Free Cotangent Calculator

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Cotangent is the reciprocal of tangent, cot(θ) = cos(θ)/sin(θ) = 1/tan(θ), and it appears throughout physics and engineering wherever a slope or rate is more naturally expressed as "run over rise" rather than "rise over run." It shows up in optics when describing the angle of a refracted ray, in electrical engineering when analyzing phase relationships in AC circuits, and in surveying calculations involving grade and elevation. Unlike tangent, which is undefined at 90°, cotangent is undefined at the opposite set of points — 0°, 180°, 360°, and every 180° interval — because those are exactly where sine (its denominator) is zero.

How it works

Enter an angle and choose degrees or radians. The calculator computes both sin(θ) and cos(θ), then divides cos(θ) by sin(θ), returning "undefined" instead of an error whenever sin(θ) is zero. Equivalently, this is the same as taking 1/tan(θ), except cotangent is defined at points (like 0°) where tangent is zero and would otherwise force a division by zero in the reciprocal.

Enter angle (θ).
Enter angle unit.
Click Calculate to see your results.

Examples

cot(45°)

sin(45°) = cos(45°) ≈ 0.7071, so cot(45°) = 0.7071/0.7071 = 1.

cot(150°)

cos(150°) ≈ -0.8660 and sin(150°) = 0.5, so cot(150°) = -0.8660/0.5 ≈ -1.732 (which is -√3).

cot(0°)

cot(0°) is undefined, because sin(0°) = 0.

Common mistakes to avoid

Entering an angle of 0°, 180°, or 360° and expecting a numeric result instead of "undefined".
Assuming cotangent is undefined at the same angles as tangent (90°, 270°) — it is not, those are actually where cotangent equals zero.
Confusing cotangent (cos/sin) with cosecant (1/sin) — the names sound similar but the formulas differ.
Forgetting that cotangent can be negative, positive, or zero depending on the quadrant, unlike secant and cosecant which never fall between -1 and 1.

Frequently asked questions

Cotangent is the reciprocal of tangent: cot(θ) = 1/tan(θ), except at points where tan(θ) = 0 (like 0° and 180°) — there, cotangent is undefined rather than being 1/0 in the tangent sense, since cotangent is defined directly as cos(θ)/sin(θ).

Tangent is undefined where cosine is zero (90°, 270°); cotangent is undefined where sine is zero (0°, 180°, 360°) — they have swapped denominators.

No. Arctan (tan⁻¹) takes a ratio and returns an angle. Cotangent takes an angle and returns cos(θ)/sin(θ) — a completely different operation despite the similar name.

Unlike secant and cosecant, cotangent can take any real number value — its graph passes through every horizontal line exactly once per period.

On the unit circle, cotangent equals the x-coordinate divided by the y-coordinate of the point at angle θ — the reciprocal of the slope of the line from the origin to that point.

Because sine and cosine are equal at 45°, their ratio is always exactly 1, making it one of a handful of angles with a clean, memorable cotangent value.

Cotangent grows without bound in magnitude as the angle approaches these points from either side, since the denominator (sine) shrinks toward zero while cosine stays near its maximum magnitude.