Cotangent Calculator - cot(x) Degrees Radians Gradians
Calculate the cotangent of any angle or coordinate pair instantly. Supports degrees, radians, and gradians with high-precision results.
Select angle or coordinate input mode, enter your value, and click Calculate to get the cotangent with the formula used.
Cotangent Calculator - cot(x) Degrees Radians Gradians
Calculate the cotangent of any angle or coordinate pair instantly. Supports degrees, radians, and gradians with high-precision results.
About the cotangent calculator
The cotangent function is one of the six fundamental trigonometric functions. It is defined as the reciprocal of the tangent: cot(θ) = 1/tan(θ) = cos(θ)/sin(θ). Equivalently, in a right triangle, cotangent is the ratio of the adjacent side to the opposite side — the inverse of the tangent ratio.
The cotangent function is periodic with period π (180°), meaning cot(θ + π) = cot(θ) for all valid θ. Its domain excludes all integer multiples of π (i.e., 0°, 180°, 360°, and so on), because sin(θ) = 0 at those points and division by zero is undefined. The function has vertical asymptotes at these excluded values.
For the standard angles memorized in trigonometry courses: cot(30°) = √3 ≈ 1.732, cot(45°) = 1, cot(60°) = 1/√3 ≈ 0.577, and cot(90°) = 0. These values follow directly from the known sine and cosine values for those angles.
The cotangent function appears in a wide variety of engineering and physics applications. In structural engineering, it is used to analyze the slope angle of surfaces: if a roof rises by a vertical height h over a horizontal run r, then tan(θ) = h/r and cot(θ) = r/h, which is the reciprocal of the roof slope. In physics, cotangent appears in analyses of harmonic oscillators, wave propagation, and quantum mechanics. In navigation and surveying, cotangent relationships help calculate horizontal distances from observed angles of elevation.
This calculator supports three angle measurement systems. Degrees divide a full rotation into 360 equal parts and are used in everyday geometry and navigation. Radians define angle by the ratio of arc length to radius, making them natural for calculus and physics because derivatives of sine and cosine have the simplest form in radians. Gradians (also called grads or gon) divide a full rotation into 400 equal parts, making a right angle exactly 100 gradians — a system used in some European surveying traditions.
The coordinate mode computes cotangent from a point (x, y) in the plane. If a ray from the origin passes through (x, y), the angle θ it makes with the positive x-axis satisfies tan(θ) = y/x, so cot(θ) = x/y. This is useful when you know the coordinates of a point but not the explicit angle.
Cotangent calculator examples
Standard angles and a coordinate example showing the cotangent function in action.
| Input | cot(θ) | Notes |
|---|---|---|
| 45° (degrees) | 1 | cot(45°) = cos(45°)/sin(45°) = (√2/2)/(√2/2) = 1. The tangent and cotangent are equal at 45°. |
| 30° (degrees) | 1.732051 | cot(30°) = cos(30°)/sin(30°) = (√3/2)/(1/2) = √3 ≈ 1.732. A classic standard angle result. |
| π/6 radians | 1.732051 | π/6 rad = 30°. The result is identical to the degree calculation above, confirming the unit conversion. |
| Coordinates: X=3, Y=4 | 0.75 | cot(θ) = X/Y = 3/4 = 0.75. The point (3, 4) gives angle θ = arctan(4/3) ≈ 53.13°. |
How to use the cotangent calculator
- Choose your input mode: 'Angle' if you have an angle value, or 'Coordinates' if you have a point (x, y) in the plane.
- For angle mode: enter the angle value and select the unit — Degrees, Radians, or Gradians.
- For coordinate mode: enter the X (adjacent/horizontal) and Y (opposite/vertical) coordinate values.
- Click 'Calculate Cotangent'. The result appears below with the formula for reference.
- If cotangent is undefined for your input (e.g., 0°, 180°), the calculator shows an explanation instead of a number.
Cotangent calculator FAQ
What is the cotangent function?
Cotangent is a trigonometric function defined as cot(θ) = cos(θ)/sin(θ) = 1/tan(θ). In a right triangle it equals the ratio of the adjacent side to the opposite side. The cotangent is the reciprocal of the tangent and complements the tangent in much the same way that cosecant complements sine and secant complements cosine.
Where is cotangent undefined?
Cotangent is undefined wherever sin(θ) = 0, which occurs at θ = 0°, 180°, 360°, and any integer multiple of 180° (or 0, π, 2π in radians). At these points the cotangent has vertical asymptotes — the function value approaches positive or negative infinity from either side, but the function itself has no finite value at the asymptote.
How does cotangent differ from tangent?
Tangent is defined as sin(θ)/cos(θ) and equals the ratio of the opposite side to the adjacent side in a right triangle. Cotangent is the exact reciprocal: cos(θ)/sin(θ), or adjacent over opposite. They are undefined at different angles: tangent is undefined at 90° and 270°, while cotangent is undefined at 0° and 180°. Their graphs are reflections of each other across the line y = x.
How do I convert between angle units?
To convert degrees to radians, multiply by π/180. To convert radians to degrees, multiply by 180/π. To convert degrees to gradians, multiply by 10/9. The calculator handles all conversions internally — just select the unit your angle is in and enter the value directly.
What is the cotangent of a right angle (90°)?
cot(90°) = cos(90°)/sin(90°) = 0/1 = 0. At a right angle the tangent is undefined (because cos(90°) = 0 in the denominator), but the cotangent is perfectly well defined and equals zero. This is why cot(90°) = 0 even though tan(90°) is undefined.
How is cotangent used in engineering?
Engineers use cotangent in slope and grade calculations, structural analysis, and signal processing. For example, the grade of a road is expressed as rise over run (the tangent), but the horizontal distance per unit of vertical climb is the cotangent. In electromagnetic theory, cotangent functions appear in transmission line calculations and waveguide boundary conditions.