# Tan 5pi/6 — Exact Value, Unit Circle, Methods

## TL;DR
Tan 5pi/6 is −1/√3, which rationalises to −√3/3 (about −0.5774), because 5π/6 lands at 150° in the second quadrant where tangent is negative. This article finds the value through the degree conversion, the π/6 reference angle, and the sine-over-cosine quotient.

**Quick Answer:**  
**Result:** tan(5π/6) = −1/√3 = −√3/3 ≈ −0.5774  
**Notation:** rationalised exact form −√3/3 (equivalently −1/√3)  
**Method shown:** degree conversion + reference angle + sin/cos quotient  
**Degree equivalent:** tan 150°  
**Sign:** negative (second quadrant)

## Quick Reference Table
Neighbouring angles in both notations, with their tangent values.

| Angle (radians) | Angle (degrees) | Quadrant | tan⁡(angle) value |  
| --- | --- | --- | --- |  
| π/6 | 30° | I | √3/3 |  
| π/3 | 60° | I | √3 |  
| 2π/3 | 120° | II | −√3 |  
| 5π/6 | 150° | II | −√3/3 |  
| π | 180° | — | 0 |  
| 7π/6 | 210° | III | √3/3 |

## What Tangent of an Angle Means
Tangent is the ratio of sine to cosine: tan⁡θ=sin⁡θ/cos⁡θ. On the unit circle it equals the y-coordinate divided by the x-coordinate of the terminal point, which is why tangent reads as the slope of the radius.

A **quadrant** is one of the four regions the axes divide the plane into. Tangent is positive where sine and cosine share a sign (quadrants I and III) and negative where they differ (II and IV). The angle 5π/6 is in the second quadrant, so its tangent is negative. These sign patterns come straight from the [reciprocal identities](/content/math/trigonometry/reciprocal-identities/index.html) and basic ratio definitions.

## Methods to Find Tan 5pi/6
How do you find tan 5pi/6 without a calculator? Each method below lands on the same value.

### **Method 1: Convert radians to degrees**
5π/6 × 180°/π = 5 × 180°/6 = 150°  
So tan⁡(5π/6) = tan(150°). If the conversion factor feels shaky, the [radian-to-degree relationship](/content/math/trigonometry/1-radian-to-degrees/index.html) lays it out.
**Final answer:** 150°.

### **Method 2: Reference angle**
For a second-quadrant angle, the reference angle is π minus the angle.
π−5π/6=π/6.
The reference angle is π/6 (30°), and tan⁡(π/6)=1/√3.
Quadrant II makes tangent negative, so:
tan⁡(5π/6)=−tan⁡(π/6)=−1/√3=−√3/3.
**Final answer:** −√3/3.

### **Method 3: Sine over cosine**
At 150°, the unit-circle point is (−√3/2, 1/2). Tangent is y over x:
tan⁡(5π/6)=sin⁡(5π/6)/cos⁡(5π/6)=1/(-√3/2)=−1/√3=−√3/3.
**Final answer:** −√3/3.

Tan 5pi/6 is the radian twin of tan 150°; both describe the same 150° direction, so the two pages differ only in how the angle is written, not in the value.

## Common Mistakes With Tan 5pi/6
### **Mistake 1: Leaving the answer as −1/√3 when rationalised form is expected**  
**Where it slips in:** At the final line, when −1/√3 looks finished.  
**Don't do this:** Hand in −1/√3 on a paper that asks for a rationalised denominator.  
**The correct way:** Multiply top and bottom by √3 to get −√3/3.

### **Mistake 2: Dropping the negative sign**  
**Where it slips in:** After computing the reference value tan⁡(π/6)=1/√3, which is positive.  
**Don't do this:** Report tan⁡(5π/6)=√3/3.  
**The correct way:** The reference angle gives the magnitude; the second quadrant supplies a negative sign.

### **Mistake 3: Using 30° as the reference but adding instead of subtracting**  
**Where it slips in:** Confusing the second-quadrant rule with the third-quadrant one.  
**Don't do this:** Compute 5π/6−π to get a reference angle.  
**The correct way:** In quadrant II, reference angle = π − angle.

## Frequently Asked Questions
Is tan 5pi/6 positive or negative?  
Negative. In the second quadrant, cosine is negative and sine is positive, so their quotient — the tangent — is negative.

What is tan 5pi/6 as a decimal?  
About −0.5774. The exact value −√3/3 avoids rounding error.

What is the reference angle for 5pi/6?  
π/6, or 30°. The terminal side makes a 30° angle with the negative x-axis.

Are −1/√3 and −√3/3 the same value?  
Yes. Rationalising −1/√3 gives −√3/3 — identical numbers, just written differently.

How does tan 5pi/6 compare to tan 2pi/3?  
Both are negative second-quadrant tangents.