Sine, denoted by \(\sin\), is one of the primary trigonometric functions, describing the ratio between the length of the side opposite to an acute angle and the hypotenuse in a right-angled triangle. This function is integral to the study of wave patterns, oscillations, and circular motions, offering a mathematical framework for analyzing periodic phenomena. Sine values range from \(-1\) to \(1\), enabling the depiction of sine waves that model sound, light, and other waveforms in physics and engineering. Beyond its geometric origins, sine is also applied in complex analysis, signal processing, and electrical engineering to describe alternating current waveforms and signal modulation. The versatility of the sine function makes it indispensable in both theoretical and applied mathematics, facilitating solutions to a wide array of problems involving angles and periodicity.