Tan 330 Degrees: Exact Value, Radians & Unit Circle

What Is The Value Of Tan 330 Degrees?

Tan 330 Degrees is $-\dfrac{1}{\sqrt{3}}$, which rationalises to $-\dfrac{\sqrt{3}}{3}$ and works out to about $-0.5774$. In radians the angle is written $\tan\dfrac{11\pi}{6}$, because $330^\circ = \dfrac{11\pi}{6}$.

$$\tan 330^\circ = \tan\frac{11\pi}{6} = -\frac{1}{\sqrt{3}} = -\frac{\sqrt{3}}{3} \approx -0.5774$$

The value is exact, not a rounded approximation. The decimal $-0.5774$ is only the first four places of a number that never ends, so the surd form $-\dfrac{1}{\sqrt{3}}$ is the one to quote in any proof or exam answer. For the wider family of these results, see the trigonometric ratios of specific angles.

How Do You Find Tan 330 Degrees Using The Reference Angle?

The fastest route uses two facts: the quadrant of the angle and its reference angle. A reference angle is the acute angle between the terminal arm and the horizontal axis, and it is always positive.

Follow three steps.

• Locate the quadrant. $330^\circ$ lies between $270^\circ$ and $360^\circ$, so its terminal arm is in the fourth quadrant.

• Find the reference angle. In the fourth quadrant the reference angle is $360^\circ - 330^\circ = 30^\circ$.

• Attach the sign. By the ASTC rule, only cosine is positive in the fourth quadrant, so tangent is negative there.

Putting those together:

$$\tan 330^\circ = -\tan 30^\circ = -\frac{1}{\sqrt{3}}$$

The size of the answer comes from $\tan 30^\circ$, and the minus sign comes from the quadrant. If you want the full base value first, our page on tan 30 degrees works it out from a right triangle.

Where Does 330 Degrees Sit On The Unit Circle?

On the unit circle, an angle is measured anticlockwise from the positive $x$-axis. An angle of $330^\circ$ stops just $30^\circ$ short of a full turn, so its terminal point lands in the fourth quadrant, low and to the right.

The coordinates of that point are $\left(\cos 330^\circ, \sin 330^\circ\right) = \left(\dfrac{\sqrt{3}}{2}, -\dfrac{1}{2}\right)$, which is about $(0.866, -0.5)$. Tangent is the $y$-coordinate divided by the $x$-coordinate.

$$\tan 330^\circ = \frac{\sin 330^\circ}{\cos 330^\circ} = \frac{-\tfrac{1}{2}}{\tfrac{\sqrt{3}}{2}} = -\frac{1}{\sqrt{3}}$$

The $x$-coordinate is positive and the $y$-coordinate is negative, so their ratio is negative. That single sign split is the whole reason Tan 330 Degrees comes out below zero.

How Do You Derive Tan 330 Degrees From The Difference Formula?

The reference-angle shortcut can be proved directly with the tangent difference formula, treating $330^\circ$ as $360^\circ - 30^\circ$. The formula is:

$$\tan(A - B) = \frac{\tan A - \tan B}{1 + \tan A \tan B}$$

Set $A = 360^\circ$ and $B = 30^\circ$. Since $\tan 360^\circ = 0$:

$$\tan(360^\circ - 30^\circ) = \frac{\tan 360^\circ - \tan 30^\circ}{1 + \tan 360^\circ \tan 30^\circ}$$

$$= \frac{0 - \tfrac{1}{\sqrt{3}}}{1 + 0} = -\frac{1}{\sqrt{3}}$$

The same value drops out as the unit-circle route, which is the point: every method has to agree, because the angle is one fixed direction. For the identities behind this step, see the trigonometric identities.

You can also read Tan 330 Degrees from a right triangle. Build a $30^\circ$ reference triangle with the opposite side $1$ and the adjacent side $\sqrt{3}$, so $\tan 30^\circ = \dfrac{1}{\sqrt{3}}$, then flip the sign for the fourth quadrant. Triangle and circle give the identical result, which is exactly the double anchor the tangent function is built on.

What Are The Related Tangent Values Around 330 Degrees?

Tan 330 Degrees belongs to a family of angles that all share the reference angle $30^\circ$. Their tangents are the same size and differ only in sign, set by the quadrant.

Table: Tangent values for angles sharing the 30° reference angle, in degrees and radians.

The pattern is worth memorising: tangent is positive in the first and third quadrants and negative in the second and fourth. To see every standard angle at once, use the trigonometric table.

Why Is Tan 330 Degrees Negative?

The minus sign is not a rule to memorise blindly. It falls out of where the angle points, and it can be reasoned in two ways.

• The ASTC sign rule. Starting from the first quadrant and going anticlockwise, the functions that stay positive are All, then Sine, then Tangent, then Cosine. The fourth quadrant is the C, so cosine is positive and both sine and tangent are negative.

• The sign of the coordinates. In the fourth quadrant the point on the unit circle has a positive $x$-value and a negative $y$-value. Tangent is $\dfrac{y}{x}$, and a negative divided by a positive is negative.

Both explanations describe the same picture. The terminal arm of $330^\circ$ dips below the horizontal axis while staying to the right of the vertical axis, so the height is negative and the width is positive. For the full set of these rules, see the basic properties of trigonometric ratios.

Who Discovered The Tangent And These Angle Values?

