Tan 5pi/3: Exact Value, Steps & Unit Circle

What Is The Value Of Tan 5pi/3?

Tan 5pi/3 is equal to $-\sqrt{3}$, or about $-1.7321$ as a decimal. Written with the angle in both forms, $\tan\frac{5\pi}{3} = \tan 300^\circ = -\sqrt{3}$.

$$\tan\frac{5\pi}{3} = -\sqrt{3} \approx -1.7321$$

The value is negative for one reason: the angle sits in Quadrant IV of the coordinate plane, and tangent is negative there. The size of the number, $\sqrt{3}$, comes straight from the tangent function evaluated at the reference angle $\frac{\pi}{3}$. The rest of this article shows where that comes from, two independent ways.

How Do You Find The Value Of Tan 5pi/3?

Finding this value takes three short steps: convert the angle, find its reference angle, then fix the sign from the quadrant.

Step 1: Convert to degrees (optional but clarifying).

$$\frac{5\pi}{3} = \frac{5}{3} \times 180^\circ = 300^\circ$$

Step 2: Find the reference angle. In Quadrant IV, the reference angle is $360^\circ$ minus the angle:

$$360^\circ - 300^\circ = 60^\circ = \frac{\pi}{3}$$

Step 3: Fix the sign with the quadrant. Use the ASTC rule (also called CAST). In Quadrant IV, only cosine and secant are positive, so tangent is negative. The reference angle gives the size, and the quadrant gives the sign:

$$\tan 300^\circ = -\tan 60^\circ = -\sqrt{3}$$

You can read the size straight from a 30-60-90 right triangle. In that triangle the side opposite $60^\circ$ is $\sqrt{3}$ and the side adjacent is $1$, so $\tan 60^\circ = \frac{\sqrt{3}}{1} = \sqrt{3}$. That triangle value is confirmed on the tan 60 degrees and tan π/3 pages, and the reference-angle method itself is set out under trigonometric ratios of specific angles.

Where Does 5pi/3 Sit On The Unit Circle?

On the unit circle, the angle $\frac{5\pi}{3}$ points into the lower-right quarter (Quadrant IV). The point where its ray meets the circle has coordinates $\left(\frac{1}{2}, -\frac{\sqrt{3}}{2}\right)$.

Tangent on the unit circle is the y-coordinate divided by the x-coordinate:

$$\tan\frac{5\pi}{3} = \frac{y}{x} = \frac{-\frac{\sqrt{3}}{2}}{\frac{1}{2}} = -\sqrt{3}$$

This is the second, independent route to the same answer. The right triangle gave the size $\sqrt{3}$; the unit circle gives both the size and the sign at once, because in Quadrant IV the y-coordinate is negative while the x-coordinate is positive. A negative divided by a positive is negative. For a fuller reader-driven view of tangent on the circle, see unit circle with tangent, and for the meaning of the angle measure itself, what is a radian.

How Do You Confirm Tan 5pi/3 From Sine And Cosine?

Tangent is defined as sine over cosine, so a third check uses the sine and cosine of the same angle. At $\frac{5\pi}{3}$:

$$\sin\frac{5\pi}{3} = -\frac{\sqrt{3}}{2}, \qquad \cos\frac{5\pi}{3} = \frac{1}{2}$$

Dividing one by the other returns the tangent:

$$\tan\frac{5\pi}{3} = \frac{\sin\frac{5\pi}{3}}{\cos\frac{5\pi}{3}} = \frac{-\frac{\sqrt{3}}{2}}{\frac{1}{2}} = -\sqrt{3}$$

All three methods agree on $-\sqrt{3}$. The cosine value here is worked out on the cos 5pi/3 page, and the general relationship between the three ratios is set out under sin cos tan.

What Are The Tangent Values Around 5pi/3?

The angle $\frac{5\pi}{3}$ belongs to a family that all share the reference angle $60^\circ$. Their tangents are $\sqrt{3}$ or $-\sqrt{3}$, and only the quadrant decides which.

Table: Tangent values for the reference-angle-60° family, in degrees and radians.

Reading down the table, tangent is positive in Quadrants I and III and negative in Quadrants II and IV. The value for $\frac{2\pi}{3}$ is confirmed on the tan 2π/3 page. A full grid of standard angles lives in the trigonometric table, and the radian versions of these ratios are collected under trigonometric ratios in radians.

Why Is Tan 5pi/3 Negative?

The negative sign is not arbitrary. It follows from where the angle lands and how tangent is built.

• The quadrant sets the signs. At $300^\circ$ the terminal ray is in Quadrant IV, where points have a positive x-coordinate and a negative y-coordinate.

• Tangent is height over width. Since $\tan\theta = \frac{y}{x}$, a negative height over a positive width gives a negative result.

• ASTC confirms it. The ASTC rule states that only cosine (and its reciprocal secant) stay positive in Quadrant IV, so both sine and tangent are negative there.

Put together, the size of the number comes from the reference angle $60^\circ$, and the minus sign comes from Quadrant IV. That is the whole reason $\tan\frac{5\pi}{3}$ reads as $-\sqrt{3}$ rather than $+\sqrt{3}$.

Who Discovered The Values Behind Tan 5pi/3?

