Tan 6 Degrees: Value, Exact Form, Approximations And Examples

What Does Tan 6 Degrees Mean?

In a right triangle, $\tan\theta = \frac{\text{opposite}}{\text{adjacent}}$, so $\tan 6^\circ$ is how far a line rises for each unit it runs when it leans $6^\circ$ above horizontal. On the unit circle, it is $\frac{\sin 6^\circ}{\cos 6^\circ}$, which also equals the height where the extended radius crosses the vertical tangent line $x = 1$.

Because $6^\circ$ lies in Quadrant I (the region where $x$ and $y$ are both positive), the value is positive. No reference-angle or sign work is needed, unlike the larger angles in the same family.

This value is a calculator and approximation skill rather than a memorisation target. School tables stop at $0^\circ$, $30^\circ$, $45^\circ$, $60^\circ$ and $90^\circ$; $\tan 6^\circ$ matters because it shows how the less famous angles are reached.

Standard-Angle Tangent Reference Table

Small angles near $6^\circ$ are listed below with their radian measure, so you can see how closely $\tan\theta$ tracks the angle in radians. The exact column shows which of these angles have a short closed form.

The last two columns agree closely for tiny angles and drift apart as the angle grows. At $6^\circ$ they differ only in the fourth decimal place, which is the basis of the small-angle shortcut later in this article.

Where Does Tan 6 Degrees Show Up, And Why Does It Matter?

Road signs give steepness as a percentage grade, rise divided by run times 100, and a $6^\circ$ incline is $\tan 6^\circ \times 100 \approx 10.5$. That is a steep road; the slope formula for a line and the grade on a sign are the same tangent ratio.

Six degrees is also one minute on a clock face, since $360^\circ \div 60 = 6^\circ$. In surveying, a $6^\circ$ angle of elevation measured from 50 m away puts the top of a hill about $50 \times 0.1051 \approx 5.26$ m above eye level.

Can Tan 6 Degrees Be Written As An Exact Fraction?

No ordinary fraction of whole numbers equals $\tan 6^\circ$; it is irrational. It can, however, be written exactly with square roots, because $6^\circ$ is a multiple of $3^\circ$, and an integer number of degrees has a square-root form exactly when it is a multiple of $3$ (Wikipedia, exact trigonometric values).

The OEIS entry A019904 lists $\tan 6^\circ$ as an algebraic integer of degree 8, a root of $x^8 - 28x^6 + 134x^4 - 92x^2 + 1 = 0$. That degree is why the surd is long: no short expression can satisfy an eighth-degree equation that does not factor further.

How Do You Find The Exact Value Of Tan 6 Degrees?

Write $6^\circ = 36^\circ - 30^\circ$ and apply the tangent difference formula. The two ingredients come from the regular pentagon ($36^\circ$) and the equilateral triangle ($30^\circ$).

Method 1: Tangent Difference Formula

The tan (A − B) formula states:

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

Here $A = 36^\circ$, $B = 30^\circ$. The values needed are $\tan 36^\circ = \sqrt{5 - 2\sqrt{5}} \approx 0.7265$, which comes from $\cos 36^\circ = \frac{1 + \sqrt{5}}{4}$ in the angles in a pentagon, and tan 30 degrees $= \frac{1}{\sqrt{3}}$.

$$\tan 6^\circ = \frac{\sqrt{5 - 2\sqrt{5}} - \frac{1}{\sqrt{3}}}{1 + \frac{\sqrt{5 - 2\sqrt{5}}}{\sqrt{3}}}$$

Multiply the top and bottom by $\sqrt{3}$:

$$\tan 6^\circ = \frac{\sqrt{3}\sqrt{5 - 2\sqrt{5}} - 1}{\sqrt{3} + \sqrt{5 - 2\sqrt{5}}}$$

Numerically, $\frac{1.732051 \times 0.726543 - 1}{1.732051 + 0.726543} = \frac{0.258409}{2.458593} \approx 0.1051$, matching the calculator to ten decimal places.

