What Is The Value Of Cot 3pi/2?
Cot 3pi/2 is $0$. Written as a decimal it is exactly $0.0000$, because the value is a clean whole number, not a rounded one. In radians the angle is $\frac{3\pi}{2}$; in degrees it is $270^\circ$, and both name the same rotation.
Here is the part that trips almost everyone up. At this angle, tangent and cotangent behave in opposite ways.
$$\cot\frac{3\pi}{2} = \frac{\cos\frac{3\pi}{2}}{\sin\frac{3\pi}{2}} = \frac{0}{-1} = 0$$
$$\tan\frac{3\pi}{2} = \frac{\sin\frac{3\pi}{2}}{\cos\frac{3\pi}{2}} = \frac{-1}{0} = \text{undefined}$$
The cotangent is defined as $\frac{\cos}{\sin}$, so the sine sits in the denominator. At $270^\circ$ the sine is $-1$, which is not zero, so the division is perfectly legal and the answer is $0$. Tangent is $\frac{\sin}{\cos}$, so the cosine sits in the denominator, and the cosine at $270^\circ$ is zero, which makes tangent undefined. Same angle, opposite outcome, because a different function is doing the dividing.
Table: The six trigonometric functions evaluated at 3π/2 (270°).
Function at $\frac{3\pi}{2}$ | Value | Why |
|---|---|---|
$\sin\frac{3\pi}{2}$ | $-1$ | $y$-coordinate of $(0, -1)$ |
$\cos\frac{3\pi}{2}$ | $0$ | $x$-coordinate of $(0, -1)$ |
$\tan\frac{3\pi}{2}$ | undefined | $\frac{\sin}{\cos}=\frac{-1}{0}$ |
$\cot\frac{3\pi}{2}$ | $0$ | $\frac{\cos}{\sin}=\frac{0}{-1}$ |
$\csc\frac{3\pi}{2}$ | $-1$ | $\frac{1}{\sin}=\frac{1}{-1}$ |
$\sec\frac{3\pi}{2}$ | undefined | $\frac{1}{\cos}=\frac{1}{0}$ |
How Do You Find Cot 3pi/2?
The safest route is the ratio definition, because it never breaks the way the reciprocal shortcut can.
Step 1: Convert the angle so you can picture it. The angle $\frac{3\pi}{2}$ radians equals $\frac{3 \times 180^\circ}{2} = 270^\circ$. If the radian-to-degree swap feels shaky, the walkthrough at what is a radian rebuilds it from scratch.
Step 2: Read off sine and cosine. At $270^\circ$ the point on the unit circle is $(0, -1)$, so $\cos\frac{3\pi}{2} = 0$ and $\sin\frac{3\pi}{2} = -1$. You can confirm each on its own page: cos 3pi/2 and sin 3pi/2.
Step 3: Divide cosine by sine.
$$\cot\frac{3\pi}{2} = \frac{\cos\frac{3\pi}{2}}{\sin\frac{3\pi}{2}} = \frac{0}{-1} = 0$$
A word of warning about the popular shortcut. Many students reach for $\cot\theta = \frac{1}{\tan\theta}$, but $\tan\frac{3\pi}{2}$ is undefined, so $\frac{1}{\text{undefined}}$ leaves them stuck. The reciprocal identities still hold in spirit, yet the clean way to see it is that cotangent approaches $0$ exactly where tangent blows up. Use $\frac{\cos}{\sin}$ and the problem disappears.
There is also a one-line check using the period. Cotangent repeats every $\pi$ radians ($180^\circ$), so $\cot\frac{3\pi}{2} = \cot\left(\frac{3\pi}{2} - \pi\right) = \cot\frac{\pi}{2} = 0$. That matches the sibling result at cot pi/2.
Where Does 3pi/2 Sit On The Unit Circle?
An angle of $\frac{3\pi}{2}$ is three-quarters of a full turn, measured anticlockwise from the positive $x$-axis. Its terminal side points straight down along the negative $y$-axis, so the terminal point is $(0, -1)$.
This is a quadrantal angle, meaning it lands exactly on an axis rather than inside a quadrant. That single fact explains why the usual ASTC or CAST quadrant-sign rule does not apply here: the angle is not in Quadrant I, II, III, or IV at all, it is on the boundary between III and IV. There is no reference triangle to draw, because the "triangle" has collapsed flat onto the axis.
