TRIGONOMETRYTrigonometryMathematics Calculator
cot

The Cotangent Function

Cotangent is the reciprocal of tangent: cot(θ) = cos(θ)/sin(θ) = 1/tan(θ). It represents adjacent/opposite in a right triangle and is undefined where sin(θ) = 0.

Concept Fundamentals
cot(θ) = cos/sin = 1/tan
Definition
0°, 180°, 360°
Undefined at
1 + cot²θ = csc²θ
Identity
π (180°)
Period

Did our AI summary help? Let us know.

Cotangent shares the same sign as tangent — positive in Q1 and Q3, negative in Q2 and Q4. cot(30°)=√3, cot(45°)=1, cot(60°)=1/√3 — the reciprocal of tan at those angles. Cotangent has period π like tangent; both repeat every 180°.

Key quantities
cot(θ) = cos/sin = 1/tan
Definition
Key relation
0°, 180°, 360°
Undefined at
Key relation
1 + cot²θ = csc²θ
Identity
Key relation
π (180°)
Period
Key relation

Ready to run the numbers?

Why: Cotangent appears in calculus (∫ csc² dx = -cot), electrical engineering (impedance), and when working with reciprocal trig identities.

How: cot(θ) = cos(θ)/sin(θ). When sin(θ) = 0, cotangent is undefined. The identity 1+cot²θ = csc²θ comes from dividing sin²+cos²=1 by sin².

Cotangent shares the same sign as tangent — positive in Q1 and Q3, negative in Q2 and Q4.cot(30°)=√3, cot(45°)=1, cot(60°)=1/√3 — the reciprocal of tan at those angles.
Sources:Wolfram MathWorld — CotangentKhan Academy — Trigonometry

Run the calculator when you are ready.

Start CalculatingEnter an angle to compute cot(θ) — undefined at 0°, 180°, 360°

Examples — Click to Load

cotangent.sh
CALCULATED
$ cot --angle 45° --all-functions
cot(θ)
1
tan(θ)
1
sin(θ)
0.70710678
Quadrant
Q1
cos(θ)
0.70710678
csc(θ)
1.41421356
sec(θ)
1.41421356
Ref Angle
45°
Share:
Cotangent Calculator Result
cot(45°)
1
Q1ref 45°1+cot² = 2
numbervibe.com/calculators/mathematics/trigonometry/cotangent-calculator

Trig Value Breakdown

All 6 Trig Functions

sin² vs cos² (Pythagorean Identity)

Calculation Breakdown

CONVERSION
Input Angle
45°
Convert to Radians
0.78539816 rad
45° × π/180
Normalized Angle
45°
\text{theta} mod 360^{circ}
Quadrant
Q1
45.0° is in quadrant 1
Reference Angle
45°
ext{Acute} ext{angle} ext{to} x- ext{axis}
PRIMARY RESULT
COTANGENT VALUE
1
cot(45°) = cos(θ)/sin(θ)
RELATED VALUES
Sine
0.70710678
sin(45°)
Cosine
0.70710678
cos(45°)
Tangent
1
tan(45°) = sin/cos
Cosecant
1.41421356
1/\text{sin}(\text{theta} )
Secant
1.41421356
1/\text{cos}(\text{theta} )
IDENTITY
1 + cot²(θ)
2
csc²(θ)
2
1 + ext{cot}^{2} = ext{csc}^{2}

For educational and informational purposes only. Verify with a qualified professional.

🧮 Fascinating Math Facts

↗️

Cotangent is the reciprocal of tangent — cot(θ) = 1/tan(θ).

— Wolfram MathWorld

∞

Cotangent has vertical asymptotes at 0°, 180°, 360° — where sine equals zero.

— Paul's Online Notes

Key Takeaways

  • • cot(θ) = cos(θ)/sin(θ) = adjacent/opposite in a right triangle; cotangent is the reciprocal of tangent
  • • Cotangent is undefined at 0°, 180°, 360° where sin(θ) = 0. Range: all real numbers — unlike sec/csc, cot can be any value
  • • Cotangent is an odd function: cot(-θ) = -cot(θ) with period π (180°) — half the period of sine and cosine
  • • The Pythagorean identity 1 + cot²(θ) = csc²(θ) always holds when cot is defined
  • • Cotangent is positive in Q1 and Q3 (where sin and cos have same sign), negative in Q2 and Q4

Did You Know?

🔄Cotangent is the reciprocal of tangent: cot(θ) = 1/tan(θ). The 'co-' prefix indicates it complements tangent in the co-function identity cot(θ) = tan(90° - θ)Source: Wolfram MathWorld
📐In surveying, cotangent gives the horizontal run per unit rise — the inverse of slope. A 45° slope has cot(45°)=1 (equal rise and run)Source: Khan Academy
📊Cotangent has the shortest period (π) among the six trig functions — its graph repeats twice as often as sine and cosineSource: MIT OpenCourseWare
🔬In quantum mechanics, cotangent appears in scattering amplitude formulas and bound-state wave functionsSource: Paul's Online Notes
🌐Navigation and GPS use cotangent in formulas for converting between bearing angles and coordinate differencesSource: IEEE Geoscience
📱Computer graphics use cotangent for perspective projection — cot(fov/2) relates field of view to projection matrix scalingSource: OpenGL Specification

How the Cotangent Function Works

The cotangent function is the ratio of cosine to sine: cot(θ) = cos(θ)/sin(θ). Equivalently, cot(θ) = 1/tan(θ). On the unit circle, it represents the x-coordinate divided by the y-coordinate.