Tables of these values are far older than the modern names for them. Long before anyone wrote "tan," astronomers were building tables of ratios to predict the positions of the stars and the moon.

Two other figures shaped these ratios:

• Hipparchus of Nicaea (c. 190–120 BCE, Greece) built the first known table of chords, the earliest trigonometric table in the Western record, to track the motion of the sun and moon.

• Muhammad ibn Musa al-Khwarizmi (c. 780–850 CE, Baghdad) refined and spread these sine and tangent tables through the Islamic world, work that later reached Europe and set the stage for modern trigonometry.

Where Is Tan 330 Degrees Used In The Real World?

A single tangent value looks abstract until you meet the angle in the field. The direction $330^\circ$ and its slope turn up across several trades.

• Navigation and bearings: a heading of $330^\circ$ is north-north-west, and the tangent of that bearing sets the ratio of sideways drift to forward travel for a ship or aircraft.

• Electrical engineering: alternating-current voltages and currents are described by phase angles, and a phase of $330^\circ$ carries exactly this negative tangent when engineers compute lead and lag.

• Computer graphics: rotating a sprite or a 3D model by $330^\circ$ is the same as rotating it $-30^\circ$, and the rotation matrix uses this tangent-linked value.

• Surveying and construction: slopes measured as angles from a baseline use the tangent to convert a bearing into a rise-over-run gradient.

One value, read off a circle, quietly does work in ships, circuits, screens, and building sites. That reach is why the special angles are worth knowing cold rather than reaching for a calculator each time.

What Are The Most Common Mistakes With Tan 330 Degrees?

These four slips account for most of the wrong answers on fourth-quadrant angles, and each has a clean fix.

Making the answer positive.

Where it slips in:

A student finds the reference-angle value $\tan 30^\circ = \dfrac{1}{\sqrt{3}}$ and writes it as the final answer, forgetting the quadrant.

Don't do this:

Do not report $+\dfrac{1}{\sqrt{3}}$. The fourth quadrant makes tangent negative.

The correct way:

Attach the sign from ASTC before writing the answer. In quadrant IV tangent is negative, so $\tan 330^\circ = -\dfrac{1}{\sqrt{3}}$.

Leaving the calculator in the wrong mode.

Where it slips in:

A student types $\tan(330)$ while the calculator is set to radians and reads off a value near $-8$, which is nonsense for this angle.

Don't do this:

Do not trust the display without checking the mode indicator.

The correct way:

Set the calculator to degrees for $\tan 330^\circ$, or convert first and enter $\tan\dfrac{11\pi}{6}$ in radian mode. Either way the result is $-0.5774$.

Miscounting the reference angle.

Where it slips in:

A student subtracts $270^\circ$ from $330^\circ$ to get $60^\circ$, treating the angle as if it were measured from the vertical axis.

Don't do this:

Do not use $330^\circ - 270^\circ$. The reference angle is measured from the horizontal axis.

The correct way:

In the fourth quadrant, use $360^\circ - 330^\circ = 30^\circ$. The reference angle is $30^\circ$, not $60^\circ$.

Confusing the value with its reciprocal.

Where it slips in:

A student writes $\tan 330^\circ = -\sqrt{3}$, which is actually $\cot 330^\circ$, the reciprocal.

Don't do this:

Do not flip the surd. $-\sqrt{3}$ is the cotangent, not the tangent.

The correct way:

Keep tangent as $\dfrac{\text{opposite}}{\text{adjacent}} = -\dfrac{1}{\sqrt{3}}$. Its reciprocal $\cot 330^\circ = -\sqrt{3}$ is the separate value, covered under the reciprocal identities.

Practice Problems On Tan 330 Degrees

Work each one, then check against the answer that follows.

1. Evaluate $\tan 330^\circ$ using the reference angle.
   (Answer: $-\tan 30^\circ = -\dfrac{1}{\sqrt{3}} \approx -0.5774$.)

2. Convert $330^\circ$ to radians.
   (Answer: $330 \times \dfrac{\pi}{180} = \dfrac{11\pi}{6}$.)

3. Find $\cot 330^\circ$.
   (Answer: the reciprocal of $\tan 330^\circ$, so $-\sqrt{3}$.)

4. Find $\sec 330^\circ$.
   (Answer: $\dfrac{1}{\cos 330^\circ} = \dfrac{2}{\sqrt{3}} = \dfrac{2\sqrt{3}}{3} \approx 1.1547$.)

5. Is $\tan 330^\circ$ equal to $\tan 30^\circ$?
   (Answer: no. They are equal in size but opposite in sign, since $\tan 330^\circ = -\tan 30^\circ$.)

6. Compute $\tan 330^\circ + \tan 30^\circ$.
   (Answer: $-\dfrac{1}{\sqrt{3}} + \dfrac{1}{\sqrt{3}} = 0$.)

Where Should You Go Next After Tan 330 Degrees?

One special angle opens several doors, and each builds the foundation the next one needs.

1. Tan 30 degrees. The base value behind this result, derived straight from a right triangle.

2. Sin cos tan. See how all three ratios connect for the same angle, so a single point on the circle gives you every function.

3. Trigonometric ratios in radians. Move fluently between $330^\circ$ and $\dfrac{11\pi}{6}$, the language every higher course uses.

If your child is building these foundations, a live Bhanzu trainer teaches the unit circle starting from the "why" behind each sign, not rote tables, in the Bhanzu trigonometry program.