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.

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 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 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.