On the unit circle, the coordinates of the terminal point are $(\cos\theta, \sin\theta)$. So reading the picture gives cosine and sine directly, and cotangent is simply the $x$-coordinate divided by the $y$-coordinate.
$$\cot\frac{3\pi}{2} = \frac{x}{y} = \frac{0}{-1} = 0$$
For a version of this diagram that also plots the tangent and cotangent segments, see unit circle with tangent.
How Does Cot 3pi/2 Compare To Cotangent At The Other Quadrantal Angles?
Cotangent has a tidy pattern at the four quadrantal angles. Wherever the point sits on the $y$-axis (top or bottom), the $x$-coordinate is $0$, so cotangent is $0$. Wherever the point sits on the $x$-axis (left or right), the $y$-coordinate is $0$, so cotangent is undefined.
Table: Cotangent at the quadrantal angles, in degrees and radians.
Angle | Radians | $\sin$ | $\cos$ | $\cot=\frac{\cos}{\sin}$ |
|---|---|---|---|---|
$0^\circ$ | $0$ | $0$ | $1$ | |
$90^\circ$ | $\frac{\pi}{2}$ | $1$ | $0$ | |
$180^\circ$ | $\pi$ | $0$ | $-1$ | |
$270^\circ$ | $\frac{3\pi}{2}$ | $-1$ | $0$ | $0$ |
$360^\circ$ | $2\pi$ | $0$ | $1$ | undefined |
For the fuller grid of angles including $30^\circ$, $45^\circ$, and $60^\circ$, the trigonometric table lays every value out side by side, and trigonometric ratios in radians does the same in radian form.
Why Is Cot 3pi/2 Equal To 0?
The reason is short once the pieces are named. Cotangent measures how much horizontal reach you have per unit of vertical reach, and at $270^\circ$ there is no horizontal reach at all.
The terminal point is $(0, -1)$, so the horizontal coordinate (cosine) is exactly $0$.
Cotangent is $\frac{\cos}{\sin} = \frac{\text{horizontal}}{\text{vertical}}$, and a numerator of $0$ over any non-zero number is $0$.
The vertical coordinate (sine) is $-1$, which is safely non-zero, so nothing about the division is illegal.
Compare that with the $x$-axis angles, where the roles swap. At $0^\circ$ or $180^\circ$ the vertical coordinate is $0$, so cotangent would ask you to divide by zero, which is why it is undefined there. The value of cotangent depends entirely on which coordinate happens to be zero, and at $\frac{3\pi}{2}$ it is the top of the fraction, not the bottom.
Who Discovered The Cotangent Function?
Cotangent is older than its name by centuries. Long before anyone wrote $\cot\theta$, the quantity was measured with a stick and a patch of sunlight.
Two more names shaped the idea:
Al-Battani (858–929, Mesopotamia) used shadow-based cotangent relations in his astronomical work, sharpening the link between shadow length and angle that later tables relied on.
Edmund Gunter (1581–1626, England) introduced the modern words "cosine" and "cotangent" in 1620, naming cotangent as the tangent of the complementary angle.
Where Is Cot 3pi/2 Used In The Real World?
Cotangent shows up wherever a horizontal reach is compared against a vertical one, and its zero-versus-undefined behaviour is exactly what engineers have to handle at the edges.
Shadow reckoning and sundials: the cotangent of the sun's elevation is the shadow-length-to-height ratio, the same measurement Abu al-Wafa tabulated.
Architecture and roofing: roof pitch and ramp gradient are often stated as run over rise, which is the cotangent of the slope angle rather than the tangent.
Surveying and navigation: finding the height of a distant tower from a measured horizontal distance uses cotangent relationships between the baseline and the line of sight.
Computer graphics: the perspective-projection matrix inside a 3D game or CAD engine scales the view by $\cot\left(\frac{\text{field of view}}{2}\right)$, so the function that reads $0$ at a quadrantal angle is literally shaping what appears on screen.
Physics and optics: cotangent terms appear in wave, diffraction, and lens equations where an angle's horizontal and vertical components trade off.
One quiet ratio, first read off a shadow, now helps place a rooftop, a survey marker, and a rendered pixel.
What Are The Most Common Mistakes With Cot 3pi/2?
