Cot Calculator

Free Online Cot Calculator — Solve cot(θ) Instantly

This cot calculator (cotangent calculator) is a complete trigonometry tool that solves cotangent problems in every direction you're likely to face — plug in an angle to get cot(θ), plug in the opposite and adjacent sides of a right triangle to get the angle, or plug in a decimal cotangent value to solve an equation like cot(θ) = 1 for θ. Every calculation is backed by a labeled diagram and a full step-by-step solution, so you can see exactly how the cotangent value or angle was derived instead of just getting a bare number.

Whether you're a student solving trigonometry homework, checking a physics or engineering problem, or just need a quick sin, cos, tan, and cot value for an angle in degrees or radians, this calculator covers the complete reciprocal-ratio picture in one place, with every step of the cotangent formula shown clearly.

What Is Cotangent (cot) in Trigonometry?

Cotangent is one of the three reciprocal trigonometric ratios, alongside cosecant and secant. For any angle θ measured in a right triangle, cotangent is defined as the ratio of the length of the side adjacent to the angle to the length of the side opposite it — the exact reverse of the tangent ratio. This ratio, written cot(θ) = Adjacent ÷ Opposite, is essentially TOA flipped upside down, and like tangent, the hypotenuse never appears in it:

• SOH — Sine = Opposite ÷ Hypotenuse
• CAH — Cosine = Adjacent ÷ Hypotenuse
• TOA — Tangent = Opposite ÷ Adjacent
• Cotangent = Adjacent ÷ Opposite = 1 ÷ Tangent

The Cotangent Formula and the Unit Circle

Cotangent can also be written as the ratio of cosine to sine: cot(θ) = cos(θ) ÷ sin(θ). This makes it the reciprocal of tan(θ) = sin(θ) ÷ cos(θ) — flip the fraction and you move from tangent to cotangent. Geometrically, this has a neat interpretation on the unit circle — the one shown in the diagram above. Draw a horizontal line touching the circle at the point (0, 1); this is the co-tangent line. Extend the ray from the center through the angle θ until it crosses this horizontal line — the x-coordinate of that crossing point is exactly cot(θ). This is exactly what the red dashed line and red dot in the diagram represent, and every value you enter updates the diagram live so you can see the number and the geometry side by side.

• cot(θ) = Adjacent ÷ Opposite (right-triangle definition)
• cot(θ) = cos(θ) ÷ sin(θ) (ratio identity)
• cot(θ) = 1 ÷ tan(θ) (reciprocal identity)
• cot(θ) = x-value where the extended angle ray crosses the horizontal co-tangent line at y = 1
• θ = cot⁻¹(x) — the inverse cotangent (arccot) function, used to solve for the angle when the ratio is known
• Unlike sine and cosine, cot(θ) has no upper or lower bound — it can be any real number

How to Use This Cot Calculator — Step by Step

Pick the mode that matches your problem, enter your numbers, and read the result along with the diagram and worked solution below it.

• cot(θ) from an angle: type an angle and choose degrees, radians, or gradians — the calculator instantly returns cot(θ), sin(θ), cos(θ), the quadrant, and the reference angle, plotted on the co-tangent-line unit circle diagram.
• θ from right-triangle sides: enter the opposite and adjacent sides of a right triangle — the calculator applies cot(θ) = Adjacent ÷ Opposite, then uses the inverse cotangent to solve for θ, and draws the triangle to scale with every side labeled.
• θ from a cotangent value: enter any non-zero real number to solve equations like cot(θ) = 1 for θ, returning the principal angle plus the next solution 180° further round.

Worked Example — Solving cot(30°)

Suppose you need cot(30°). Switch to the first mode, enter 30, choose degrees, and the calculator converts 30° to π/6 radians, finds sin(30°) = 0.5 and cos(30°) = √3/2 ≈ 0.866, then divides them: cot(30°) = 0.866 ÷ 0.5 = 1.732, which is exactly √3. In the diagram, the extended ray crosses the horizontal co-tangent line exactly at x = √3, matching this well-known result.

Now suppose you're given a right triangle with an opposite side of 3 and an adjacent side of 4 — a classic 3-4-5 triangle. Switching to the second mode and entering those values gives cot(θ) = 4 ÷ 3 ≈ 1.333, and applying the inverse cotangent (computed as tan⁻¹(3 ÷ 4)), θ = tan⁻¹(0.75) ≈ 36.87°. The calculator also finds the hypotenuse using the Pythagorean theorem: √(3² + 4²) = 5, completing the full 3-4-5 right triangle so every side and angle is visible on the diagram.