Method 2: Exact Sine Over Exact Cosine

The same difference trick gives $\sin 6^\circ$ and $\cos 6^\circ$ separately, using $\sin 36^\circ = \frac{\sqrt{10 - 2\sqrt{5}}}{4}$:

$$\sin 6^\circ = \sin 36^\circ \cos 30^\circ - \cos 36^\circ \sin 30^\circ = \frac{\sqrt{30 - 6\sqrt{5}} - \sqrt{5} - 1}{8}$$

$$\cos 6^\circ = \cos 36^\circ \cos 30^\circ + \sin 36^\circ \sin 30^\circ = \frac{\sqrt{3} + \sqrt{15} + \sqrt{10 - 2\sqrt{5}}}{8}$$

Dividing, the eights cancel:

$$\tan 6^\circ = \frac{\sqrt{30 - 6\sqrt{5}} - \sqrt{5} - 1}{\sqrt{10 - 2\sqrt{5}} + \sqrt{3} + \sqrt{15}}$$

This looks different from Method 1 but is the same number, $0.1051042353$. Both forms were checked against each other to 30 digits.

Method 3: Half-Angle From 12 Degrees

The half angle formula for tangent is $\tan\frac{\theta}{2} = \frac{\sin\theta}{1 + \cos\theta}$. With $\theta = 12^\circ$, and since $\cos 12^\circ = \sin 78^\circ$:

$$\tan 6^\circ = \frac{\sin 12^\circ}{1 + \cos 12^\circ} = \frac{0.207912}{1.978148} \approx 0.1051$$

Method 4: The Tangent Series

For $x$ in radians, $\tan x = x + \frac{x^3}{3} + \frac{2x^5}{15} + \ldots$, valid for $|x| < \frac{\pi}{2}$. With $x = \frac{\pi}{30} \approx 0.104720$:

$$\tan 6^\circ \approx 0.104720 + \frac{0.104720^3}{3} \approx 0.104720 + 0.000383 = 0.105103$$

Two terms already agree with the true value to five decimal places. Calculators use refined versions of this idea, together with range reduction, rather than the surd.

Method 5: Calculator In Degree Mode

Set DEG, enter $\tan(6)$, and read $0.1051042353$. If the screen shows $-0.2910$, it is in radian mode; that number is the tangent of 6 radians.

How Good Is The Small-Angle Approximation For Tan 6 Degrees?

For small angles in radians, $\tan\theta \approx \theta$. At $6^\circ$ that gives $\frac{\pi}{30} \approx 0.1047$, which is low by about $0.37$.

The error grows quickly as the angle opens up, so the shortcut has a clear range:

• At $1^\circ$: about $0.01$ low.

• At $5^\circ$: about $0.25$ low.

• At $6^\circ$: about $0.37$ low.

• At $10^\circ$: about $1.0$ low.

• At $15^\circ$: about $2.3$ low, and past $20^\circ$ the error passes $4$.

Adding the cubic term, $\theta + \frac{\theta^3}{3}$, cuts the error at $6^\circ$ to about $0.0016$. For hand estimates up to roughly $10^\circ$, the one-term version is fine; beyond that, use the two-term version or a calculator.

What Is Tan 6 Degrees In Radians?

In radians, $6^\circ = \frac{\pi}{30}$, so $\tan \frac{\pi}{30} \approx 0.1051$. The conversion multiplies by $\frac{\pi}{180}$, the standard rule for converting degrees to radians.

$$6^\circ \times \frac{\pi}{180^\circ} = \frac{\pi}{30} \approx 0.1047$$

A radian is the angle whose arc equals the radius, so $\frac{\pi}{30}$ is an arc just over a tenth of the radius. The tangent, $0.1051$, is slightly longer than that arc, which is the small-angle comparison seen from the circle.

How Is Tan 6 Degrees Related To Tan 84 Degrees?

