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 wherever sin(θ) = 0, which means 0°, 180°, 360° and every other multiple of 180° (every multiple of π in radians).
Unlike cosecant and secant, cotangent has no gap in its range: it takes every 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, or 180°, the same as tangent. The function falls from +∞ to −∞ across each period, crossing 0 at the 90° marks and passing through ±1 at 45° and 135°.

Note that it falls rather than rises. Tangent climbs across its period and cotangent drops across its, which trips people up when they sketch the two from memory.

The “co-” relationship: cot(θ) = tan(90° − θ). The cotangent of an angle is the tangent of its complement. That is where the whole “co-” family comes from: co-sine, co-tangent and co-secant are each the plain function 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).