosine in Quadrant IV. Quadrant III is the tangent quadrant, so $\tan 240^\circ$ is positive.

Putting the two steps together, tangent keeps the value of its reference angle and takes the sign from the quadrant:

$$\tan 240^\circ = +\tan 60^\circ = \sqrt{3}$$

You can read the value of $\tan 60^\circ$ straight from the trigonometric table or the wider list of trigonometric ratios of specific angles. The special angle $60^\circ$ gives $\tan 60^\circ = \sqrt{3}$, and that single value drives everything here.

Where Does 240° Sit On The Unit Circle?

On the unit circle, every angle picks out a point whose coordinates are $(\cos\theta, \sin\theta)$. Tangent is the ratio of those coordinates, $\tan\theta = \frac{\sin\theta}{\cos\theta} = \frac{y}{x}$, which is the same as the slope of the line from the centre to that point.

At $240^\circ$ the point is:

$$\left(\cos 240^\circ,\ \sin 240^\circ\right) = \left(-\tfrac{1}{2},\ -\tfrac{\sqrt{3}}{2}\right)$$

Both coordinates are negative, which is exactly what "third quadrant" means. Dividing $y$ by $x$ gives the tangent:

$$\tan 240^\circ = \frac{\sin 240^\circ}{\cos 240^\circ} = \frac{-\tfrac{\sqrt{3}}{2}}{-\tfrac{1}{2}} = \sqrt{3}$$

The two minus signs cancel, which is the unit-circle reason the answer comes out positive. A negative over a negative is a positive slope.

For a fuller walk-through of how the tangent behaves all the way around the circle, see unit circle with tangent.

How Do You Derive Tan 240° From Tan 60°?

There is a cleaner route that skips coordinates and uses one property of the tangent: it repeats every $180^\circ$. In symbols, $\tan(\theta + 180^\circ) = \tan\theta$. This is the period of the tangent, and it is shorter than the $360^\circ$ period of sine and cosine.

Write $240^\circ$ as $180^\circ + 60^\circ$ and apply the period:

$$\tan 240^\circ = \tan(180^\circ + 60^\circ)$$

$$= \tan 60^\circ$$

$$= \sqrt{3}$$

The same derivation works in radians, which is how the value usually appears in higher classes. Here $240^\circ = \frac{4\pi}{3}$, and the period is $\pi$:

$$\tan\frac{4\pi}{3} = \tan\left(\pi + \frac{\pi}{3}\right)$$

$$= \tan\frac{\pi}{3}$$

$$= \sqrt{3}$$

If the step from $240^\circ$ to $\frac{4\pi}{3}$ is not obvious, review what is a radian and the family of trigonometric ratios in radians. The conversion is $240 \times \frac{\pi}{180} = \frac{4\pi}{3}$.

What Are The Tangent Values Around The Unit Circle?

The angle $240^\circ$ is one of six "nice" angles spaced $60^\circ$ apart. Laying them side by side shows why $240^\circ$ and $60^\circ$ share a tangent, and why the signs alternate.

Table: Tangent values at the multiples of 60° around a full turn.

Read down the tangent column and the $180^\circ$ period jumps out: $60^\circ$ and $240^\circ$ match, and $120^\circ$ and $300^\circ$ match, because each pair differs by exactly half a turn. Two of the neighbours have their own Bhanzu pages, tan 60 degrees and tan 120 degrees, and the radian twin of $120^\circ$ is covered at tan 2pi/3.

Why Is Tan 240 Degrees Positive?

The sign is the part students most often miss, so it is worth grounding rather than memorising. Tangent is $\frac{\sin\theta}{\cos\theta}$, so its sign is decided by the signs of sine and cosine together.

• In Quadrant III, the point on the circle has a negative $x$ and a negative $y$. So both $\cos 240^\circ$ and $\sin 240^\circ$ are negative.

• Tangent divides one negative by another: $\frac{-}{-} = +$. A negative divided by a negative is positive.

• This is the same reason tangent is positive in Quadrant I, where both coordinates are positive: $\frac{+}{+} = +$. Quadrants I and III are the two "same-sign" quadrants, and they are exactly the tangent-positive ones.

That is the whole logic behind the T in ASTC. Tangent does not care that the point is in the far corner of the circle, only that $x$ and $y$ carry the same sign there. For the fuller pattern of signs across all six ratios, see basic properties of trigonometric ratios.

Who Discovered The Tangent And Its Values?

The tangent is younger than sine, and its story runs from shadow-lengths on a sundial to the value tables every student now takes for granted.

Two more mathematicians shaped the values we use:

• Aryabhata (476–550 CE, India) tabulated sine values (he called the sine jya) at close angle steps in his Aryabhatiya, giving India one of the earliest accurate sine tables and the vocabulary that, through Arabic and Latin, became our word "sine."

• Madhava of Sangamagrama (c. 1340–1425, India) found infinite series for sine, cosine, and the arctangent, roughly 250 years before Newton and Leibniz, which is how calculators actually compute a tangent to many decimal places today.