Nobody woke up one morning and defined the tangent of $300^\circ$. The values in the tangent table were built slowly, table by table, across more than a thousand years and several civilisations.

Two later figures pushed the same idea toward the tangent we use now:

• Aryabhata (476 to 550 CE, India) compiled an influential table of half-chords, the jya values that became our sine, in his work of 499 CE, giving the sine-based route that lets us find any tangent as sine over cosine.

• Georg Joachim Rheticus (1514 to 1574, Austria) produced extensive tables of all six trigonometric ratios directly from the right triangle, helping fix the tangent as a ratio in its own right rather than a by-product of chords.

Where Is Tan 5pi/3 Used In The Real World?

A single tangent value looks abstract on the page, yet the tangent of an angle near a full turn shows up wherever a steepness or a slope is measured just past the horizontal.

• Roads and ramps: the gradient of a downhill slope is a tangent, and a downward direction carries the same negative sign that Quadrant IV gives here.

• Computer graphics: tilting a virtual camera below the horizon uses tangents of angles in the fourth quadrant to work out how the view shears.

• Alternating current: the phase of a voltage or current signal is tracked around the full circle, and tangents of late-cycle angles like $300^\circ$ describe the signal near the end of each cycle.

• Navigation and surveying: bearings measured clockwise past three-quarters of a turn land in this region, and their tangents give the ratio of one direction to another.

The lesson repeats across fields: once a rotation passes three-quarters of the way around, its tangent turns negative, and that sign carries real meaning about direction.

What Are The Most Common Mistakes With Tan 5pi/3?

These four errors account for most wrong answers on this value. Each one is quick to avoid once you have seen it.

Leaving the calculator in degree mode for a radian input.

Where it slips in:

A student types $\tan(5\pi/3)$ while the calculator is set to degrees, so it reads the input as $300$ radians, not $\frac{5\pi}{3}$ radians.

Don't do this:

Do not trust the display before checking the angle mode.

The correct way:

Set the mode to radians for a radian input, or convert to $300^\circ$ first and use degree mode. Either way the answer is $-\sqrt{3} \approx -1.7321$.

Forcing the tangent to be positive.

Where it slips in:

A student finds the reference value $\tan 60^\circ = \sqrt{3}$ and writes that as the final answer, forgetting the quadrant.

Don't do this:

Do not report the reference-angle value without applying the sign.

The correct way:

Check the quadrant first. Quadrant IV makes tangent negative, so the answer is $-\sqrt{3}$, not $\sqrt{3}$.

Measuring the reference angle from the wrong axis.

Where it slips in:

A student subtracts $270^\circ$ instead of $360^\circ$, getting a reference angle of $30^\circ$ and the wrong value.

Don't do this:

Do not measure a Quadrant IV reference angle from the vertical axis.

The correct way:

In Quadrant IV, measure from the positive x-axis: $360^\circ - 300^\circ = 60^\circ$, which gives the reference angle $\frac{\pi}{3}$.

Flipping the ratio in tan equals sine over cosine.

Where it slips in:

A student computes $\frac{\cos}{\sin}$ by mistake, getting $-\frac{1}{\sqrt{3}}$ instead of $-\sqrt{3}$.

Don't do this:

Do not divide cosine by sine when you want tangent.

The correct way:

Keep the order as $\tan\theta = \frac{\sin\theta}{\cos\theta}$. Here that is $\frac{-\sqrt{3}/2}{1/2} = -\sqrt{3}$.

Practice Problems On Tan 5pi/3

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

1. Convert $\frac{5\pi}{3}$ to degrees.
   (Answer: $300^\circ$.)

2. State the reference angle of $\frac{5\pi}{3}$.
   (Answer: $\frac{\pi}{3}$, or $60^\circ$.)

3. In which quadrant does $\frac{5\pi}{3}$ lie, and is tangent positive or negative there?
   (Answer: Quadrant IV, tangent negative.)

4. Find $\tan\frac{5\pi}{3}$ using the unit-circle coordinates $\left(\frac{1}{2}, -\frac{\sqrt{3}}{2}\right)$.
   (Answer: $\frac{-\sqrt{3}/2}{1/2} = -\sqrt{3}$.)

5. Evaluate $\tan\frac{5\pi}{3} + \tan\frac{2\pi}{3}$.
   (Answer: $-\sqrt{3} + (-\sqrt{3}) = -2\sqrt{3} \approx -3.4641$.)

6. True or false: $\tan\frac{5\pi}{3} = \tan\frac{\pi}{3}$.
   (Answer: False. They have the same size but opposite signs, since $\tan\frac{\pi}{3} = \sqrt{3}$.)

Where Should You Go Next After Tan 5pi/3?

Once this value makes sense, a few natural doors open from here.

1. Tangent function. See how tangent behaves across the whole circle, including where it climbs without bound.

2. Trigonometric ratios of specific angles. Lock in the standard-angle values so any quadrant becomes a quick reference-plus-sign check.

3. Trigonometric ratios in radians. Get comfortable working in radians directly, without converting to degrees each time.

If your child is building these foundations, a live Bhanzu trainer teaches angle values starting from the reason behind the sign, not the memorised table, in the Bhanzu trigonometry program.