The value of tan 12 degrees is approximately $0.2126$ ($0.21255656$ to eight places). Unlike $\tan 30°$ or $\tan 45°$, the angle $12°$ is not a special angle, so $\tan 12°$ has no simple exact surd — it is read from a calculator, a trig table, or rewritten as the cofunction $\cot 78°$.
Quick Answer:
Result: $\tan 12° \approx 0.2126$
In radians: $\tan\left(\frac{\pi}{15}\right) = \tan(0.20944) \approx 0.2126$
Notation: decimal approximation — $0.21255656$ (8 dp)
Method shown: calculator (degree mode), $\tan 12° = \frac{\sin 12°}{\cos 12°}$, the cofunction $\cot 78°$, and table interpolation
Exact form: none simple — $12°$ is not a special angle, so no clean radical exists
Quick Reference — Tangent Near 12 Degrees
Tan 12° sits below the first special landmark $\tan 30°$. The table places it among its small-angle neighbours.
Angle (degrees) | Angle (radians) | $\tan\theta$ | Special angle? |
|---|---|---|---|
$0°$ | $0$ | $0.0000$ | Yes (exact $0$) |
$10°$ | $\frac{\pi}{18}$ | $0.1763$ | No |
$11°$ | $\frac{11\pi}{180}$ | $0.1944$ | No |
$12°$ | $\frac{\pi}{15}$ | $0.2126$ | No — decimal only |
$15°$ | $\frac{\pi}{12}$ | $0.2679$ | No (but exact $2-\sqrt3$) |
$30°$ | $\frac{\pi}{6}$ | $0.5774$ | Yes ($\tfrac{1}{\sqrt3}$) |
The nearest exact landmark above is $\tan 30° = \frac{1}{\sqrt3} \approx 0.5774$, and $\tan 12°$ sits well below it — tangent grows slowly near $0$.
What Does Tan 12 Degrees Mean?
Tangent of an angle is the ratio of sine to cosine: $\tan\theta = \dfrac{\sin\theta}{\cos\theta}$. On the unit circle, that is the $y$-coordinate divided by the $x$-coordinate of the point at angle $\theta$.
A quadrant is one of the four regions the axes cut the plane into, numbered anticlockwise from the top right; $12°$ lands in Quadrant I, where sine and cosine are both positive, so tangent is positive.
Because $12°$ is a shallow angle, the numerator $\sin 12°$ is small while the denominator $\cos 12°$ stays close to $1$, so the ratio is small — about $0.2126$.
How Do You Find the Value of Tan 12 Degrees?
Because $12°$ is not a special angle, there is no surd to simplify to. So how do you find tan 12 degrees without a calculator? You build it from sine and cosine, swap to a cofunction, or interpolate — here are the honest routes.
Method 1: Calculator (set to degree mode)
Type $\tan(12)$ with the calculator in DEG mode.
$$\tan 12° = 0.21255656\ldots \approx 0.2126$$
In radian mode the same keystrokes give $\tan(12\ \text{rad}) \approx -0.636$ — a completely different number, so the mode matters.
Method 2: Sine over cosine
Using $\sin 12° = 0.2079$ and $\cos 12° = 0.9781$:
$$\tan 12° = \frac{\sin 12°}{\cos 12°} = \frac{0.2079}{0.9781} = 0.2126$$
Method 3: Cofunction identity
Tangent and cotangent are cofunctions: $\tan\theta = \cot(90° - \theta)$.
$$\tan 12° = \cot(90° - 12°) = \cot 78° = \frac{1}{\tan 78°}$$
Since $\tan 78° \approx 4.7046$, this gives $\dfrac{1}{4.7046} \approx 0.2126$ — the same value, a third way.
Method 4: Table interpolation
If a trig table lists $\tan 10° = 0.1763$ and $\tan 15° = 0.2679$, estimate $\tan 12°$ by linear interpolation:
$$\tan 12° \approx 0.1763 + \frac{12 - 10}{15 - 10},(0.2679 - 0.1763) = 0.1763 + 0.4(0.0916) = 0.2129$$
That lands within $0.0003$ of the true $0.2126$ — interpolation works well here because tangent is nearly straight near small angles.
