The tangent of an angle in a right-angled triangle is defined as the ratio of the length of the opposite side to the length of the adjacent side.

In a right-angled triangle, each angle exhibits a specific relationship with the sides of the triangle. The tangent function, often abbreviated as “tan”, is a trigonometric function that captures this relationship. For a given angle $\theta$, the tangent is calculated by dividing the length of the side opposite the angle by the length of the side adjacent to the angle. This relationship can be mathematically expressed as:

For instance, consider a right-angled triangle where one of the angles (other than the right angle) is $\theta$, and the lengths of the sides opposite and adjacent to $\theta$ are 3 units and 4 units, respectively. In this case, the tangent of the angle $\theta$ can be calculated as follows:

Understanding the tangent function is essential for solving various problems in trigonometry, such as determining unknown side lengths or angles in right-angled triangles. Additionally, the tangent function is widely utilized in real-world applications, including fields like engineering, physics, and navigation.

To effectively use the tangent function, it is important to accurately identify the opposite and adjacent sides in relation to the angle in question. The opposite side is the one directly across from the angle, while the adjacent side is the one that is next to the angle, excluding the hypotenuse (the longest side of the triangle).

By mastering the concept of the tangent function, you will be well-prepared to address a variety of trigonometric problems and enhance your understanding of right-angled triangles.