Common Cotangent Values Table

These standard angles come up constantly in trigonometry, and it helps to recognize them at a glance rather than recalculating them each time:

• cot(0°) = undefined
• cot(30°) = √3 ≈ 1.732
• cot(45°) = 1
• cot(60°) = 1/√3 ≈ 0.577
• cot(90°) = 0
• cot(180°) = undefined
• cot(270°) = 0
• cot(360°) = undefined

Why Is cot(0°) Undefined?

Since cot(θ) = cos(θ) ÷ sin(θ), cotangent becomes undefined whenever sin(θ) = 0 — which happens at 0°, 180°, 360°, and every multiple of 180°. Geometrically, at exactly 0° the angle's ray points straight along the x-axis, running parallel to the horizontal co-tangent line rather than crossing it, so there's no intersection point and no finite cotangent value. On a graph of cot(θ), these points appear as vertical asymptotes, where the curve rockets toward positive or negative infinity. This calculator detects these angles automatically and reports 'undefined' instead of a misleading number.

Understanding the Sign of cot(θ) by Quadrant

Because cot(θ) = cos(θ) ÷ sin(θ), its sign depends on whether cosine and sine share the same sign in a given quadrant — exactly the same pattern as tangent, since cotangent is its reciprocal. This calculator automatically detects and displays the quadrant for any angle you enter:

• Quadrant 1 (0°-90°): both sin and cos are positive, so cot(θ) is positive
• Quadrant 2 (90°-180°): sin is positive, cos is negative, so cot(θ) is negative
• Quadrant 3 (180°-270°): both sin and cos are negative, so cot(θ) is positive again
• Quadrant 4 (270°-360°): sin is negative, cos is positive, so cot(θ) is negative

Cotangent vs Tangent — How They Relate

Cotangent and tangent are reciprocals of each other: cot(θ) = 1 ÷ tan(θ), and tan(θ) = 1 ÷ cot(θ). This means whenever tan(θ) is large, cot(θ) is small and close to zero, and whenever tan(θ) is close to zero, cot(θ) grows very large. It also explains why cot(θ) is undefined exactly where tan(θ) = 0 (at 0°, 180°, 360°), and why cot(θ) = 0 exactly where tan(θ) is undefined (at 90°, 270°). Both functions share the same period of 180°, meaning cot(θ) = cot(θ + 180°) for any angle θ.

Degrees vs Radians vs Gradians

Angles can be measured in more than one unit, and this calculator supports the three most common ones. Degrees split a full circle into 360 equal parts and are the most familiar unit in everyday geometry. Radians measure angle by arc length relative to the radius, making a full circle equal to 2π radians — this is the unit calculus and most programming languages use internally. Gradians (or gons) split a full circle into 400 parts and appear mostly in surveying and some European engineering contexts. Whichever unit your problem uses, simply select it from the dropdown and the calculator handles the conversion automatically.

Where Cotangent Calculations Are Used in Real Life

Cotangent isn't just a classroom exercise — it shows up anywhere the reciprocal of a slope or rate of change matters:

• Construction and roofing: expressing a roof's run-over-rise ratio, which is exactly what cotangent represents when tangent gives rise-over-run.
• Surveying and land measurement: finding a horizontal distance from a known height and an angle of elevation, without needing tangent directly.
• Physics and engineering: analyzing circuits, wave problems, and rotational systems where reciprocal trigonometric ratios simplify the math.
• Calculus: cotangent appears constantly in integration and differentiation problems involving trigonometric substitution.
• Computer graphics and optics: cotangent is used in lens and field-of-view calculations, and in projection matrices for 3D rendering.

Tips for Getting Accurate Results

Always double-check which angle unit your problem is stated in before entering a value — a common mistake is entering a radian value while the unit dropdown is still set to degrees, which will give a wildly different (and wrong) answer. When solving for an angle from a cotangent value or triangle sides, remember that inverse cotangent only returns one principal solution, conventionally between 0° and 180°; since cotangent repeats every 180° (not 360° like sine and cosine), always check whether your problem expects a second solution 180° further around the circle. Also remember that cot(θ) is simply 1 ÷ tan(θ), so if you already know an angle's tangent, you can find its cotangent instantly by flipping the fraction — no calculator required for that last step.