# 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.

| Angle (degrees) | Angle (radians) | tanθ | Special angle? |
| --- | --- | --- | --- |
| 0° | 0 | 0.0000 | Yes (exact 0) |
| 10° | π/18 | 0.1763 | No |
| 11° | 11π/180 | 0.1944 | No |
| **12°** | **π/15** | **0.2126** | No — decimal only |
| 15° | π/12 | 0.2679 | No (but exact 2−3) |
| 30° | π/6 | 0.5774 | Yes (1/√3) |

## 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](/content/math/geometry/unit-circle/index.html), 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](/content/math/trigonometry/cofunction-identities/index.html): \(\tanθ=\cot(90°−θ)\).

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

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.