The sine function in the context of a right-angled triangle is defined as follows: the sine of an angle is equal to the length of the side opposite that angle divided by the length of the hypotenuse.

In a right-angled triangle, the sine function is one of the fundamental trigonometric ratios that connect the angles to the lengths of the sides. For a given angle $\theta$, the sine of $\theta$ is expressed as the ratio of the length of the opposite side to the length of the hypotenuse. The hypotenuse is the longest side of the triangle and is located opposite the right angle.

To represent this mathematically, if we have a right-angled triangle with an angle $\theta$, we denote the length of the side opposite this angle as “opposite” and the length of the hypotenuse as “hypotenuse.” The formula can be stated as:

For instance, consider a right-angled triangle in which the length of the side opposite the angle $\theta$ is $3$ units and the length of the hypotenuse is $5$ units. To calculate the sine of $\theta$, we can substitute these values into the formula:

Grasping this ratio is essential for solving a variety of problems in trigonometry, including determining unknown side lengths or angles in right-angled triangles. Additionally, it serves as a foundational concept for more advanced topics in trigonometry and calculus. It is important to note that the sine function applies specifically to right-angled triangles and that its values range between $-1$ and $1$.