Explore cotangent from angles or coordinate basis matrices. See steps, warnings, and rounded results instantly. Download tables as CSV or PDF with one click.
| Example | Inputs | Expected Output |
|---|---|---|
| Trig: 45° | Angle=45, Unit=Degrees | cot(θ)=1.000000 |
| Trig: 30° | Angle=30, Unit=Degrees | cot(θ)=1.732051 |
| Trig: 180° | Angle=180, Unit=Degrees | cot(θ)=undefined (sin(θ)=0) |
| Dual basis (n=2) | V = [[1,0],[0,2]] | W = [[1,0],[0,0.5]] |
| Dual basis (n=3) | V = identity matrix | W = identity matrix |
This calculator provides two complementary views. The trigonometric mode computes cot(θ) alongside sin(θ), cos(θ), tan(θ), sec(θ), and csc(θ), with a configurable precision from 0–12 decimals. The cotangent-space mode treats your matrix V as a tangent basis and returns the dual basis W that satisfies WTV = I.
Angle inputs can be entered in degrees, radians, or gradians. Internally, values are converted to radians for evaluation, using π/180 for degrees and π/200 for gradians. The results table also reports the degree-equivalent angle to help compare periodic behavior over 0–360° in the graph.
Cotangent becomes undefined whenever sin(θ)=0, occurring at integer multiples of π. The calculator detects near-zero sine values with a small tolerance and labels outputs as undefined instead of returning misleading large numbers. The Plotly curve intentionally breaks at those points, preventing artificial spikes from connecting across asymptotes.
For an n-dimensional basis, V must be full rank. The tool computes V-1 using pivoted Gaussian elimination, then forms W=(V-1)T. Each column of W represents a covector that pairs with the input tangent basis via the Kronecker delta, giving an immediately checkable identity WTV=I.
In trigonometric mode, the interactive plot shows cot(θ) over one full rotation. You can zoom to inspect neighborhoods around steep regions near 0°, 180°, and 360°. In dual mode, the heatmap visualizes W entry magnitudes, helping you spot scaling, near-singularity, and sensitivity patterns across rows and columns.
CSV export stores every key-value pair from the computed result block, including matrix data, which supports audits and spreadsheet analysis. PDF export produces a compact single-page report suitable for assignments, lab notes, or quick sharing. Recompute with different precision to create consistent, reproducible numeric summaries.
cot(θ)=cos(θ)/sin(θ). When sin(θ)=0, division is not defined. The calculator detects that case and reports “undefined” to avoid unstable, misleading values.
The dual basis W contains covectors wi that satisfy wi(vj)=δij. It is a concrete coordinate model for the cotangent space relative to your tangent basis.
Enter V by rows in the grid. Conceptually, its columns are the tangent basis vectors v1…vn. The tool uses that convention when computing W=(V-1)T.
The dual-basis mode supports n=2 to n=5 for clarity and speed in browsers. Larger sizes are possible in principle, but UI and numeric conditioning become harder to manage.
The plot uses null points where sin(θ)≈0, which creates a break in the line. This prevents the curve from incorrectly drawing across cotangent asymptotes.
If det(V)=0, the inverse does not exist and no dual basis is defined. Adjust the basis vectors so they become linearly independent, then recompute.
Important Note: All the Calculators listed in this site are for educational purpose only and we do not guarentee the accuracy of results. Please do consult with other sources as well.