Right Triangle

This figure illustrates the right triangle.
Figure 1: Right Triangle

A right triangle is a triangle where one of the three angles is a perpendicular angle. There are three sides of a right triangle: the adjacent, opposite and hypotenuse sides. The sides are labeled relative to the angle of the right triangle denoted with the symbol (theta). Since one of the three angles of the triangle will always be a right angle, only one angle is needed to describe the shape of the right triangle.

The geometry of the right triangle is of particular importance since the shape defines a point in both the Cartesian Coordinate System and the Polar Coordinate System, albeit by different means. This connection is deepened by the Trigonometric Functions which relate the angle of a right triangle to the ratio of its sides. The importance of the right triangle and the functions associated with it is demonstrated by study of the unit circle.

Pythagorean’s Theorem

A well known property of the right triangle is the Pythagorean Theorem which relates the squared-length of the adjacent, opposite, and hypotenuse sides together.

The theorem is used throughout mathmatics. For example, the equation can be used to find the unknown length of a right triangle, verify that a triangle is a right triangle and more.

Pythagorean's Theorem

Special Right Triangles

There are two special right triangles which have simple, unique properties. The triangles are “special” since their angles perfectly divide a full rotation by eight and twelve respectively. The first special triangle is the 45 45 90.

Special Right Triangle 45 45 90

The 45° 45° 90° right triangle is a special triangle where the adjacent and opposite sides are equal in length. The most simple version of this triangle is where the adjacent and opposite sides are length 1 and then, using Pythagorean’s theorem, the hypotenuse is equal to the .

30 60 90 Triangle

Special Right Triangle 30 60 90

The 30° 60° 90° right triangle is a special triangle where the hypotenus is twice the length of the opposite side. If the opposite side is of length 1 and the hypotenuse is of length 2, then using Pythagorean’s theorem, then length of the adjacent side is equal to .

Trigonometric Functions

Right Triangle

There are three main trigonometric functions that relate the angle of a right triangle with the ratio of its sides. They symbol θ is used to denote the angle.

Cosine Function
Cosine | Function

The cosine function returns the cosine of an angle provided in radians. In geometric terms, the function returns the ratio of the right-triangle's adjacent side over its hypotenuse.

Sine Function
Sine | Function

The sine function returns the sine of a number provided in radians. In geometric terms, the function returns the vertical component of the point formed by the angle on the unit circle.

Tangent Function
Tangent | Function

Given the angle of a right triangle as input returns the ratio of the triangle's opposite side over its adjacent side.

Further Reading

The right-triangle connects to many other concepts. It is the shape that makes up a 2D point in the Cartesian Coordinate System. It is the shape that is formed by a 2D vector and so on. Here are some links.

Cartesian Coordinate System
Cartesian Coordinate System | Concept

The Cartesian Coordinate System describes space of one, two, and three dimensions. Each point in space is represented by its distance relative to the origin of the system.

Unit Circle
Unit Circle | Concept

The unit circle is a circle of radius one placed at the origin of the coordinate system. This article discusses how the unit circle represents the output of the trigonometric functions for all real numbers.

Trigonometric Identities
Trigonometric Identities | Concept

The trigonometric identites are a set of equations derived from the properties of the right triangle and the circle.