What is the definition of sin in trigonometry?

In trigonometry, the sine function, often abbreviated as sin, is defined as the ratio of the length of the side opposite an angle in a right triangle to the length of the hypotenuse of that triangle. In this context, if we denote the opposite side as ( y ) and the hypotenuse as ( r ), the sine of the angle is expressed mathematically as ( \sin(\theta) = \frac{y}{r} ). This definition is fundamental because it establishes a relationship between the angles of the triangle and the ratios of its sides, enabling calculations involving trigonometric functions.

This definition is critical in various applications, including physics, engineering, and other fields that involve the analysis of periodic phenomena or geometric relationships. Hence, the response indicating ( y/r ) captures the essence of the sine function and its role in trigonometric calculations.