These four errors account for most wrong answers on this exact value, and each traces back to the SERP questions readers actually ask.
Assuming cot 3pi/2 is undefined because tan 3pi/2 is undefined.
Where it slips in:
A student sees that $\tan\frac{3\pi}{2}$ is undefined, remembers that cotangent is "tangent's partner," and concludes cotangent must be undefined too.
Don't do this:
Do not carry the undefined verdict across from tangent. The two functions divide by different things.
The correct way:
Use $\cot\theta = \frac{\cos\theta}{\sin\theta}$. At $270^\circ$ the denominator is $\sin = -1$, which is non-zero, so the value is defined and equals $0$.
Leaving the calculator in the wrong mode, or missing the cot key.
Where it slips in:
A student types $270$ while the calculator is set to radians, or types $\frac{3\pi}{2} \approx 4.712$ while it is set to degrees, and most calculators have no dedicated cotangent button.
Don't do this:
Do not trust a reading before checking the angle MODE, and do not assume $\frac{1}{\tan}$ will work here.
The correct way:
Match the mode to the angle form, then compute $\frac{\cos}{\sin}$ directly. The $\frac{1}{\tan}$ route fails at this angle because $\tan\frac{3\pi}{2}$ is undefined.
Trying to build a reference triangle for a quadrantal angle.
Where it slips in:
A student reaches for a reference angle and a right triangle, as they would for $210^\circ$ or $300^\circ$, then cannot form one because the angle lands on an axis.
Don't do this:
Do not force a triangle where none exists. At $\frac{3\pi}{2}$ the terminal side lies on the negative $y$-axis, so the triangle collapses.
The correct way:
Read the coordinates of the terminal point $(0, -1)$ straight off the unit circle, and divide cosine by sine.
Confusing the reciprocal identity with the co-function identity.
Where it slips in:
A student mixes up $\cot\theta = \frac{1}{\tan\theta}$ (reciprocal) with $\cot\theta = \tan\left(\frac{\pi}{2} - \theta\right)$ (co-function) and applies the wrong one.
Don't do this:
Do not guess which identity to use when tangent misbehaves.
The correct way:
The co-function form is safe here: $\cot\frac{3\pi}{2} = \tan\left(\frac{\pi}{2} - \frac{3\pi}{2}\right) = \tan(-\pi) = 0$, which agrees with the ratio method.
Practice Problems On Cot 3pi/2
Work each one, then check the answer beside it.
Evaluate $\cot\frac{3\pi}{2}$.
(Answer: $0$, since $\frac{\cos\frac{3\pi}{2}}{\sin\frac{3\pi}{2}} = \frac{0}{-1} = 0$.)Is $\cot\frac{3\pi}{2}$ defined or undefined? Justify in one line.
(Answer: defined, because the denominator $\sin\frac{3\pi}{2} = -1$ is non-zero.)Evaluate $\tan\frac{3\pi}{2}$ and explain how it differs from part 1.
(Answer: undefined, because $\tan = \frac{\sin}{\cos}$ and $\cos\frac{3\pi}{2} = 0$ sits in the denominator.)Use the period of cotangent to relate $\cot\frac{3\pi}{2}$ to $\cot\frac{\pi}{2}$.
(Answer: cotangent has period $\pi$, so $\cot\frac{3\pi}{2} = \cot\frac{\pi}{2} = 0$.)Find $\csc\frac{3\pi}{2}$.
(Answer: $\frac{1}{\sin\frac{3\pi}{2}} = \frac{1}{-1} = -1$.)Convert $\frac{3\pi}{2}$ to degrees, then state the terminal point on the unit circle.
(Answer: $270^\circ$, terminal point $(0, -1)$.)
Where Should You Go Next After Cot 3pi/2?
This value opens onto the rest of the reciprocal-function family and the wider unit circle.
Cosecant, secant, and cotangent functions. See how all three reciprocal functions are built, and when each one is defined or undefined.
Sin, cos, tan. Go back to the three parent ratios that cotangent is built from, with the right-triangle picture.
Reciprocal identities. Master the $\cot = \frac{1}{\tan}$ relationship and the exact places, like $\frac{3\pi}{2}$, where it needs care.
If your child is building unit-circle fluency, a live Bhanzu trainer teaches these values starting from the "why" behind each coordinate at the Bhanzu trigonometry program.
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