Cotangent Calculator

What is Cotangent?

The cotangent function, written as $\cot(\theta)$, is one of the six fundamental trigonometric functions. It represents the reciprocal of the tangent function and describes the ratio of the adjacent side to the opposite side in a right-angled triangle.

In terms of circular functions and unit circle trigonometry, cotangent is defined as the quotient of cosine and sine:

$$\cot(\theta) = \frac{\cos(\theta)}{\sin(\theta)} = \frac{1}{\tan(\theta)}$$

Cotangent is widely used in calculus, surveying, wave mechanics, physics, and computer graphics whenever analyzing slopes, harmonic motions, or polar coordinates.

Cotangent Formulas and Definitions

Depending on whether you are working with right triangles, coordinate geometry, or analytical identities, cotangent can be defined in multiple equivalent ways:

Right Triangle Ratio: In a right triangle with acute angle $\theta$, adjacent side $b$, and opposite side $a$: $$\cot(\theta) = \frac{\text{Adjacent}}{\text{Opposite}} = \frac{b}{a}$$
Reciprocal Identity: Cotangent is the exact multiplicative inverse of tangent: $$\cot(\theta) = \frac{1}{\tan(\theta)}$$
Quotient Identity: On the unit circle with coordinates $(x, y) = (\cos\theta, \sin\theta)$: $$\cot(\theta) = \frac{x}{y} = \frac{\cos(\theta)}{\sin(\theta)}$$
Pythagorean Identity: Cotangent relates to cosecant through: $$1 + \cot^2(\theta) = \csc^2(\theta)$$

Exact Values of Cotangent for Common Angles

The table below lists standard exact angles and their corresponding cotangent values:

Angle (Degrees)	Angle (Radians)	$\cot(\theta)$ Exact Value	Decimal Approximation
$0^\circ$	$0$	Undefined	Undefined (Asymptote)
$30^\circ$	$\frac{\pi}{6}$	$\sqrt{3}$	1.73205
$45^\circ$	$\frac{\pi}{4}$	$1$	1.00000
$60^\circ$	$\frac{\pi}{3}$	$\frac{1}{\sqrt{3}} = \frac{\sqrt{3}}{3}$	0.57735
$90^\circ$	$\frac{\pi}{2}$	$0$	0.00000
$180^\circ$	$\pi$	Undefined	Undefined (Asymptote)
$270^\circ$	$\frac{3\pi}{2}$	$0$	0.00000

Properties and Behavior of the Cotangent Graph

The cotangent function has distinctive graphical and analytical properties:

Domain: All real numbers except integer multiples of $\pi$, i.e., $\theta \neq k\pi$ for any integer $k$. At these points, $\sin(\theta) = 0$, creating vertical asymptotes.
Range: All real numbers, $(-\infty, +\infty)$.
Periodicity: The fundamental period of $\cot(\theta)$ is $\pi$ ($180^\circ$), meaning $\cot(\theta + \pi) = \cot(\theta)$.
Odd Function: Cotangent is symmetric with respect to the origin: $\cot(-\theta) = -\cot(\theta)$.
Decreasing Nature: The cotangent curve strictly decreases across every continuous open interval between its vertical asymptotes.

Related Trigonometric Tools

Explore other helpful calculators on OnlineMiniTools:

Tan Calculator: Compute the tangent of any angle in degrees or radians.
Cos Calculator: Calculate cosine values and solve adjacent ratios.
Right Triangle Calculator: Solve missing sides, angles, and hypotenuse of right triangles.

Frequently Asked Questions

Why is cotangent undefined at 0 degrees and 180 degrees?

Cotangent is defined as $\cos(\theta) / \sin(\theta)$. At $0^\circ$, $180^\circ$, $360^\circ$ (and any integer multiple of $\pi$), $\sin(\theta) = 0$. Dividing by zero is undefined, which produces vertical asymptotes at these angles.

How do you convert between radians and degrees?

To convert degrees to radians, multiply by $\pi / 180$. To convert radians to degrees, multiply by $180 / \pi$. This calculator handles degree, radian, pi radian, and gradian inputs automatically.

What is the difference between cotangent and arctangent?

Cotangent ($\cot\theta$) is the reciprocal trigonometric function equal to $1/\tan\theta$. Arctangent ($\arctan x$ or $\tan^{-1}x$) is the inverse trigonometric function that gives the angle whose tangent equals $x$.

Can cotangent be greater than 1 or negative?

Yes. Unlike sine and cosine which are bounded between $-1$ and $+1$, the range of cotangent is all real numbers from $-\infty$ to $+\infty$. It is positive in Quadrants I and III, and negative in Quadrants II and IV.

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