Tan Pi/6 : Exact Value 1/√3, Rationalized, and How to Find It

What Does Tan Pi/6 Mean?

The angle $\frac{\pi}{6}$ is written in radians, where a full turn is $2\pi$, so $\frac{\pi}{6}$ is one-twelfth of a circle. Readers new to the unit can start with what a radian is before returning here.

Tangent is one of the three core trigonometric ratios: on the unit circle it is the $y$-coordinate divided by the $x$-coordinate of the angle's point, and equally $\frac{\sin\theta}{\cos\theta}$. The degree-first companion, tan 30 degrees, builds the same value straight from the $30$-$60$-$90$ triangle, which is the easier entry if degrees feel more natural.

Where Does Tan Pi/6 Show Up?

A $\frac{\pi}{6}$ radian angle is the gentle $30^\circ$ incline, and its tangent of about $0.5774$ is the gradient of a shallow ramp or a low roof pitch, where the rise is a little over half the run. Surveyors and roofers reach for this ratio whenever a $30^\circ$ slope needs a horizontal measurement turned into a height.

The value also sits inside the $30$-$60$-$90$ triangle that structures hexagons, equilateral-triangle bisectors, and the standard set of drafting angles. Because $\tan\frac{\pi}{6}$ and $\tan\frac{\pi}{3}$ are reciprocals, one measurement at $30^\circ$ immediately gives you the matching $60^\circ$ relationship without a second calculation.

Standard-Angle Tangent Reference Table

Seeing the whole first-quadrant sweep makes $\frac{\pi}{6}$ easier to place. The unit circle gives each of these as a $y$-over-$x$ ratio.

Notice the mirror: $\tan\frac{\pi}{6} = \frac{1}{\sqrt{3}}$ and $\tan\frac{\pi}{3} = \sqrt{3}$ are reciprocals of each other, because $30^\circ$ and $60^\circ$ are complementary.

How Do You Find The Exact Value Of Tan Pi/6?

Two routes reach the value, and a third step cleans up how it is written.

Method 1: Sine over cosine.

At $\frac{\pi}{6}$ the sine and cosine are known standard values:

$$\sin\frac{\pi}{6} = \frac{1}{2}, \qquad \cos\frac{\pi}{6} = \frac{\sqrt{3}}{2}$$

Divide them:

$$\tan\frac{\pi}{6} = \frac{\sin(\pi/6)}{\cos(\pi/6)} = \frac{1/2}{\sqrt{3}/2} = \frac{1}{\sqrt{3}}$$

Method 2: The unit circle.

Rotate a radius of length $1$ through $\frac{\pi}{6}$. It lands at $\left(\frac{\sqrt{3}}{2}, \frac{1}{2}\right)$, a point low and wide because the angle is shallow.

$$\tan\frac{\pi}{6} = \frac{y}{x} = \frac{1/2}{\sqrt{3}/2} = \frac{1}{\sqrt{3}}$$

Method 3: Rationalizing the denominator.

Leaving a root in the denominator is correct but not the standard written form. Multiply top and bottom by $\sqrt{3}$:

$$\frac{1}{\sqrt{3}} \times \frac{\sqrt{3}}{\sqrt{3}} = \frac{\sqrt{3}}{3}$$

So $\frac{1}{\sqrt{3}}$ and $\frac{\sqrt{3}}{3}$ are the same number, about $0.5774$; the second is the rationalized form most exams expect.

Examples Of Tan Pi/6

Example 1

Evaluate $6\tan\dfrac{\pi}{6}$.

$$6\tan\frac{\pi}{6} = 6 \times \frac{1}{\sqrt{3}} = \frac{6}{\sqrt{3}} = \frac{6\sqrt{3}}{3} = 2\sqrt{3} \approx 3.464$$

Example 2

Write $\tan\dfrac{\pi}{6}$ as a decimal and then as a rationalized fraction. A student writes $0.577$ and stops. Is that the exact value?

Wrong reading. Copying the calculator, the student records $\tan\frac{\pi}{6} = 0.577$ as the exact answer.

That cannot be the exact value, because $0.577$ is a rounded decimal, and $\frac{\pi}{6}$ is a standard angle with an exact surd form.

Correct. The exact value is $\frac{1}{\sqrt{3}}$, rationalized to $\frac{\sqrt{3}}{3}$. The decimal $0.5774$ is only an approximation of it.

Example 3

Simplify $\tan\dfrac{\pi}{6} \times \tan\dfrac{\pi}{3}$.

