A trigonometric function, the complement of sine, describing a smooth periodic oscillation.The cosine function, written cos(θ), is a fundamental trigonometric function that describes a smooth, periodic oscillation. It is defined as the ratio of the adjacent side to the hypotenuse in a right triangle, but its true power lies in its wave-like behavior: it repeats every 360° (or 2π radians) and oscillates between −1 and 1. Cosine is the complement of sine, meaning cos(θ) = sin(θ + 90°). It appears throughout physics, from the motion of pendulums and alternating currents to the interference patterns of light.
The cosine function was developed in ancient India and later formalized in Islamic mathematics. The modern notation 'cos' was introduced by Leonhard Euler in the 18th century, who also established its connection to complex exponentials via Euler's formula.
On the unit circle, the cosine of an angle θ is the x-coordinate of the point where the terminal side of the angle intersects the circle. As θ increases, this coordinate traces a smooth wave. In physics, cosine functions often describe the projection of circular motion onto a line, such as the horizontal position of a point on a rotating wheel. Its derivative is the negative sine, which links it to harmonic motion: the second derivative is proportional to the negative of the function itself, the hallmark of simple harmonic oscillators.
Cosine is used in signal processing (e.g., Fourier transforms), electrical engineering for AC circuits, and in physics to model wave phenomena like sound and light.
The cosine of an angle in radians is the real part of the complex exponential e^{iθ}, a relationship that underpins quantum mechanics and signal analysis.
| z | variable | redshift parameter — redshift |
| v | variable | relative velocity between source and observer — velocity metre per second |
| θ | variable | angle between velocity vector and line of sight — angle radian |
| c | constant | speed of light in vacuum — speed of light = 299792458 metre per second |
| cos | operator | trigonometric cosine function — cosine |
| e | variable | Orbital eccentricity — eccentricity |
| ω | variable | Argument of periastron — argument of periapsis radian |
| Δt | variable | Time offset in eclipse timing — time offset second |
| P | variable | Orbital period — period second |
| cos | operator | Computes the cosine of an angle — cosine |
| F | variable | Correlation function — correlation function |
| F_0 | variable | Normalization constant — correlation function |
| C_QE | variable | Degree of quantum entanglement — entanglement degree |
| A(\theta_1) | variable | Angular modulation factor at angle theta_1 — modulation factor |
| A(\theta_2) | variable | Angular modulation factor at angle theta_2 — modulation factor |
| \theta_1 | variable | Scattering angle for first particle — angle radian |
| \theta_2 | variable | Scattering angle for second particle — angle radian |
| \phi | variable | Azimuthal angle — angle radian |
| \cos | operator | Trigonometric cosine function — cosine |
| 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 |
| B | variable | bispectrum — bispectrum cubic metre |
| k_1 | variable | first wavenumber — wavenumber reciprocal metre |
| k_2 | variable | second wavenumber — wavenumber reciprocal metre |
| k_3 | variable | third wavenumber — wavenumber reciprocal metre |
| \Delta | variable | spectral tilt — dimensionless |
| \omega | variable | oscillation frequency in log space — dimensionless |
| \propto | operator | indicates the left side equals the right side times a constant factor — proportionality |
| \ln | operator | logarithm to base e — natural logarithm |
| \cos | operator | trigonometric cosine — cosine |
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