Tan 10 Degrees: Value, Radians & How To Find It

What Is The Value Of Tan 10 Degrees?

Tan 10 degrees equals approximately $0.1763$, and to more places it is $0.17632698$. In radians the angle is $\frac{\pi}{18}$, so the same fact is written $\tan\frac{\pi}{18} \approx 0.1763$.

The tangent of an angle is a ratio, not a rounded decimal that someone chose. For $10^\circ$ that ratio does not simplify to a neat expression with square roots, so the decimal is the honest working value. Here are the three forms you will actually use:

• Degrees: $\tan 10^\circ \approx 0.1763$.

• Radians: $\tan\frac{\pi}{18} \approx 0.1763$, since $10^\circ = \frac{\pi}{18}$ rad $\approx 0.1745$ rad.

• As a ratio of two other values: $\tan 10^\circ = \dfrac{\sin 10^\circ}{\cos 10^\circ} = \dfrac{0.1736}{0.9848} \approx 0.1763$.

Notice how close the angle in radians ($0.1745$) is to its own tangent ($0.1763$). For small angles the tangent and the radian measure almost agree, and that near-match is a useful sanity check you can carry into other small-angle work.

How Do You Find Tan 10 Degrees?

To find tan 10 degrees you can read it two ways: from a right triangle, and from the unit circle. Both give the same $0.1763$, and Bhanzu teaches both so the value never feels like it came from nowhere.

From a right triangle. In a right triangle, the tangent of an angle is the side opposite the angle divided by the side next to it:

$$\tan 10^\circ = \frac{\text{opposite}}{\text{adjacent}}$$

Draw a right triangle with the $10^\circ$ angle at one corner. If the adjacent side has length $1$, the opposite side comes out to about $0.1763$, so $\tan 10^\circ = \frac{0.1763}{1} = 0.1763$. The tangent function is exactly this opposite-over-adjacent ratio for every angle, not only for $10^\circ$.

From sine and cosine. Because tangent is sine divided by cosine, you can also build it from two values many students already know how to look up:

$$\tan 10^\circ = \frac{\sin 10^\circ}{\cos 10^\circ} = \frac{0.1736}{0.9848} \approx 0.1763$$

Since $10^\circ$ is a first-quadrant angle, its reference angle is simply $10^\circ$ itself, and every ratio in Quadrant I is positive. That is why the answer carries a plus sign with no extra work. For the sign rules in the other quadrants, the trigonometric ratios page walks through the full ASTC pattern.

Where Does 10 Degrees Sit On The Unit Circle?

On the unit circle, $10^\circ$ sits just above the positive x-axis, and the point on the circle at that angle has coordinates $(\cos 10^\circ, \sin 10^\circ) = (0.9848,\ 0.1736)$. The tangent is the y-coordinate divided by the x-coordinate:

$$\tan 10^\circ = \frac{y}{x} = \frac{0.1736}{0.9848} \approx 0.1763$$

The unit circle also shows why tangent grows slowly here. Near $0^\circ$ the point barely lifts off the x-axis, so the height is tiny compared with the width, and the ratio stays small. Compare this with tan 0 degrees, which is exactly $0$ because the point sits right on the axis. For a fuller picture of how the tangent segment stretches as the angle opens, see unit circle with tangent.

Does Tan 10 Degrees Have An Exact Value?

No. Tan 10 degrees has no simple exact value in square roots, because $10^\circ$ is a non-constructible angle. This is the honest answer, and it is worth understanding rather than papering over with a fake formula.

Angles like $30^\circ$, $45^\circ$, and $60^\circ$ come from triangles you can build with a compass and straightedge, so their tangents simplify to clean surds such as $\tan 30^\circ = \frac{1}{\sqrt{3}}$. The angle $10^\circ$ cannot be constructed that way, and its tangent is a root of a cubic that does not break down into real square roots.

