Which formula represents the ratio used to define the tangent in trigonometry?

In trigonometry, 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 means that for a given angle, if you take the length of the side opposite that angle and divide it by the length of the side adjacent to the same angle, you get the value of the tangent for that angle.

The formula that captures this relationship is written as:

[

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

]

This specific ratio is fundamental to many trigonometric applications, including those involving right triangles. Understanding this definition is essential because it serves as the foundation for solving various problems in trigonometry, including finding unknown side lengths and angles based on known values.

The other choices present ratios that correspond to other trigonometric functions. For example, the sine function is defined as the ratio of the opposite side to the hypotenuse, and the cosine function is defined as the ratio of the adjacent side to the hypotenuse. Recognizing these distinctions enables a clearer understanding of trigonometric identities and their applications in broader mathematical contexts.