Cot 5pi/6 Exact Value: Steps & Unit Circle

What Is The Value Of Cot 5pi/6?

Cot 5pi/6 equals $-\sqrt{3}$, or about $-1.7321$. In symbols, $\cot\frac{5\pi}{6} = -\sqrt{3}$, and in degrees the same statement reads $\cot 150^\circ = -\sqrt{3}$.

The cotangent of an angle is cosine divided by sine, so it is the reciprocal of the tangent. At $\frac{5\pi}{6}$ the cosine is $-\frac{\sqrt{3}}{2}$ and the sine is $\frac{1}{2}$, and dividing the two gives the exact surd $-\sqrt{3}$.

Both forms of the angle matter. As a radian measure it is $\frac{5\pi}{6}$; as a degree measure it is $150^\circ$. Every method below carries both so the value never depends on which unit your calculator happens to be set to.

How Do You Find Cot 5pi/6?

Finding cot 5pi/6 takes three quick decisions: locate the quadrant, find the reference angle, then fix the sign.

• Quadrant. $\frac{5\pi}{6}$ is $150^\circ$, which lands between $90^\circ$ and $180^\circ$. That is Quadrant II.

• Reference angle. The reference angle is the acute gap to the horizontal axis: $\pi - \frac{5\pi}{6} = \frac{\pi}{6}$, or $180^\circ - 150^\circ = 30^\circ$.

• Sign. The CAST rule records which functions are positive in each quadrant. In Quadrant II only sine (and its reciprocal cosecant) is positive, so cotangent is negative here.

Now read the magnitude off the special angle. The cotangent of the reference angle $\frac{\pi}{6}$ is $\sqrt{3}$, so cot 5pi/6 keeps that size and takes the negative sign:

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

You can anchor the same magnitude in a right triangle. A 30-60-90 triangle has legs in the ratio $1 : \sqrt{3} : 2$, and the cotangent of the $30^\circ$ angle is the adjacent leg over the opposite leg, $\frac{\sqrt{3}}{1} = \sqrt{3}$. The quadrant then supplies the minus sign the triangle alone cannot show.

Where Does 5pi/6 Sit On The Unit Circle?

On the unit circle, $\frac{5\pi}{6}$ is the point $\left(-\frac{\sqrt{3}}{2}, \frac{1}{2}\right)$, which is roughly $(-0.866,\ 0.5)$. The first coordinate is the cosine and the second is the sine.

Cotangent on the unit circle is the $x$-coordinate divided by the $y$-coordinate:

$$\cot\frac{5\pi}{6} = \frac{x}{y} = \frac{-\frac{\sqrt{3}}{2}}{\frac{1}{2}} = -\sqrt{3}$$

The point sits in the upper-left of the circle. Its $x$ is negative and its $y$ is positive, and a negative divided by a positive is negative, which is the whole reason the answer carries a minus sign.

How Can You Check That Cot 5pi/6 Is −√3?

Three independent routes reach the same value, which is the surest way to trust it. Each one is worth keeping, because different problems hand you different starting points.

Method 1: cosine over sine.

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

Method 2: reciprocal of tangent.

Since $\tan\frac{5\pi}{6} = -\frac{1}{\sqrt{3}}$, flip it:

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

Method 3: the cofunction shift.

Cotangent of an angle equals tangent of its complement, $\cot\theta = \tan\left(\frac{\pi}{2} - \theta\right)$:

$$\cot\frac{5\pi}{6} = \tan\left(\frac{\pi}{2} - \frac{5\pi}{6}\right) = \tan\left(-\frac{\pi}{3}\right) = -\sqrt{3}$$

All three agree on $-\sqrt{3}$. If a fourth attempt ever gives $-\frac{1}{\sqrt{3}}$, you flipped the ratio the wrong way, a slip the mistakes section returns to below.

What Are The Related Values Around 5pi/6?

Reading the six functions of $\frac{5\pi}{6}$ together shows how cotangent fits the family, and lists the reciprocal pairs side by side.

Table 1: The six trigonometric functions at 5π/6 (150°), exact and to four decimal places.

Every angle that shares the reference angle $\frac{\pi}{6}$ has a cotangent of the same size, $\sqrt{3}$, and only the sign changes with the quadrant.

Table 2: The cotangent across all four angles with reference angle π/6.

For the sibling values at this same angle, see sin 5pi/6, cos 5pi/6, and tan 5pi/6. The reciprocal relationships behind the table are collected in the reciprocal identities.

Why Is Cot 5pi/6 Negative?

Cot 5pi/6 is negative because $\frac{5\pi}{6}$ lands in Quadrant II, where the two coordinates that build cotangent carry opposite signs. Here is the reasoning in three short steps.

• Cosine turns negative. Past $90^\circ$, the point on the unit circle has moved to the left of the vertical axis, so its $x$-coordinate, the cosine, is negative.

• Sine stays positive. The point is still above the horizontal axis, so its $y$-coordinate, the sine, is positive.

• The ratio inherits the sign. Cotangent is $\frac{\cos}{\sin}$, and a negative over a positive is negative.

The magnitude is a separate question from the sign. The size $\sqrt{3}$ comes from the reference angle $\frac{\pi}{6}$, which behaves exactly like the familiar $30^\circ$ corner of a set square. The quadrant only decides whether that size is written with a plus or a minus, and Quadrant II writes it with a minus.

Who Discovered How To Compute Cot 5pi/6?

Cotangent did not begin as a ratio on a circle. It began as a shadow. Ancient astronomers measured the length of the shadow a vertical stick threw on the ground, and that shadow-over-height ratio is exactly what we now call cotangent.

