Tan 12 Degrees — Value of tan(12°) and How to Find It

TL;DR

The value of tan 12 degrees is approximately 0.2126 — it is not a special-angle exact value, so there is no clean surd for it. This article shows how to find tan 12° honestly (calculator, sine over cosine, the cofunction cot 78°, and interpolation), gives the radian form, and places it on the unit circle.

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°≈0.2126
In radians: (\tan(\frac{\pi}{15})=\tan(0.20944)\approx0.2126)
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.

What Does Tan 12 Degrees Mean?

Tangent of an angle is the ratio of sine to cosine: (\tanθ=\frac{\sinθ}{\cosθ}). On the unit circle, that is the y-coordinate divided by the x-coordinate of the point at angle θ.

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…≈0.2126)

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: (\tanθ=\cot(90°−θ)).

(\tan 12°=\cot(90°−12°)=\cot 78°=\frac{1}{\tan 78°}).

Since (\tan 78°≈4.7046), this gives (\frac{1}{4.7046}≈0.2126).

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°≈0.1763+\frac{12−10}{15−10}(0.2679−0.1763)=0.2129)

What is tan 12 degrees in radians?

The angle converts to (\frac{\pi}{15}≈0.2094) rad, but the value of the tangent is the same number, ≈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

Find tan 12° from sin 12° and cos 12°. Wrong attempt. A student divides the larger by the smaller: (\tan 12°=\frac{\cos 12°}{\sin 12°}=4.705).

Correct. (\tan 12°=\frac{\sin 12°}{\cos 12°}=0.2126).

Example 3

A ramp rises at 12°. How high is it after a 555 m horizontal run? Rise = 5 × (\tan 12° = 5 × 0.2126 = 1.063 m).

Example 4

Compare tan 12° with tan 30°. (\tan 12°=0.2126; \tan 30°=0.5774).

Example 5

Verify tan 12°=cot 78° on a calculator. (\tan 12°=0.21256) and (\cot 78°=1/\tan 78°=0.21256).

Tan 12 Degrees — Where Students Lose the Mark

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°=\frac{\cos 12°}{\sin 12°}=4.705).

Mistake 2: Hunting for an exact surd

Don't do this: trying to write (\tan 12°) as a simple radical.

Mistake 3: Forgetting the calculator's angle mode

Don't do this: reading tan(12)=−0.636 and reporting it as tan 12°. Check DEG mode for tan 12°; −0.636 is for radians.

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, sine over cosine, or a trigonometric table.
• In radians the angle is (\frac{\pi}{15}), but the tangent value stays ≈0.2126.
• The biggest slip is flipping the ratio and computing cot 12° instead.