Cot 180 Degrees: Why It Is Undefined, Graph, Examples

Where Does Cot 180 Degrees Show Up, And Why Does It Matter?

On the graph of $y = \cot x$, the angle $180^\circ$ is a vertical asymptote, a line the curve approaches but never touches. Solving any equation with $\cot x$ in it means excluding $x = 180^\circ$ (and every multiple of it) from the domain first.

It also shows up in code. Because a computer stores $\pi$ only approximately, 1/tan(pi) in Python returns about $-8.17 \times 10^{15}$ instead of an error, a huge number that is really a disguised "undefined".

Standard-Angle Cotangent Reference Table

The table covers the angles students meet most, from $0^\circ$ to $360^\circ$. Every value was recomputed with exact arithmetic.

Cotangent is undefined exactly where sine is zero ($0^\circ$, $180^\circ$, $360^\circ$) and equals zero exactly where cosine is zero ($90^\circ$, $270^\circ$). Those two rows are the most swapped facts in the table.

What Kind Of Angle Is 180 Degrees On The Unit Circle?

The unit circle is a circle of radius $1$ centred at the origin; an angle's point on it is $(\cos\theta, \sin\theta)$, and cotangent is the ratio $\frac{x}{y}$ of those coordinates. At $180^\circ$ the point is $(-1, 0)$, the far left of the circle.

A quadrant is one of the four regions the axes cut the plane into. An angle like $180^\circ$ whose terminal side lies on an axis belongs to no quadrant and is called a quadrantal angle; the others are $0^\circ$, $90^\circ$, $270^\circ$ and $360^\circ$.

In radians, $180^\circ = \pi$. The radian twin of this page, cot pi, starts from the radian measure and the arc length of $\pi$, but the conclusion is identical: undefined.

Why Is Cot 180 Degrees Undefined?

Because every route to it ends in a division by zero. Here are the four routes, each built from facts you can check.

Method 1: Cosine Over Sine

Cotangent is defined as $\cot\theta = \frac{\cos\theta}{\sin\theta}$. Use the known values sin 180 degrees $= 0$ and cos 180 degrees $= -1$.

$$\cot 180^\circ = \frac{\cos 180^\circ}{\sin 180^\circ}$$

$$= \frac{-1}{0}$$

No number times $0$ gives $-1$, so the quotient does not exist.

Method 2: The Reciprocal Of Tan 180 Degrees

The reciprocal identities give $\cot\theta = \frac{1}{\tan\theta}$ wherever $\tan\theta \neq 0$. Here $\tan 180^\circ = \frac{0}{-1} = 0$, so $\frac{1}{\tan 180^\circ} = \frac{1}{0}$, again undefined.

Method 3: The Right-Triangle View Breaks

In a right triangle, $\cot\theta = \frac{\text{adjacent}}{\text{opposite}}$. At $180^\circ$ the "triangle" collapses into a flat line with opposite side $0$, so there is no triangle and no ratio.

This is why the unit circle, not the triangle, is the home definition for angles beyond $90^\circ$.

Method 4: Periodicity

Cotangent repeats every $180^\circ$, so it is a periodic function with period $\pi$. That makes $\cot 180^\circ = \cot(0^\circ + 180^\circ) = \cot 0^\circ$, and $\cot 0^\circ = \frac{1}{0}$ is undefined, so the two angles fail together.

Standard trigonometry textbooks generalise this: $\sin(n \times 180^\circ) = 0$ for every integer $n$, so $\cot(n \times 180^\circ)$ is undefined for every integer $n$.

What Happens To $\cot\theta$ Just Before And After 180 Degrees?

The values blow up in opposite directions. Just below $180^\circ$ (Quadrant II) cosine is negative and sine is a tiny positive number, so cotangent is a huge negative number; just above (Quadrant III) both are negative, so cotangent is a huge positive number.

In limit notation, $\lim_{\theta \to 180^{\circ-}} \cot\theta = -\infty$ and $\lim_{\theta \to 180^{\circ+}} \cot\theta = +\infty$. Since the two sides disagree, not even "infinity" is a fair answer; undefined is the only honest one.

How Do $\cot(180^\circ - \theta)$ And $\cot(180^\circ + \theta)$ Work?

These two identities come up whenever $180^\circ$ appears inside a cotangent. Both follow from Method 1 with $\sin(180^\circ - \theta) = \sin\theta$, $\cos(180^\circ - \theta) = -\cos\theta$, $\sin(180^\circ + \theta) = -\sin\theta$ and $\cos(180^\circ + \theta) = -\cos\theta$.

$$\cot(180^\circ - \theta) = \frac{-\cos\theta}{\sin\theta} = -\cot\theta$$

$$\cot(180^\circ + \theta) = \frac{-\cos\theta}{-\sin\theta} = \cot\theta$$

The second line is periodicity in action. Both identities hold for every $\theta$ that is not a multiple of $180^\circ$; at $\theta = 0^\circ$ each side is undefined, which is just the headline fact again.

The reference angle, the acute angle between the terminal side and the $x$-axis, explains the signs: $180^\circ - \theta$ lands in Quadrant II where cotangent is negative, and $180^\circ + \theta$ lands in Quadrant III where it is positive.

Examples Of Cot 180 Degrees

Example 1

Find $\cot 150^\circ$ using $\cot(180^\circ - \theta)$.

$$\cot 150^\circ = \cot(180^\circ - 30^\circ) = -\cot 30^\circ$$

$$= -\sqrt{3} \approx -1.7321$$

Example 2

Given $\tan 180^\circ = 0$, find $\cot 180^\circ$.

First attempt. Cotangent is the reciprocal of tangent, and "the reciprocal of zero is zero", so $\cot 180^\circ = 0$.

