In a right triangle, what does the tangent function represent?

In a right triangle, the tangent function specifically represents the ratio of the length of the side opposite the angle to the length of the side adjacent to that angle. This relationship can be summarized by the formula:

[

\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}

]

This definition is crucial in trigonometry, particularly for solving problems involving right triangles. The tangent function is particularly useful because it helps in calculating angles and side lengths when only certain measurements are known. By understanding this ratio, students can effectively utilize the tangent function in various applications, such as in physics when analyzing forces or in engineering when designing structures.