Overview
In Euclidean geometry, complementary angles are two angles whose measures add up to a right angle. In degree measure this means their sum is 90°; in radian measure it is π/2. Complementary angles arise naturally when studying right triangles and the relationships between sine and cosine.
Definition and basic properties
Given an angle θ, a complement is any angle φ such that θ + φ = 90° (or θ + φ = π/2). The complement of θ is usually written as 90° − θ. For angles measured in radians the complement is π/2 − θ. Typical properties include:
- Both angles that form a complementary pair are usually acute (between 0° and 90°), though one can be 0° and the other 90° in degenerate cases.
- If two adjacent angles are complementary, their noncommon sides form a straight line perpendicular to the common side, yielding a right angle at the vertex.
Computation and examples
To find a complement, subtract the given measure from 90°. For example, the complement of 30° is 60°, since 30° + 60° = 90°. In radians, the complement of π/6 is π/3 because π/6 + π/3 = π/2. If an angle exceeds 90°, it has no positive complement in the usual sense.
Trigonometric significance
Complementary angles underlie a key trigonometric identity: sine of an angle equals cosine of its complement, sin(θ) = cos(90° − θ). This symmetry connects cofunctions and simplifies many calculations in right-triangle trigonometry and analytic work. See the standard definition for more on angle measures and refer to related identities at trigonometric cofunctions.
Distinctions and uses
Complementary angles are distinct from supplementary angles, which sum to 180°. They are commonly used in geometry problems, engineering applications involving right angles, and in proofs where right-angle relationships are required. Historically the concept is fundamental to classical geometry and appears in elementary education as part of angle classification.