Since $6^\circ$ and $84^\circ$ add to $90^\circ$, the co-function identity gives $\cot 84^\circ = \tan 6^\circ \approx 0.1051$ and $\tan 84^\circ = \cot 6^\circ \approx 9.5144$. The cofunction identities swap tangent and cotangent across complementary angles.

The two tangents are reciprocals, $\tan 6^\circ \times \tan 84^\circ = 1$. So once you know one, the other costs a single division: $\frac{1}{0.1051042} \approx 9.5144$.

Examples Of Tan 6 Degrees

Example 1

A road rises at $6^\circ$. Express its steepness as a percentage grade.

$$\text{grade} = \tan 6^\circ \times 100$$

$$= 0.1051 \times 100 = 10.51$$

Final answer: about $10.5$

Example 2

A student writes $\tan 6^\circ = \tan(36^\circ - 30^\circ) = \tan 36^\circ - \tan 30^\circ$. Find the correct value.

The tempting step distributes tangent over subtraction, giving $0.72654 - 0.57735 = 0.14919$.

That cannot be right: the calculator says $0.1051$, and even the rough estimate $\frac{\pi}{30} \approx 0.105$ is far from $0.149$. Tangent does not split across a minus sign.

The rescue is the difference formula, with its denominator:

$$\tan 6^\circ = \frac{0.72654 - 0.57735}{1 + 0.72654 \times 0.57735}$$

$$= \frac{0.14919}{1.41947} \approx 0.1051$$

Final answer: $\tan 6^\circ \approx 0.1051$

Example 3

From a point 50 m from the foot of a hill, the angle of elevation to the top is $6^\circ$. How high is the top above eye level?

$$h = 50 \times \tan 6^\circ$$

$$= 50 \times 0.1051042 \approx 5.26$$

Final answer: about $5.26$ m

Example 4

Given $\tan 6^\circ \approx 0.1051$, find $\cot 84^\circ$ and $\tan 84^\circ$.

$$\cot 84^\circ = \tan(90^\circ - 84^\circ) = \tan 6^\circ \approx 0.1051$$

$$\tan 84^\circ = \frac{1}{\tan 6^\circ} \approx 9.5144$$

Final answer: $\cot 84^\circ \approx 0.1051$, $\tan 84^\circ \approx 9.5144$

Example 5

Show that $\tan 6^\circ + \tan 39^\circ + \tan 6^\circ \tan 39^\circ = 1$.

Since $6^\circ + 39^\circ = 45^\circ$ and $\tan 45^\circ = 1$:

$$\frac{\tan 6^\circ + \tan 39^\circ}{1 - \tan 6^\circ \tan 39^\circ} = 1$$

$$\tan 6^\circ + \tan 39^\circ = 1 - \tan 6^\circ \tan 39^\circ$$

$$\tan 6^\circ + \tan 39^\circ + \tan 6^\circ \tan 39^\circ = 1$$

Final answer: the identity holds (numerically $0.99999\ldots$ on a calculator, exactly $1$)

Example 6

Show that $\tan 6^\circ \tan 42^\circ \tan 66^\circ \tan 78^\circ = 1$.

Use the identity $\tan\theta \tan(60^\circ - \theta) \tan(60^\circ + \theta) = \tan 3\theta$ twice.

With $\theta = 6^\circ$:

$$\tan 6^\circ \tan 54^\circ \tan 66^\circ = \tan 18^\circ$$

With $\theta = 18^\circ$:

$$\tan 18^\circ \tan 42^\circ \tan 78^\circ = \tan 54^\circ$$

Multiply the two lines and cancel $\tan 18^\circ \tan 54^\circ$ from both sides:

$$\tan 6^\circ \tan 42^\circ \tan 66^\circ \tan 78^\circ = 1$$

Final answer: the product is exactly $1$ (calculator: $0.9999999999999996$, rounding noise)

These four angles are exactly the positive roots of the eighth-degree equation in the OEIS entry, which is a satisfying way to see why their product is $1$.

