How is tangent defined in relation to a right triangle?

The tangent of an angle in a right triangle is defined as the ratio of the length of the side opposite the angle to the length of the side adjacent to the angle. This fundamental relationship is derived from the definitions of the trigonometric functions based on the geometry of a right triangle.

For example, if you have a right triangle with one angle (let's call it ( \theta )), the side opposite to ( \theta ) is the side that does not touch the angle, while the adjacent side is the one that forms the angle along with the hypotenuse, which is the longest side of the triangle. Therefore, the formula for tangent is expressed as:

[ \text{tan}(\theta) = \frac{\text{Opposite}}{\text{Adjacent}} ]

This definition is crucial for solving problems in trigonometry as it connects the angle measures to the ratios of the sides, allowing one to calculate unknown lengths or angles given the other measurements in a right triangle. Understanding this concept is foundational in trigonometry and is widely used in various applications, such as physics, engineering, and architecture.