The two are reciprocals:

$$\frac{1}{\sqrt{3}} \times \sqrt{3} = 1$$

Example 4

A ramp rises at $\dfrac{\pi}{6}$ radians. If the horizontal run is $9$ m, how high does it climb?

Height equals run times the gradient $\tan\frac{\pi}{6}$:

$$\text{rise} = 9 \times \frac{1}{\sqrt{3}} = \frac{9}{\sqrt{3}} = 3\sqrt{3} \approx 5.196 \text{ m}$$

Example 5

Verify $\tan^2\dfrac{\pi}{6} + 1 = \sec^2\dfrac{\pi}{6}$.

Left side: $\left(\frac{1}{\sqrt{3}}\right)^2 + 1 = \frac{1}{3} + 1 = \frac{4}{3}$.

Right side: $\sec\frac{\pi}{6} = \frac{1}{\cos(\pi/6)} = \frac{2}{\sqrt{3}}$, so $\sec^2\frac{\pi}{6} = \frac{4}{3}$. Both sides equal $\frac{4}{3}$.

Where Students Trip Up On Tan Pi/6

Mistake 1: Leaving 1/√3 unrationalized when the exact form is asked

Where it slips in: Final answers on exam questions that say "give the exact value," where a root is left sitting in the denominator.

Don't do this: Writing $\tan\frac{\pi}{6} = \frac{1}{\sqrt{3}}$ and assuming it will always be marked complete.

The correct way: Rationalize to $\frac{\sqrt{3}}{3}$. The habit of multiplying by $\frac{\sqrt{3}}{\sqrt{3}}$ the moment a root lands in a denominator is what keeps this mark.

Mistake 2: Swapping tan pi/6 and tan pi/3

Where it slips in: Recall of the shallow-versus-steep values, where $\frac{1}{\sqrt{3}}$ and $\sqrt{3}$ get attached to the wrong angle.

Don't do this: Writing $\tan\frac{\pi}{6} = \sqrt{3}$, which is actually $\tan\frac{\pi}{3}$.

The correct way: The shallower angle has the smaller tangent. At $\frac{\pi}{6}$ the slope is gentle, so $\tan\frac{\pi}{6} = \frac{1}{\sqrt{3}} \approx 0.58$; the steep $\frac{\pi}{3}$ gives $\sqrt{3} \approx 1.73$.

Mistake 3: Reading the radian angle in degree mode

Where it slips in: Calculator entry, when $\frac{\pi}{6} \approx 0.524$ is typed while the mode is set to degrees.

Don't do this: Trusting the near-$0.009$ that a degree-mode screen prints for $\tan(0.524)$.

The correct way: Switch to radian mode first. The exact ratio $\frac{\sin(\pi/6)}{\cos(\pi/6)} = \frac{1}{\sqrt{3}}$ is the check that confirms the mode.

Key Takeaways

• Tan pi/6 equals $\frac{1}{\sqrt{3}}$, which rationalizes to $\frac{\sqrt{3}}{3}$ and rounds to about $0.5774$.

• On the unit circle, $\frac{\pi}{6}$ lands at $\left(\frac{\sqrt{3}}{2}, \frac{1}{2}\right)$, a low, wide point that makes the tangent small.

• The radian angle $\frac{\pi}{6}$ and the degree angle $30^\circ$ are identical, with the same tangent.

• $\tan\frac{\pi}{6}$ and $\tan\frac{\pi}{3}$ are reciprocals, because $30^\circ$ and $60^\circ$ are complementary.

• To take this further with a teacher, explore a trigonometry tutor, a high school math tutor, or math classes online.

Practice These Before Moving On

1. Rationalize and simplify $\dfrac{4}{\tan(\pi/6)}$.

2. A $30^\circ$ ramp runs $12$ m horizontally. Use $\tan\dfrac{\pi}{6}$ to find its height, in rationalized form.

3. Show that $\tan\dfrac{\pi}{6} + \tan\dfrac{\pi}{3}$ equals $\dfrac{4}{\sqrt{3}}$, then rationalize the result.

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Read More

• Tan pi/4 exact value — the neighbouring radian angle, where tangent is exactly $1$.

• Tan pi/3 exact value — the reciprocal partner, where tangent is $\sqrt{3}$.

• Trigonometric ratios of specific angles — how the standard-angle values are built.

• Trigonometric ratios in radians — the full radian version of the ratio table.