Here is where that cubic comes from. Use the triple-angle formula with $\theta = 10^\circ$, so that $3\theta = 30^\circ$:

$$\tan 3\theta = \frac{3\tan\theta - \tan^{3}\theta}{1 - 3\tan^{2}\theta}$$

Set $3\theta = 30^\circ$, where $\tan 30^\circ = \frac{1}{\sqrt{3}}$, and let $t = \tan 10^\circ$:

$$\frac{3t - t^{3}}{1 - 3t^{2}} = \frac{1}{\sqrt{3}}$$

Cross-multiplying and tidying gives a cubic equation:

$$\sqrt{3},t^{3} - 3t^{2} - 3\sqrt{3},t + 1 = 0$$

This cubic has three real roots ($\tan 10^\circ$, $\tan 130^\circ$, and $\tan 250^\circ$). A cubic with three real irrational roots falls into the case that cannot be rewritten with real square roots alone, so no tidy surd exists for $\tan 10^\circ$. That leaves three practical ways to get the number:

• A trigonometric table. Historical trigonometric table makers computed values like this once, by hand, and everyone else looked them up.

• A calculator or a power series. A calculator uses a series such as $\tan x = x + \frac{x^{3}}{3} + \frac{2x^{5}}{15} + \dots$ with $x = \frac{\pi}{18}$, which converges to $0.176327$ after only a few terms.

• The cofunction relation. Since $\tan 10^\circ = \cot 80^\circ$, you can also read it from the cotangent of the complementary angle.

What Are The Tan Values Of Nearby Angles?

Tan 10 degrees is one step in a family of first-quadrant tangents. Seeing it beside its neighbours makes clear which angles simplify and which, like $10^\circ$, stay as decimals.

Table: Tangent values for common first-quadrant angles, in degrees and radians.

The pattern to notice: tangent rises as the angle opens, and only the constructible angles ($0^\circ$, $30^\circ$, $45^\circ$) land on clean forms. For the smallest step of all, compare tan 1 degrees, which is about $0.0175$.

Why Is Tan 10 Degrees Positive?

Tan 10 degrees is positive because $10^\circ$ lands in the first quadrant, where both coordinates of the point on the unit circle are positive. Tangent is $y \div x$, and a positive divided by a positive is positive.

• Quadrant position. At $10^\circ$ the point $(0.9848,\ 0.1736)$ sits in the upper-right region, so $x > 0$ and $y > 0$.

• Sign of the ratio. Since $\tan\theta = \frac{\sin\theta}{\cos\theta}$, and both $\sin 10^\circ$ and $\cos 10^\circ$ are positive, the quotient is positive.

• The ASTC check. In the first quadrant All ratios are positive, so sine, cosine, and tangent all carry a plus sign there.

The same reasoning flips the sign elsewhere. At $190^\circ$, for instance, the reference angle is again $10^\circ$, but the point sits in the third quadrant where $y$ and $x$ are both negative, and their ratio comes back positive too. Sign is decided by the quadrant, not by the reference angle alone. That distinction is exactly what the sin cos tan overview drills before students meet larger angles.

Who Discovered How To Compute Tan 10 Degrees?

Values like tan 10 degrees were not found with calculators. They were built by hand, over centuries, by people who needed the sky mapped and the seas crossed.

Two more names shaped how these values are found and named:

• Aryabhata (476–550 CE, India) tabulated sine values at intervals and gave the trigonometric table its Indian foundation.

• Madhava of Sangamagrama (c. 1340–1425 CE, India) discovered the power series for sine, cosine, and arctangent roughly two centuries before similar work in Europe.

Where Is Tan 10 Degrees Used In The Real World?

A shallow tangent like $0.1763$ shows up wherever a small angle turns into a slope, a height, or a distance.

• Ramps and accessibility: a wheelchair ramp near $10^\circ$ rises about $0.176$ metres for every metre of run, which is why building codes cap ramp angles so tightly.

• Road and rail gradients: engineers quote a $10^\circ$ climb as roughly a $17.6$ grade, since grade is the tangent written as a percentage.

• Surveying and heights: pointing a sight line up $10^\circ$ at a tower lets a surveyor find its height from the horizontal distance, multiplying the distance by $0.1763$.

• Camera and screen tilt: a display tilted $10^\circ$ shifts its top edge back by the depth times $\tan 10^\circ$, a calculation used in ergonomic design.

