In geometry, complementary angles are angle

Angle

In geometry, an angle is the figure formed by two rays sharing a common endpoint, called the vertex of the angle.Angles are usually presumed to be in a Euclidean plane with the circle taken for standard with regard to direction. In fact, an angle is frequently viewed as a measure of an circular arc...

s whose measures sum to 90°. If the two complementary angles are adjacent (i.e. have a common vertex

Vertex (geometry)

In geometry, a vertex is a special kind of point that describes the corners or intersections of geometric shapes.-Of an angle:...

 and share just one side) their non-shared sides form a right angle

Angle

In geometry, an angle is the figure formed by two rays sharing a common endpoint, called the vertex of the angle.Angles are usually presumed to be in a Euclidean plane with the circle taken for standard with regard to direction. In fact, an angle is frequently viewed as a measure of an circular arc...

.

In Euclidean geometry

Euclidean geometry

Euclidean geometry is a mathematical system attributed to the Alexandrian Greek mathematician Euclid, which he described in his textbook on geometry: the Elements. Euclid's method consists in assuming a small set of intuitively appealing axioms, and deducing many other propositions from these...

, the two acute angles

Angle

In geometry, an angle is the figure formed by two rays sharing a common endpoint, called the vertex of the angle.Angles are usually presumed to be in a Euclidean plane with the circle taken for standard with regard to direction. In fact, an angle is frequently viewed as a measure of an circular arc...

 in a right triangle

Right triangle

A right triangle or right-angled triangle is a triangle in which one angle is a right angle . The relation between the sides and angles of a right triangle is the basis for trigonometry.-Terminology:The side opposite the right angle is called the hypotenuse...

 are complementary, because the sum of internal angles of a triangle is 180 degrees, and the right angle itself accounts for ninety degrees.

The adjective complementary is from Latin

Latin

Latin is an Italic language originally spoken in Latium and Ancient Rome. It, along with most European languages, is a descendant of the ancient Proto-Indo-European language. Although it is considered a dead language, a number of scholars and members of the Christian clergy speak it fluently, and...

 complementum, associated with the verb complere, "to fill up". An acute angle is "filled up" by its complement to form a right angle.

Trigonometric ratios

The sine

Sine

In mathematics, the sine function is a function of an angle. In a right triangle, sine gives the ratio of the length of the side opposite to an angle to the length of the hypotenuse.Sine is usually listed first amongst the trigonometric functions....

of an angle equals the cosine of its complementary angle. It is therefore true that, if angles A and B are complementary, , and .

The tangent

Tangent

In geometry, the tangent line to a plane curve at a given point is the straight line that "just touches" the curve at that point. More precisely, a straight line is said to be a tangent of a curve at a point on the curve if the line passes through the point on the curve and has slope where f...

of an angle equals the cotangent of its complementary angle. The tangents of complementary angles are reciprocals

Multiplicative inverse

In mathematics, a multiplicative inverse or reciprocal for a number x, denoted by 1/x or x−1, is a number which when multiplied by x yields the multiplicative identity, 1. The multiplicative inverse of a fraction a/b is b/a. For the multiplicative inverse of a real number, divide 1 by the...

 of each other.

The secant of an angle equals the cosecant of its complementary angle.

The prefix "co-" in the names of some trigonometric ratios refers to the word "complementary".

External links

• Animated demonstration - Interactive applet and explanation of the characteristics of complementary angles.