The study of triangles in plane geometry led to trigonometry. Originally trigonometry was concerned with the measurement of angles and the determination of three parts or a triangle (sides or angles) when the remaining three parts were known. If we know two angles and the length of one side of a triangle, for example, we can compute the other angle and the length of the remaining sides. Trigonometry uses triangles because all shapes in plane geometry can be broken down into triangles.

The relationships between the sides and angles of triangles can be expressed as ratios called trigonometric functions and used in calculations. Similar triangles—triangles with the same angles—have the same trigonometric functions because the lengths of their sides are in the same ratio. Right triangles (triangles with one angle of 90 degrees) are used to define three important trigonometric functions: sine (usually abbreviated sin), cosine (cos), and tangent (tan). As mathematics progressed the properties and applications of the trigonometric functions, or ratios associated with angles, became more important. The relationships between the ratios have many applications in the fields of physics and engineering. More complex applications result from the periodic (regularly recurring) properties of trigonometric functions and apply to physical phenomena, such as light, sound, and electricity.

Most of the elementary applications of trigonometry make use of triangles in a plane. Three-dimensional trigonometry is concerned with relationships between triangles drawn on the surface of a sphere and with solid angles—that is, volumes that extend from angles on the surface on a sphere.