Two threads meet in that story. Centuries earlier, around 150 BCE, the Greek astronomer Hipparchus had built the first known table of chords, the ancestor of every sine and cosine table. The shadow tradition and the chord tradition eventually merged into the trigonometry that lets us pin down cot 5pi/6 as an exact surd.

Where Is Cot 5pi/6 Used In The Real World?

The cotangent value behind $150^\circ$ is not only a homework answer. The same ratio shows up wherever an angle turns into a "sideways-per-upward" measurement.

• 3D graphics and games. Every perspective camera builds a projection matrix whose scaling terms are $\cot\left(\frac{\text{fov}}{2}\right)$, where fov is the field of view. Change the angle, and cotangent decides how much of the world fits on screen.

• Surveying and roads. A gradient is rise over run, so its inverse, run over rise, is a cotangent. Engineers use it to lay out gentle ramps and road grades from a target angle.

• Sundials and architecture. Shadow length for a given sun angle is a cotangent, the direct descendant of the gnomon tables, and it still guides how far a roof overhang must reach to shade a window.

• Alternating-current circuits. Phase relationships between voltage and current are described with tangent and cotangent of the phase angle, which sets how a circuit stores and returns energy.

One ratio, read off a single $150^\circ$ angle, threads through screens, roads, buildings, and power lines. The math a student meets on the unit circle is the same math running the tools around them.

What Are The Most Common Mistakes With Cot 5pi/6?

These four errors account for most wrong answers on cot 5pi/6, and each has a clean fix.

Making the answer positive.

Where it slips in:

A student finds the reference-angle value $\sqrt{3}$ and stops, forgetting that $\frac{5\pi}{6}$ is in Quadrant II.

Don't do this:

Do not report $\sqrt{3}$. In Quadrant II only sine is positive, so cotangent must be negative.

The correct way:

Take the reference value $\cot\frac{\pi}{6} = \sqrt{3}$, then apply the CAST sign for Quadrant II to get $-\sqrt{3}$.

Confusing cotangent with tangent.

Where it slips in:

A student writes $\cot\frac{5\pi}{6} = -\frac{1}{\sqrt{3}}$, which is actually the tangent, because they used the reciprocal in the wrong direction.

Don't do this:

Do not report the tangent's value for the cotangent. $\cot\theta = \frac{\cos\theta}{\sin\theta}$, not $\frac{\sin\theta}{\cos\theta}$.

The correct way:

Divide cosine by sine, or take the reciprocal of $\tan\frac{5\pi}{6} = -\frac{1}{\sqrt{3}}$ to land on $-\sqrt{3}$.

Working in the wrong angle mode.

Where it slips in:

A student types $\frac{5\pi}{6}$ into a calculator set to degrees, or types $150$ with the calculator set to radians, and reads off a meaningless number.

Don't do this:

Do not mix the unit of the angle with the mode of the calculator. Many calculators also have no dedicated cot button.

The correct way:

Match the mode to the angle, and compute cotangent as $\frac{1}{\tan}$ or $\frac{\cos}{\sin}$ when there is no cot key.

Misreading the reference angle.

Where it slips in:

A student uses $\frac{5\pi}{6}$ itself, or subtracts from the wrong axis, instead of taking $\pi - \frac{5\pi}{6}$.

Don't do this:

Do not treat the given angle as its own reference angle in Quadrant II.

The correct way:

In Quadrant II the reference angle is $\pi - \theta$, so $\pi - \frac{5\pi}{6} = \frac{\pi}{6}$.

Practice Problems On Cot 5pi/6

Work each one, then check against the answer beside it.

1. Evaluate $\cot\frac{5\pi}{6}$.
   (Answer: $-\sqrt{3} \approx -1.7321$.)

2. Given $\tan\frac{5\pi}{6} = -\frac{1}{\sqrt{3}}$, use the reciprocal to find $\cot\frac{5\pi}{6}$.
   (Answer: $-\sqrt{3}$.)

3. State $\csc\frac{5\pi}{6}$ and $\sec\frac{5\pi}{6}$.
   (Answer: $\csc\frac{5\pi}{6} = 2$ and $\sec\frac{5\pi}{6} = -\frac{2}{\sqrt{3}} \approx -1.1547$.)

4. Use the cofunction rule to confirm cot 5pi/6, starting from $\cot\theta = \tan\left(\frac{\pi}{2} - \theta\right)$.
   (Answer: $\tan\left(-\frac{\pi}{3}\right) = -\sqrt{3}$.)

5. Compare $\cot\frac{\pi}{6}$ with $\cot\frac{5\pi}{6}$.
   (Answer: $\sqrt{3}$ and $-\sqrt{3}$; same size, opposite sign, because Quadrant II flips it.)

6. Convert $\frac{5\pi}{6}$ to degrees and give its reference angle.
   (Answer: $150^\circ$, reference angle $30^\circ$.)

Where Should You Go Next After Cot 5pi/6?

A single value opens onto the wider machinery of trigonometry, and a few natural doors lead out from here.

1. The unit circle with tangent. See how every angle's coordinates generate sine, cosine, tangent, and cotangent at once.

2. Cosecant, secant, and cotangent functions. Go deeper on the three reciprocal functions and where each one is defined.

3. Trigonometric table. Keep the standard-angle values, in degrees and radians, in one reference.

4. What is a radian. Firm up why $\frac{5\pi}{6}$ and $150^\circ$ are the same angle.

If your child is building this fluency, a live Bhanzu trainer teaches the unit circle from the reasoning up in the Bhanzu trigonometry program.