Where Is Tan 240 Degrees Used In The Real World?

A tangent measures a slope or a ratio of "rise to run," and angles past $180^\circ$ appear wherever direction keeps turning past halfway.

• Navigation and bearings: a heading of $240^\circ$ points into the south-west, and the tangent of that heading feeds the ratio between east-west and north-south travel used in dead-reckoning and autopilot code.

• Ramps and roads: slope is the tangent of the incline angle, so engineers convert between an angle and a gradient (a $60^\circ$ reference slope is a steep $173$ grade) using exactly this value.

• Computer graphics and games: rotating a sprite or camera past half a turn lands it at angles like $240^\circ$, and the renderer uses the signed tangent to keep the projection correct in every quadrant.

• Alternating current: voltage and current in an AC circuit are described by sine and cosine over a full rotation, and phase relationships at angles like $\frac{4\pi}{3}$ are read straight off these values.

• Astronomy and surveying: the field Hipparchus started still runs on tangents, from measuring a star's altitude to fixing a boundary line from two known points.

One small value, $\sqrt{3}$, quietly connects a spotlight's slope, a ship's heading, and the maths inside a game engine.

What Are The Most Common Mistakes With Tan 240 Degrees?

These three errors account for most wrong answers on this angle, and each has a clean fix.

Giving the wrong sign.

Where it slips in:

A student finds the reference angle $60^\circ$ correctly, then writes $\tan 240^\circ = -\sqrt{3}$, copying the sign pattern from sine or cosine, which are negative here.

Don't do this:

Do not assume every ratio is negative in the third quadrant. Sine and cosine are negative there, but tangent is positive.

The correct way:

Read the sign from ASTC. Quadrant III is the tangent quadrant, so $\tan 240^\circ = +\tan 60^\circ = \sqrt{3}$.

Leaving the calculator in the wrong mode.

Where it slips in:

A student types tan(240) while the calculator is set to radians and reads off $\approx 1.62$, which is the tangent of $240$ radians, not $240$ degrees.

Don't do this:

Do not trust the display until the angle mode matches the angle. $240$ degrees and $240$ radians are completely different rotations.

The correct way:

Set the calculator to DEG for degree questions (or enter $\frac{4\pi}{3}$ in RAD mode). Correctly done, both give $\sqrt{3} \approx 1.7321$.

Taking the wrong reference angle.

Where it slips in:

A student computes the reference angle as $240^\circ - 90^\circ = 150^\circ$ or $270^\circ - 240^\circ = 30^\circ$, mixing up the rule for Quadrant III.

Don't do this:

Do not subtract from $90^\circ$ or $270^\circ$ for a third-quadrant angle. The reference angle is measured from the nearest horizontal axis, not the vertical one.

The correct way:

For a Quadrant III angle, subtract $180^\circ$. Here $240^\circ - 180^\circ = 60^\circ$, so the reference angle is $60^\circ$.

Practice Problems On Tan 240 Degrees

Work each one, then check against the answer. Values are exact where a surd is cleaner than a decimal.

1. Evaluate $\tan 240^\circ + \tan 60^\circ$.
   (Answer: $\sqrt{3} + \sqrt{3} = 2\sqrt{3} \approx 3.4641$.)

2. Simplify $\dfrac{\tan 240^\circ}{\tan 60^\circ}$.
   (Answer: $\dfrac{\sqrt{3}}{\sqrt{3}} = 1$.)

3. Find $\cot 240^\circ$.
   (Answer: $\dfrac{1}{\tan 240^\circ} = \dfrac{1}{\sqrt{3}} = \dfrac{\sqrt{3}}{3} \approx 0.5774$.)

4. Evaluate $\sin 240^\circ + \cos 240^\circ$.
   (Answer: $-\dfrac{\sqrt{3}}{2} - \dfrac{1}{2} = -\dfrac{\sqrt{3}+1}{2} \approx -1.3660$.)

5. Verify that $\tan 240^\circ = \dfrac{\sin 240^\circ}{\cos 240^\circ}$.
   (Answer: $\dfrac{-\sqrt{3}/2}{-1/2} = \sqrt{3}$, which matches.)

6. Convert $240^\circ$ to radians and state its tangent.
   (Answer: $240^\circ = \dfrac{4\pi}{3}$, and $\tan\dfrac{4\pi}{3} = \sqrt{3}$.)

Where Should You Go Next After Tan 240 Degrees?

One value opens onto the whole tangent function, so pick the door that matches what you want next.

1. Tangent function. See how $\tan\theta$ behaves as a curve, where it climbs to infinity, and why its period is $180^\circ$ rather than $360^\circ$.

2. Sin cos tan. Tie the three core ratios together from the right triangle and the unit circle in one place.

3. Trigonometric identities. Move from single values to the rules that let you rewrite and simplify whole expressions.

If your child is building these foundations, a live Bhanzu trainer teaches angle values starting from the "why" (the reference angle and the unit circle, not memorised tables) in the Bhanzu trigonometry program.