sine — from formula anatomyThe sine function is a fundamental mathematical tool that describes smooth, repetitive oscillations. It maps an angle to the ratio of two sides of a right triangle, but its true power lies in modeling any periodic phenomenon — from sound waves and alternating current to the vibrations of a guitar string. In physics, sine waves are the building blocks of all waveforms: any complex signal can be decomposed into a sum of sine waves of different frequencies and amplitudes, a principle known as Fourier analysis.
The sine function was formalized in ancient India and later refined by Islamic mathematicians, but the modern notation and treatment as a function of a real variable emerged in the 17th century with Euler and others.
Imagine a point moving around a circle at constant speed. The sine of the angle it sweeps is the vertical coordinate of that point, plotted against time. This produces a smooth, wavy curve that repeats every 360 degrees (or 2π radians). The function takes an angle as input and outputs a value between -1 and 1, with the shape determined by the circle's radius and the speed of rotation.
Sine waves are used in signal processing, acoustics, electrical engineering (AC power), and even in medical imaging (MRI) to encode spatial information.
The sine of an angle in radians is also the length of the shadow cast by a vertical stick of unit length when the sun is at that angle above the horizon.
| D_L | variable | luminosity distance — photometric distance metre |
| z | variable | redshift — redshift |
| χ | variable | comoving distance — comoving distance metre |
| Ω_k | variable | curvature density parameter — curvature density parameter |
| c | constant | speed of light in vacuum — speed of light = 299792458 metre per second |
| H_0 | constant | current expansion rate of the universe — hubble constant = 2.2e-18 per second |
| sinh | operator | hyperbolic sine function — hyperbolic sine |
| sin | operator | sine function — sine |
| √ | operator | square root — square root |
| theta | variable | deflection angle — angle radian |
| q | variable | distance of closest approach in absence of gravity — distance metre |
| r | variable | radial distance from the gravitating body — distance metre |
| M | variable | mass of the gravitating body — mass kilogram |
| v_infinity | variable | speed of the test particle at infinity — velocity metre per second |
| G | constant | Universal constant of gravitation — gravitational constant = 6.67430e-11 cubic metre per kilogram per second squared |
| sin | operator | trigonometric sine function — sine |
| sqrt | operator | takes the square root of the argument — square root |
| x | variable | x-coordinate of a point on the line — x coordinate metre |
| y | variable | y-coordinate of a point on the line — y coordinate metre |
| \phi | variable | angle of the normal to the line with respect to the x-axis — angle radian |
| p | variable | perpendicular distance from the origin to the line — distance metre |
| \cos | operator | computes the cosine of an angle — cosine |
| \sin | operator | computes the sine of an angle — sine |
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