Check it against the definition of reciprocals: $\tan\theta \times \cot\theta$ should equal $1$. With both equal to $0$, the product is $0 \times 0 = 0$, not $1$, so the answer cannot be right.

Correct route. Zero has no reciprocal. $\cot 180^\circ = \frac{1}{0}$, which is undefined.

Example 3

List every angle in $[0^\circ, 360^\circ]$ where $\cot\theta$ is undefined.

Cotangent is undefined where $\sin\theta = 0$.

$$\sin\theta = 0 \text{ at } \theta = 0^\circ, 180^\circ, 360^\circ$$

So $\cot\theta$ is undefined at $0^\circ$, $180^\circ$ and $360^\circ$.

Example 4

Solve $\cot\theta = 0$ for $0^\circ \leq \theta < 360^\circ$. Is $180^\circ$ a solution?

$\cot\theta = 0$ needs $\cos\theta = 0$ with $\sin\theta \neq 0$.

$$\theta = 90^\circ \text{ or } \theta = 270^\circ$$

$180^\circ$ is not a solution: there the numerator is $-1$ and the denominator is $0$, the opposite situation.

Example 5

Evaluate $\cot 179^\circ$ and $\cot 181^\circ$ with a calculator and explain the signs.

$$\cot 179^\circ = \frac{1}{\tan 179^\circ} \approx -57.29$$

$$\cot 181^\circ = \frac{1}{\tan 181^\circ} \approx 57.29$$

$179^\circ$ is in Quadrant II, where cosine is negative and sine positive; $181^\circ$ is in Quadrant III, where both are negative. One degree either side of the asymptote already gives a value above $57$ in size.

Example 6

State the domain of $y = \cot x$ in radians and use it to decide whether $\cot 540^\circ$ exists.

$$\text{Domain: } x \neq n\pi, \quad n \text{ an integer}$$

$540^\circ = 3\pi$, a multiple of $\pi$, so $\cot 540^\circ$ is undefined. The domain and range of trigonometric functions list this exclusion for cotangent; the range is all real numbers.

Where Do Students Go Wrong With Cot 180 Degrees?

Mistake 1: Writing Cot 180° = 0

Where it slips in: Right after computing $\tan 180^\circ = 0$, or when the $0$ in $\frac{-1}{0}$ gets read as the answer.

Don't do this: Write $\cot 180^\circ = 0$. That value belongs to $\cot 90^\circ$.

The correct way: Ask which coordinate is zero: zero on top gives $0$, zero on the bottom gives undefined. The rusher reaches for "zero in, zero out" without looking at where the zero sits, and that one look fixes it.

Mistake 2: Trusting A Radian-Mode Calculator

Where it slips in: Typing "cot 180" into a device set to radians, which is why autosuggest shows $0.7470$ for the query.

Don't do this: Accept $\cot 180 \approx 0.7470$. That is the cotangent of $180$ radians, about $10313^\circ$.

The correct way: Switch to degrees, where most calculators then show an error for $\frac{1}{\tan 180}$. A defined, ordinary-looking answer at a quadrantal angle is the warning sign.

Mistake 3: Mixing Up Cot 180° With Tan 90° And Cot 90°

Where it slips in: Memorised tables where "undefined" and "0" rows sit side by side.

Don't do this: Recall "cot is undefined at $90^\circ$" because tangent is.

The correct way: Tangent and cotangent swap their zeros and their gaps: $\tan 90^\circ$ is undefined and $\cot 90^\circ = 0$, while $\tan 180^\circ = 0$ and $\cot 180^\circ$ is undefined. The memorizer who learned the table as a picture gets this backwards; rebuilding each value from $\frac{x}{y}$ on the unit circle takes ten seconds and never fails.

A real version of the "undefined becomes a number" trap lives in software: floating-point $\pi$ is slightly off, so code returns about $-8.17 \times 10^{15}$ for $\cot \pi$ rather than an error, as the Python floating-point notes explain for decimal values in general.

Where Can You Get Help With Trigonometry Values Like Cot 180 Degrees?

Undefined values feel slippery until someone makes you draw the asymptote and test numbers on each side. A live trigonometry tutor can build that habit with you.

A high school math tutor ties quadrantal angles to graphs and equations, and math tutoring offers steady practice with feedback.

Key Takeaways

• Cot 180 degrees is undefined: $\frac{\cos 180^\circ}{\sin 180^\circ} = \frac{-1}{0}$, with no exact value and no decimal.

• The same holds for $\cot \pi$ and for every multiple of $180^\circ$, since sine is zero there.

• Just below $180^\circ$ cotangent falls to $-\infty$; just above it comes down from $+\infty$, giving a vertical asymptote.

• Do not confuse it with $\cot 90^\circ = 0$; check whether the zero is on top or on the bottom.

Practice These Before Moving On

1. Find $\cot 135^\circ$ using $\cot(180^\circ - \theta)$.

2. Find $\cot 210^\circ$ using $\cot(180^\circ + \theta)$.

3. Is $\cot 900^\circ$ defined?

4. Explain why $\cot\theta$ is positive just after $180^\circ$.

Answers:

1. $\cot 135^\circ = -\cot 45^\circ = -1$.

2. $\cot 210^\circ = \cot 30^\circ = \sqrt{3} \approx 1.7321$.

3. No. $900^\circ = 5 \times 180^\circ$, so $\sin 900^\circ = 0$ and the cotangent is undefined.

4. Just after $180^\circ$ the angle is in Quadrant III, where cosine and sine are both negative, so their ratio is positive.

If Question 3 stalls, return to Method 4 on periodicity. Practice these to solidify your understanding.

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Read More

• Cot 0 degrees

• Cot pi over 2

• Tan 90 degrees

• Cosecant, secant and cotangent functions

• Trigonometric table