How is tangent defined in terms of the sides of a right triangle?

Tangent is defined in a right triangle as the ratio of the length of the side opposite the angle to the length of the side adjacent to that angle. This definition stems from the fundamental relationships between the sides of a right triangle and the angles they form.

When you focus on a specific angle in a right triangle, the side opposite that angle is the one directly across from it, while the adjacent side is the one that is next to the angle. The tangent function, therefore, provides a way to express the relationship between these two sides purely in terms of their lengths, making it an essential concept in trigonometry. This ratio is foundational for solving problems related to angles and distances in a variety of applications, including physics, engineering, and architecture.

Recognizing this relationship allows for more complex applications of trigonometric identities and functions, as well as helping in the understanding of circles, waves, and oscillatory motion through the unit circle concept where tangent extends beyond right triangles.