What is tan 12 degrees in radians?
The angle converts to $\frac{\pi}{15} \approx 0.2094$ rad, but the value of the tangent is the same number, $\approx 0.2126$. Converting the angle does not change the tangent; it only relabels it.
Examples Using Tan 12 Degrees
Example 1
State $\tan 12°$ to four decimal places.
From a calculator in degree mode, $\tan 12° = 0.2126$.
Example 2 (wrong path first)
Find $\tan 12°$ from $\sin 12°$ and $\cos 12°$.
Wrong attempt. A student divides the larger by the smaller out of habit: $\tan 12° = \dfrac{\cos 12°}{\sin 12°} = \dfrac{0.9781}{0.2079} = 4.705$.
Why it breaks. That flips the ratio — $\tfrac{\cos}{\sin}$ is cotangent, not tangent. The answer $4.705$ is actually $\cot 12°$, which would mean a $78°$ slope, not a shallow $12°$ one.
Correct. $\tan 12° = \dfrac{\sin 12°}{\cos 12°} = \dfrac{0.2079}{0.9781} = 0.2126$.
Example 3
A ramp rises at $12°$. How high is it after a $5$ m horizontal run?
Rise $= 5 \times \tan 12° = 5 \times 0.2126 = 1.063$ m.
Example 4
Compare $\tan 12°$ with $\tan 30°$.
$\tan 12° = 0.2126$; $\tan 30° = 0.5774$. The smaller angle gives a much smaller gradient, as expected near the flat end of the curve.
Example 5
Verify $\tan 12° = \cot 78°$ on a calculator.
$\tan 12° = 0.21256$ and $\cot 78° = \tfrac{1}{\tan 78°} = 0.21256$ — identical, confirming the cofunction identity.
Tan 12 Degrees — Where Students Lose the Mark
Most errors on a small non-special tangent come from a few repeatable habits.
Mistake 1: Flipping the ratio to cotangent
Where it slips in: building tangent from sine and cosine without checking which goes on top.
Don't do this: writing $\tan 12° = \dfrac{\cos 12°}{\sin 12°} = 4.705$.
The correct way: tangent is $\tfrac{\sin\theta}{\cos\theta}$, sine on top. The habit that fixes this is to write "opposite over adjacent" before plugging numbers; the learner who flips it has computed $\cot 12°$ and will report a steep slope for a shallow angle.
Mistake 2: Hunting for an exact surd
Where it slips in: assuming every small angle has a clean value like $\tan 15° = 2 - \sqrt3$.
Don't do this: trying to write $\tan 12°$ as a simple radical.
The correct way: $12°$ is not a special angle, so $\tan 12°$ is given as the decimal $0.2126$. Some nearby angles like $15°$ do have surd forms, but $12°$ does not — the honest answer is the calculator value or a cofunction.
Mistake 3: Forgetting the calculator's angle mode
Where it slips in: the calculator was left in radian mode.
Don't do this: reading $\tan(12) = -0.636$ and reporting it as $\tan 12°$.
The correct way: check DEG mode for $\tan 12°$; $-0.636$ is $\tan(12\ \text{radians})$, an angle of nearly two full turns where tangent can be negative.
Key Takeaways
Tan 12 degrees is approximately $0.2126$ — a decimal, not a clean surd.
$12°$ is not a special angle, so the value comes from a calculator, $\tfrac{\sin 12°}{\cos 12°}$, the cofunction $\cot 78°$, or interpolation.
The value is small because a shallow angle has a small sine over a near-$1$ cosine.
In radians the angle is $\frac{\pi}{15}$, but the tangent value stays $\approx 0.2126$.
The biggest slip is flipping the ratio and computing $\cot 12°$ instead.
To take tangent values further with a teacher, explore Bhanzu's trigonometry tutor, a high school math tutor, or math classes online.
Practice These Before Moving On
State $\tan 12°$ to four decimal places.
Compute $\tan 12°$ from $\sin 12° = 0.2079$ and $\cos 12° = 0.9781$.
Use $\tan 10° = 0.1763$ and $\tan 15° = 0.2679$ to interpolate $\tan 12°$.
Want a live trainer to walk through more tangent-value problems? Book a free demo class.
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