Cotangent Calculator

Cotangent is the reciprocal of tangent, and also the ratio of cosine to sine:

cot(θ) = cos(θ) / sin(θ) = 1 / tan(θ)

It is undefined when sin(θ) = 0 — at 0°, 180°, 360°, and every multiple of 180° (every multiple of π in radians). Unlike the cosecant and secant, cotangent has no restriction on its range: it can take any real value.

Key values:

• cot(30°) = √3 ≈ 1.732
• cot(45°) = 1 (cos and sin are equal here)
• cot(60°) = 1/√3 ≈ 0.577
• cot(90°) = 0 (cos = 0 at 90°)

Period: π (180°) — the same as tangent. The function drops from +∞ to −∞ across each period, passing through 0 at the 90° marks and through ±1 at the 45° and 135° marks.

The “co-” relationship: cot(θ) = tan(90° − θ). Cotangent of an angle equals the tangent of the complementary angle. This is where all the “co-” prefix names in trigonometry come from — co-sine, co-tangent, co-secant are all functions of the complementary angle.

Where cotangent appears in practice: in slope calculations (cotangent of the angle of inclination equals the horizontal run per unit of rise), in physics problems involving wedges and inclined planes, in some formulations of conic section geometry, and in the calculation of the area of a regular polygon where the formula involves cot(π/n).

The Pythagorean identity for cotangent: csc²(θ) = 1 + cot²(θ), analogous to sec²(θ) = 1 + tan²(θ).