cofunctions

Cofunctions are trigonometric functions evaluated at complementary angles. If two angles A and B satisfy A + B = π/2 (90 degrees), then the trigonometric values of one angle relate directly to those of the other. The standard cofunction pairs are sin A = cos B, cos A = sin B, tan A = cot B, cot A = tan B, sec A = csc B, and csc A = sec B. Equivalently, using a single variable x, the identities sin(π/2 − x) = cos x, cos(π/2 − x) = sin x, tan(π/2 − x) = cot x, cot(π/2 − x) = tan x, sec(π/2 − x) = csc x, and csc(π/2 − x) = sec x hold for all x where the functions are defined.

Cofunctions arise from the geometry of a right triangle, where the two acute angles are complements, and from the symmetry of the unit circle under angle complementation. They are a subset of trigonometric identities and are useful for simplifying expressions, evaluating functions at complementary angles, and solving trigonometric equations. In radians, the complement is π/2; in degrees, it is 90°. The concept applies to all six primary trig functions, with sine and cosine interchanged under complementation and, similarly, tangent and cotangent, while secant and cosecant correspondingly swap roles. These identities reflect the fundamental properties of trigonometric functions under angle transformation and are widely employed in algebra, calculus, and trigonometry.