Sinuksen

Sinuksen refers to the mathematical sine function, commonly written sin(x). It assigns to an angle the ratio of the opposite side to the hypotenuse in a right triangle and, more generally, can be defined on the unit circle as the y-coordinate of the point reached by rotating (1,0) by x radians from the positive x-axis.

Key properties of the sine function include its 2π periodicity, sin(x + 2π) = sin(x), and its odd symmetry, sin(-x) = -sin(x). Its range is between -1 and 1. The derivative is cos(x), and the second derivative is -sin(x). An antiderivative is -cos(x) + C. The function has the Maclaurin series expansion sin(x) = x - x^3/3! + x^5/5! - ..., which converges for all real x.

Sinus is closely related to other trigonometric functions such as cosine, with the Pythagorean identity sin^2(x) + cos^2(x) = 1. It plays a central role in modeling periodic phenomena, including oscillatory motion, waves, and alternating current. It is fundamental in physics, engineering, signal processing, and Fourier analysis, where complex periodic signals are decomposed into sine and cosine components.

In Finnish, the sine function is called sinus, and sinuksen is the genitive form meaning “of the sine.” The term sinus derives from Latin sinus, a historical transfer from earlier mathematical traditions, and is used in many languages with similar naming. The standard notation remains sin(x) across educational and scientific contexts.