• Astronomy and navigation: small elevation angles above the horizon, like $10^\circ$, are read straight off tangent-based tables descended from the ones Hipparchus started.

One shallow ratio quietly sizes ramps, rates roads, and measures towers. The same $0.1763$ appears across fields that never talk to each other.

What Are The Most Common Mistakes With Tan 10 Degrees?

These three errors account for most wrong answers on $\tan 10^\circ$, verified against the "without a calculator" threads and the exact-value queries that surface for this angle.

Leaving the calculator in radian mode.

Where it slips in:

A student types "tan 10" expecting $0.1763$ but the calculator is set to radians, so it reads $10$ as $10$ radians and returns $0.6484$.

Don't do this:

Do not trust the number before checking the angle-mode indicator. A tangent of $10$ radians is a completely different angle from $10^\circ$.

The correct way:

Set the calculator to degree mode for $\tan 10^\circ$, or convert first: $10^\circ = \frac{\pi}{18} \approx 0.1745$ rad, then take the tangent of $0.1745$. Both routes give $0.1763$. If you are unsure what a radian is, the what is a radian page settles it.

Confusing tan 10° with cot 10°.

Where it slips in:

A student writes $\tan 10^\circ \approx 5.67$, which is actually $\cot 10^\circ$, the reciprocal.

Don't do this:

Do not swap tangent and cotangent. $\cot 10^\circ = \frac{1}{\tan 10^\circ} = \frac{1}{0.1763} \approx 5.67$, a value more than thirty times larger.

The correct way:

Keep the definitions straight: $\tan 10^\circ \approx 0.1763$ is small, and its reciprocal $\cot 10^\circ$ is large. The reciprocal pairings are laid out in reciprocal identities.

Inventing a fake exact surd.

Where it slips in:

A student copies a "closed form" from an unreliable source, often the expression that actually equals $\tan 36^\circ$, and presents it as $\tan 10^\circ$.

Don't do this:

Do not force a square-root form onto a non-constructible angle. There is no simple surd for $\tan 10^\circ$.

The correct way:

State the honest decimal $0.1763$, or express it exactly through the cofunction relation $\tan 10^\circ = \cot 80^\circ$. Honesty about what does not simplify is part of getting the answer right.

Practice Problems On Tan 10 Degrees

Work each one, then check against the answer beside it.

1. Write $10^\circ$ in radians.
   (Answer: $\frac{\pi}{18} \approx 0.1745$ rad.)

2. Use $\tan 10^\circ = \frac{\sin 10^\circ}{\cos 10^\circ}$ with $\sin 10^\circ = 0.1736$ and $\cos 10^\circ = 0.9848$ to find $\tan 10^\circ$.
   (Answer: $0.1763$.)

3. Find $\cot 10^\circ$ from $\tan 10^\circ$.
   (Answer: $\frac{1}{0.1763} \approx 5.6713$.)

4. A ramp climbs at $10^\circ$ over a horizontal run of $4$ m. How high does it rise?
   (Answer: $4 \times \tan 10^\circ \approx 0.705$ m.)

5. Which is larger, $\tan 10^\circ$ or $\tan 20^\circ$?
   (Answer: $\tan 20^\circ \approx 0.3640$ is larger, since tangent increases across the first quadrant.)

6. True or false: $\tan 10^\circ = \cot 80^\circ$.
   (Answer: True, by the cofunction relation.)

Where Should You Go Next After Tan 10 Degrees?

Tan 10 degrees opens onto the wider machinery of trigonometry, and a few natural doors lead on from here.

1. Tangent function. See how the opposite-over-adjacent ratio behaves across every angle, and why it shoots off to infinity near $90^\circ$.

2. Cofunction identities. Understand the $\tan 10^\circ = \cot 80^\circ$ relation and the full family of complementary-angle rules.

3. Trigonometric table. Keep the standard angle values in one place for quick reference and revision.

This topic appears in Class 10 trigonometry (NCERT, India) and in the US Common Core high-school functions standards, so the same value travels across curricula. If your child is building this foundation, a live Bhanzu trainer teaches angle values starting from the unit circle in the Bhanzu trigonometry program.