Where Do Students Go Wrong With Tan 6 Degrees?

Mistake 1: Splitting Tangent Across A Difference

Where it slips in: Using $6^\circ = 36^\circ - 30^\circ$ or $6^\circ = 45^\circ - 39^\circ$ to find an exact value.

Don't do this: Write $\tan(36^\circ - 30^\circ) = \tan 36^\circ - \tan 30^\circ$.

The correct way: Use $\frac{\tan A - \tan B}{1 + \tan A \tan B}$ from the sum and difference formulas. The memorizer who knows the formula's top line but not its denominator gets $0.1492$; checking against $\frac{\pi}{30} \approx 0.105$ catches it in seconds.

Mistake 2: Using The Small-Angle Shortcut In Degrees

Where it slips in: Quick estimates, where "tan of a small angle is about the angle" gets applied to the number 6.

Don't do this: Write $\tan 6^\circ \approx 6$, or $\tan 6^\circ \approx 0.06$.

The correct way: Convert first: $6^\circ = \frac{\pi}{30} \approx 0.1047$, and only then use $\tan\theta \approx \theta$. Students who write the unit beside every angle stop making this error, because "$\tan 6^\circ \approx 6^\circ$" visibly mixes a ratio with an angle.

Mistake 3: Radian Mode On The Calculator

Where it slips in: Checking the answer on a phone or calculator left in radian mode.

Don't do this: Accept $\tan 6 = -0.2910$. Google's suggestion box shows the same $-0.291006191$ for "is tan 6", because that is the tangent of 6 radians, about $344^\circ$.

The correct way: Switch to DEG, or type $\frac{\pi}{30}$ in radian mode. Any acute angle has a positive tangent, so a negative answer is an instant signal that the mode is wrong; the rusher who reads the screen without asking "should this be positive?" is the one who keeps it.

Where Can You Get Help With Trigonometry Values?

Non-standard angles like this one are where a trigonometry tutor helps most, because the difference and half-angle formulas need careful practice. A high school math tutor or regular math classes online can build the same habits across the whole course.

Key Takeaways

• Tan 6 degrees is about $0.1051$ ($0.1051042353$ to ten places) and is irrational.

• An exact square-root form exists because $6^\circ = 36^\circ - 30^\circ$, but it is long since the value has algebraic degree 8.

• The half-angle formula and the series $x + \frac{x^3}{3}$ both confirm the value quickly.

• In radians, $6^\circ = \frac{\pi}{30} \approx 0.1047$, and $\tan\theta \approx \theta$ is about $0.37$ low here.

• $\tan 6^\circ = \cot 84^\circ$, and $\tan 84^\circ = \cot 6^\circ \approx 9.5144$.

Practice These Before Moving On

1. A road climbs at $6^\circ$ over a horizontal run of 200 m. How much does it rise?

2. Use $\tan 2\theta = \frac{2\tan\theta}{1 - \tan^2\theta}$ with $\tan 6^\circ \approx 0.1051042$ to find $\tan 12^\circ$ to four decimal places.

3. Without a calculator, give $\cot 84^\circ$ to four decimal places.

4. Estimate $\tan 6^\circ$ with $x + \frac{x^3}{3}$, using $x = 0.10472$, to five decimal places.

Answers To The Practice Questions

1. $200 \times \tan 6^\circ \approx 21.02$ m

2. $\frac{2 \times 0.1051042}{1 - 0.0110469} \approx 0.2126$, which matches tan 12 degrees

3. $\cot 84^\circ = \tan 6^\circ \approx 0.1051$

4. $0.10472 + 0.00038 = 0.10510$

If question 2 goes wrong, check the denominator first, just as in Mistake 1. Want a trainer to walk through more problems like these with you? Book a free demo class.

Read More

• Tan 1 degrees and very small angles

• Tangent function and its graph

• Tan 2x formula

• Trigonometric table of standard angles

• Tan 20 degrees