Where Cotangent is Undefined

At 0°, 180°, and 360°, the terminal side is horizontal, so sin(θ) = 0. Division by zero makes cot(θ) undefined. The cotangent graph has vertical asymptotes at these angles — the same as tangent, but cot is the reciprocal so its zeros become tangent's asymptotes and vice versa.

Period π and Odd Symmetry

Cotangent has period π (180°) because cot(θ + π) = cos(θ+π)/sin(θ+π) = (-cos θ)/(-sin θ) = cot(θ). As an odd function, cot(-θ) = -cot(θ). The graph repeats every half-rotation.

1 + cot² = csc² Identity

Dividing sin²θ + cos²θ = 1 by sin²θ gives 1 + cot²θ = csc²θ. This identity connects cotangent to cosecant and is essential for integration (e.g., ∫csc²x dx = -cot x + C).

Expert Tips

Memorize Special Angles

cot(30°)=√3, cot(45°)=1, cot(60°)=1/√3. Notice cot(θ) = 1/tan(θ), so values are reciprocals. Use the Unit Circle Calculator.

Avoid 0°, 180°, 360°

Cotangent is undefined where sin = 0. Use the Sine Calculator to verify sin ≠ 0 before computing cot.

1 + cot² = csc²

If you know cot(θ), then csc(θ) = ±√(1 + cot²θ). The sign matches sin(θ). See the Trig Identities Calculator.

Cot vs Tan Relationship

cot(θ) · tan(θ) = 1 always. Cotangent has zeros where tangent has asymptotes (90°, 270°). Compare with Tangent Calculator.

Why Use This Calculator vs. Other Tools?

FeatureThis CalculatorScientific CalculatorManual Computation
All 6 trig functions at once✅❌ One at a time❌
Undefined detection (0°, 180°, 360°)✅⚠️ May show error✅ Slow
Visual charts & breakdown✅❌❌
Step-by-step explanation✅❌✅
1 + cot² = csc² identity check✅❌⚠️ Manual
Copy & share results✅❌❌
Degrees and radians✅✅✅
Preset examples✅❌❌

Frequently Asked Questions

When is cotangent undefined?

Cotangent is undefined at 0°, 180°, 360°, and all angles where sin(θ) = 0. At these angles, the terminal side is horizontal on the unit circle, so the y-coordinate (sine) is zero and division by zero occurs.

What is the range of cotangent?

Unlike secant and cosecant, cotangent has range (-∞, ∞) — all real numbers. Cot(θ) can be any value because it is cos/sin, and both can be small, giving large cot values.

Why does cotangent have period π instead of 2π?

cot(θ + π) = cos(θ+π)/sin(θ+π) = (-cos θ)/(-sin θ) = cot(θ). Both sin and cos flip sign at π, so their ratio stays the same. The graph repeats every 180°.

How do I find cotangent without a calculator?

cot(θ) = cos(θ)/sin(θ) or 1/tan(θ). Memorize: cot(30°)=√3, cot(45°)=1, cot(60°)=1/√3. For other angles, use reference angles and quadrant signs. Avoid 0°, 180°, 360°.

What is 1 + cot²(θ)?

Always equals csc²(θ) when defined. From sin²θ + cos²θ = 1, divide by sin²θ to get 1 + cot²θ = csc²θ. Used in calculus for ∫csc²x dx = -cot x + C.

Where is cotangent used in real life?

Cotangent appears in surveying (slope as rise/run inverse), navigation, computer graphics (perspective projection, FOV), and physics (quantum scattering, wave analysis).

How is cotangent related to tangent?

Cotangent is the reciprocal of tangent: cot(θ) = 1/tan(θ) = cos(θ)/sin(θ). They have opposite zeros and asymptotes: cot is undefined where tan = 0, and cot = 0 where tan is undefined.

Why is cotangent called an odd function?

cot(-θ) = cos(-θ)/sin(-θ) = cos(θ)/(-sin θ) = -cot(θ). The graph has rotational symmetry about the origin, like tangent and sine.

Cotangent Function by the Numbers

(-∞, ∞)
Output Range
180°
Period
3
Undefined Angles
Odd
Symmetry

Disclaimer: This calculator provides results based on standard IEEE 754 floating-point arithmetic. Results are accurate to approximately 15 significant digits. Cotangent is undefined at 0°, 180°, and 360° — the calculator will display an error for these inputs. For mission-critical applications, always verify with certified computational tools.

👈 START HERE
⬅️Jump in and explore the concept!
AI

Related Calculators

Cosecant Calculator

Calculate the cosecant of any angle in degrees or radians. Get csc(θ) = 1/sin(θ), undefined detection at 0°/180°, step-by-step solutions, and interactive...

Mathematics

Cosine Calculator

Calculate the cosine of any angle in degrees or radians. Get cos(θ), related sin and tan values, quadrant info, step-by-step solutions, and interactive...

Mathematics

Sine Calculator

Calculate the sine of any angle in degrees or radians. Get sin(θ), related cos and tan values, quadrant info, step-by-step solutions, and interactive charts....

Mathematics

Tangent Calculator

Calculate the tangent of any angle in degrees or radians. Get tan(θ), related sin and cos values, quadrant info, asymptote detection, step-by-step solutions...

Mathematics

Secant Calculator

Calculate the secant of any angle in degrees or radians. Get sec(θ) = 1/cos(θ), undefined detection at 90° and 270°, step-by-step solutions, and interactive...

Mathematics

Unit Circle Calculator

Find x (cos), y (sin), reference angle, quadrant for any angle. Step-by-step, charts.

Mathematics