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.
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.
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.
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
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
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.
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.
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.
Given the angle of a right triangle as input returns the ratio of the triangle's opposite side over its adjacent side.
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.
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.
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.
The trigonometric identites are a set of equations derived from the properties of the right triangle and the circle.
A Perpendicular angle, sometimes also referred to as a square angle, is exactly 90 degrees or PI fourths.
An angle is defined as the amount of rotation between two rays. Angles are measured using degrees and radians. A full rotation in degrees is 360°. A full rotation in radians is approximately 6.283 radians or τ (tau) radians.
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.
The Polar Coordinate System describes points in space using an angle and radius relative to the origin.
There are six trigonometric functions that relate to the geometry of the right-triangle sine, cosine, tangent, cosecant, secant, and cotangent. The functions take the angle of a right triangle as input and return a ratio of two of its sides.
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.
The triangle defined by the three angles: 45 degrees, 45 degrees, and 90 degrees is a special triangle that has meaningful properties in mathematics.