Coslatitude

Coslatitude refers to the cosine of geographic latitude, often written as cos(latitude). If latitude is denoted by φ, then coslatitude equals cos(φ) after φ is converted to radians for the trigonometric calculation. It is a dimensionless factor that ranges from 0 at the poles to 1 at the equator and appears frequently in spherical geometry and cartography.

In spherical Earth geometry, cos(latitude) appears in several fundamental formulas. The differential surface area of a small patch with latitude φ and longitude λ on a sphere of radius R is dA = R^2 cosφ dφ dλ. The east-west length corresponding to a small longitude change Δλ at latitude φ is R cosφ Δλ. Thus, cos(latitude) captures the compression of east-west distances toward the poles and is central to understanding distortions in map projections and spatial calculations.

The factor also plays a key role in map projections and geographic information systems. For example, in the equirectangular projection, horizontal scale is proportional to cosφ, leading to shrinking of longitude distances as latitude increases. Conversely, in the Mercator projection the scale factor varies as secφ = 1/cosφ, which causes magnification toward the poles. Consequently, coslatitude is often used to adjust or normalize data, convert grid spacing from degrees to meters, or weight computations by latitude bands in geographic datasets.

Cautions include numerical sensitivity near the poles, where cosφ approaches zero and division by cosφ can become unstable. Real-world calculations frequently employ ellipsoidal Earth models for higher accuracy, in which the simple cos(latitude) factor is supplemented by more sophisticated formulas.