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

The value of $\tan\frac{5\pi}{6}$ is $-\frac{1}{\sqrt{3}}$, usually written in rationalised form as $-\frac{\sqrt{3}}{3} \approx -0.5774$.

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: $\tan\theta = \frac{\sin\theta}{\cos\theta}$. 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, numbered I to IV anticlockwise. 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 before any arithmetic. These sign patterns come straight from the reciprocal identities and the 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

$$\frac{5\pi}{6} \times \frac{180°}{\pi} = \frac{5 \times 180°}{6} = 150°$$

So $\tan\frac{5\pi}{6} = \tan 150°$. Going the other way, 150° converts back by multiplying by $\frac{\pi}{180°}$. 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.

$$\pi - \frac{5\pi}{6} = \frac{6\pi - 5\pi}{6} = \frac{\pi}{6}$$

The reference angle is $\frac{\pi}{6}$ (30°), and $\tan\frac{\pi}{6} = \frac{1}{\sqrt{3}}$.

Quadrant II makes tangent negative, so:

$$\tan\frac{5\pi}{6} = -\tan\frac{\pi}{6} = -\frac{1}{\sqrt{3}} = -\frac{\sqrt{3}}{3}$$

Final answer: $-\frac{\sqrt{3}}{3}$.

Method 3: Sine over cosine

At 150° the unit-circle point is $\left(-\frac{\sqrt{3}}{2}, \frac{1}{2}\right)$. Tangent is $y$ over $x$:

$$\tan\frac{5\pi}{6} = \frac{\sin\frac{5\pi}{6}}{\cos\frac{5\pi}{6}} = \frac{\tfrac{1}{2}}{-\tfrac{\sqrt{3}}{2}} = -\frac{1}{\sqrt{3}} = -\frac{\sqrt{3}}{3}$$

Final answer: $-\frac{\sqrt{3}}{3}$.

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 $-\frac{1}{\sqrt{3}}$ looks finished.

Don't do this: Hand in $-\frac{1}{\sqrt{3}}$ on a paper that asks for a rationalised denominator.

The correct way: Multiply top and bottom by $\sqrt{3}$ to get $-\frac{\sqrt{3}}{3}$. Both are correct values; $-\frac{\sqrt{3}}{3}$ is the standard written form.

Mistake 2: Dropping the negative sign

Where it slips in: After computing the reference value $\tan\frac{\pi}{6} = \frac{1}{\sqrt{3}}$, which is positive.

Don't do this: Report $\tan\frac{5\pi}{6} = \frac{\sqrt{3}}{3}$.

The correct way: The reference angle gives the magnitude; the second quadrant supplies a negative sign because sine and cosine have opposite signs there. The reciprocal $\frac{1}{\tan\theta}$ and the inverse $\tan^{-1}\theta$ are different ideas — don't let one stand in for the other when checking signs.

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 $\frac{5\pi}{6} - \pi$ to get a reference angle.

The correct way: In quadrant II, reference angle = π − angle, so $\pi - \frac{5\pi}{6} = \frac{\pi}{6}$.

To build fluency with these second-quadrant values alongside a teacher, Bhanzu's trigonometry tutor and broader math tutoring walk through the reference-angle method on every quadrant.

Read More

• Sin Cos Tan — how the three ratios connect

• Trigonometric ratios and their definitions

• The trigonometric table of standard values

• Cos 2pi/3 — the radian twin of cos 120°

• What a radian is and why we use it